Optimal. Leaf size=1637 \[ \frac {b d^2 x^5 \sqrt {d-c^2 d x^2} c^5}{25 g \sqrt {1-c^2 x^2}}-\frac {b d^2 f x^4 \sqrt {d-c^2 d x^2} c^5}{16 g^2 \sqrt {1-c^2 x^2}}-\frac {d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) c^4}{4 g^2}+\frac {b d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} c^3}{9 g^3 \sqrt {1-c^2 x^2}}-\frac {b d^2 x^3 \sqrt {d-c^2 d x^2} c^3}{45 g \sqrt {1-c^2 x^2}}-\frac {b d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} c^3}{4 g^4 \sqrt {1-c^2 x^2}}+\frac {b d^2 f x^2 \sqrt {d-c^2 d x^2} c^3}{16 g^2 \sqrt {1-c^2 x^2}}-\frac {d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) c^2}{2 g^4}+\frac {d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) c^2}{8 g^2}+\frac {d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2 c}{4 b g^4 \sqrt {1-c^2 x^2}}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2 c}{2 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 f \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2 c}{16 b g^2 \sqrt {1-c^2 x^2}}+\frac {b d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} c}{g^5 \sqrt {1-c^2 x^2}}-\frac {b d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} c}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {2 b d^2 x \sqrt {d-c^2 d x^2} c}{15 g \sqrt {1-c^2 x^2}}+\frac {b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)}{g^5}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac {a d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {f x c^2+g}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {i b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt {d-c^2 d x^2} \cos ^{-1}(c x) \log \left (\frac {e^{i \cos ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {i b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt {d-c^2 d x^2} \cos ^{-1}(c x) \log \left (\frac {e^{i \cos ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}+1\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt {d-c^2 d x^2} \text {Li}_2\left (-\frac {e^{i \cos ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt {d-c^2 d x^2} \text {Li}_2\left (-\frac {e^{i \cos ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^4 (f+g x) c}-\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^6 (f+g x) \sqrt {1-c^2 x^2} c} \]
[Out]
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Rubi [A] time = 2.73, antiderivative size = 1637, normalized size of antiderivative = 1.00, number of steps used = 37, number of rules used = 28, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.903, Rules used = {4778, 4768, 4648, 4642, 30, 4678, 4698, 4708, 266, 43, 4690, 12, 4766, 683, 4758, 6742, 725, 204, 1654, 4800, 4798, 8, 4774, 3321, 2264, 2190, 2279, 2391} \[ \frac {b d^2 x^5 \sqrt {d-c^2 d x^2} c^5}{25 g \sqrt {1-c^2 x^2}}-\frac {b d^2 f x^4 \sqrt {d-c^2 d x^2} c^5}{16 g^2 \sqrt {1-c^2 x^2}}-\frac {d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) c^4}{4 g^2}+\frac {b d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} c^3}{9 g^3 \sqrt {1-c^2 x^2}}-\frac {b d^2 x^3 \sqrt {d-c^2 d x^2} c^3}{45 g \sqrt {1-c^2 x^2}}-\frac {b d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} c^3}{4 g^4 \sqrt {1-c^2 x^2}}+\frac {b d^2 f x^2 \sqrt {d-c^2 d x^2} c^3}{16 g^2 \sqrt {1-c^2 x^2}}-\frac {d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) c^2}{2 g^4}+\frac {d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) c^2}{8 g^2}+\frac {d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2 c}{4 b g^4 \sqrt {1-c^2 x^2}}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2 c}{2 b g^5 \sqrt {1-c^2 x^2}}+\frac {d^2 f \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2 c}{16 b g^2 \sqrt {1-c^2 x^2}}+\frac {b d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} c}{g^5 \sqrt {1-c^2 x^2}}-\frac {b d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} c}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {2 b d^2 x \sqrt {d-c^2 d x^2} c}{15 g \sqrt {1-c^2 x^2}}+\frac {b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)}{g^5}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac {a d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {f x c^2+g}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {i b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt {d-c^2 d x^2} \cos ^{-1}(c x) \log \left (\frac {e^{i \cos ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {i b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt {d-c^2 d x^2} \cos ^{-1}(c x) \log \left (\frac {e^{i \cos ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}+1\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,-\frac {e^{i \cos ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,-\frac {e^{i \cos ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^4 (f+g x) c}-\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^6 (f+g x) \sqrt {1-c^2 x^2} c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 12
Rule 30
Rule 43
Rule 204
Rule 266
Rule 683
Rule 725
Rule 1654
Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 3321
Rule 4642
Rule 4648
Rule 4678
Rule 4690
Rule 4698
Rule 4708
Rule 4758
Rule 4766
Rule 4768
Rule 4774
Rule 4778
Rule 4798
Rule 4800
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right )}{f+g x} \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^{5/2} \left (a+b \cos ^{-1}(c x)\right )}{f+g x} \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-\frac {c^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )}{g^4}+\frac {c^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )}{g^3}-\frac {c^4 f x^2 \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )}{g^2}+\frac {c^4 x^3 \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )}{g}+\frac {\left (-c^2 f^2+g^2\right )^2 \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )}{g^4 (f+g x)}\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {\left (c^4 d^2 f \sqrt {d-c^2 d x^2}\right ) \int x^2 \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{g^2 \sqrt {1-c^2 x^2}}+\frac {\left (c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{g \sqrt {1-c^2 x^2}}-\frac {\left (c^2 d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{g^4 \sqrt {1-c^2 x^2}}+\frac {\left (c^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int x \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{g^3 \sqrt {1-c^2 x^2}}+\frac {\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )}{f+g x} \, dx}{g^4 \sqrt {1-c^2 x^2}}\\ &=-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{4 g^2}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac {\left (c^4 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 \left (a+b \cos ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{4 g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b c^5 d^2 f \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {\left (b c^5 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {-2-c^2 x^2+3 c^4 x^4}{15 c^4} \, dx}{g \sqrt {1-c^2 x^2}}-\frac {\left (c^2 d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cos ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 g^4 \sqrt {1-c^2 x^2}}-\frac {\left (b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{2 g^4 \sqrt {1-c^2 x^2}}-\frac {\left (b c d^2 \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right ) \, dx}{3 g^3 \sqrt {1-c^2 x^2}}+\frac {\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-g-2 c^2 f x-c^2 g x^2\right ) \left (a+b \cos ^{-1}(c x)\right )^2}{(f+g x)^2} \, dx}{2 b c g^4 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{3 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2}}{4 g^4 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2}}{9 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {1-c^2 x^2}}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{4 g^2}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}+\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^4 \sqrt {1-c^2 x^2}}-\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^5 \sqrt {1-c^2 x^2}}-\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac {\left (c^2 d^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cos ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {\left (b c^3 d^2 f \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{8 g^2 \sqrt {1-c^2 x^2}}+\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-2-c^2 x^2+3 c^4 x^4\right ) \, dx}{15 g \sqrt {1-c^2 x^2}}+\frac {\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (\frac {1}{f+g x}-\frac {c^2 \left (g x+\frac {f^2}{f+g x}\right )}{g^2}\right ) \left (a+b \cos ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{g^4 \sqrt {1-c^2 x^2}}\\ &=-\frac {2 b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{3 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2}}{4 g^4 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 g \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2}}{9 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 g \sqrt {1-c^2 x^2}}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{4 g^2}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}+\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^4 \sqrt {1-c^2 x^2}}-\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^5 \sqrt {1-c^2 x^2}}-\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}+\frac {\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-\frac {a \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right )}{g^2 (f+g x) \sqrt {1-c^2 x^2}}-\frac {b \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \cos ^{-1}(c x)}{g^2 (f+g x) \sqrt {1-c^2 x^2}}\right ) \, dx}{g^4 \sqrt {1-c^2 x^2}}\\ &=-\frac {2 b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{3 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2}}{4 g^4 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 g \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2}}{9 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 g \sqrt {1-c^2 x^2}}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{4 g^2}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}+\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^4 \sqrt {1-c^2 x^2}}-\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^5 \sqrt {1-c^2 x^2}}-\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac {\left (a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^6 \sqrt {1-c^2 x^2}}-\frac {\left (b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \cos ^{-1}(c x)}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^6 \sqrt {1-c^2 x^2}}\\ &=\frac {a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {2 b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{3 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2}}{4 g^4 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 g \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2}}{9 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 g \sqrt {1-c^2 x^2}}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{4 g^2}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}+\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^4 \sqrt {1-c^2 x^2}}-\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^5 \sqrt {1-c^2 x^2}}-\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac {\left (a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {c^2 g^2 \left (c^2 f^2-g^2\right )}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{c^2 g^8 \sqrt {1-c^2 x^2}}-\frac {\left (b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {c^2 g x \cos ^{-1}(c x)}{\sqrt {1-c^2 x^2}}+\frac {\left (c^2 f^2-g^2\right ) \cos ^{-1}(c x)}{(f+g x) \sqrt {1-c^2 x^2}}\right ) \, dx}{g^6 \sqrt {1-c^2 x^2}}\\ &=\frac {a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {2 b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{3 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2}}{4 g^4 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 g \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2}}{9 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 g \sqrt {1-c^2 x^2}}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{4 g^2}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}+\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^4 \sqrt {1-c^2 x^2}}-\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^5 \sqrt {1-c^2 x^2}}-\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac {\left (b c^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \cos ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{g^5 \sqrt {1-c^2 x^2}}-\frac {\left (a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^6 \sqrt {1-c^2 x^2}}-\frac {\left (b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\cos ^{-1}(c x)}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^6 \sqrt {1-c^2 x^2}}\\ &=\frac {a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {2 b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{3 g^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2}}{4 g^4 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 g \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2}}{9 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 g \sqrt {1-c^2 x^2}}+\frac {b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)}{g^5}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{4 g^2}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}+\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^4 \sqrt {1-c^2 x^2}}-\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^5 \sqrt {1-c^2 x^2}}-\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}+\frac {\left (b c d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int 1 \, dx}{g^5 \sqrt {1-c^2 x^2}}+\frac {\left (a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-c^2 f^2+g^2-x^2} \, dx,x,\frac {g+c^2 f x}{\sqrt {1-c^2 x^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {\left (b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{c f+g \cos (x)} \, dx,x,\cos ^{-1}(c x)\right )}{g^6 \sqrt {1-c^2 x^2}}\\ &=\frac {a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {2 b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{3 g^3 \sqrt {1-c^2 x^2}}+\frac {b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2}}{g^5 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2}}{4 g^4 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 g \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2}}{9 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 g \sqrt {1-c^2 x^2}}+\frac {b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)}{g^5}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{4 g^2}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}+\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^4 \sqrt {1-c^2 x^2}}-\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^5 \sqrt {1-c^2 x^2}}-\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac {a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {\left (2 b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{2 c e^{i x} f+g+e^{2 i x} g} \, dx,x,\cos ^{-1}(c x)\right )}{g^6 \sqrt {1-c^2 x^2}}\\ &=\frac {a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {2 b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{3 g^3 \sqrt {1-c^2 x^2}}+\frac {b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2}}{g^5 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2}}{4 g^4 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 g \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2}}{9 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 g \sqrt {1-c^2 x^2}}+\frac {b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)}{g^5}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{4 g^2}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}+\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^4 \sqrt {1-c^2 x^2}}-\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^5 \sqrt {1-c^2 x^2}}-\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac {a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {\left (2 b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{2 c f+2 e^{i x} g-2 \sqrt {c^2 f^2-g^2}} \, dx,x,\cos ^{-1}(c x)\right )}{g^5 \sqrt {1-c^2 x^2}}-\frac {\left (2 b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{2 c f+2 e^{i x} g+2 \sqrt {c^2 f^2-g^2}} \, dx,x,\cos ^{-1}(c x)\right )}{g^5 \sqrt {1-c^2 x^2}}\\ &=\frac {a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {2 b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{3 g^3 \sqrt {1-c^2 x^2}}+\frac {b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2}}{g^5 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2}}{4 g^4 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 g \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2}}{9 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 g \sqrt {1-c^2 x^2}}+\frac {b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)}{g^5}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{4 g^2}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}+\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^4 \sqrt {1-c^2 x^2}}-\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^5 \sqrt {1-c^2 x^2}}-\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac {a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac {e^{i \cos ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac {e^{i \cos ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {\left (i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+\frac {2 e^{i x} g}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {\left (i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+\frac {2 e^{i x} g}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{g^6 \sqrt {1-c^2 x^2}}\\ &=\frac {a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {2 b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{3 g^3 \sqrt {1-c^2 x^2}}+\frac {b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2}}{g^5 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2}}{4 g^4 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 g \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2}}{9 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 g \sqrt {1-c^2 x^2}}+\frac {b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)}{g^5}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{4 g^2}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}+\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^4 \sqrt {1-c^2 x^2}}-\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^5 \sqrt {1-c^2 x^2}}-\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac {a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac {e^{i \cos ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac {e^{i \cos ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {\left (b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 g x}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {\left (b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 g x}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )}{g^6 \sqrt {1-c^2 x^2}}\\ &=\frac {a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}}{g^5}-\frac {2 b c d^2 x \sqrt {d-c^2 d x^2}}{15 g \sqrt {1-c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{3 g^3 \sqrt {1-c^2 x^2}}+\frac {b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2}}{g^5 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 f x^2 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2}}{4 g^4 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 g \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2}}{9 g^3 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 f x^4 \sqrt {d-c^2 d x^2}}{16 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 g \sqrt {1-c^2 x^2}}+\frac {b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)}{g^5}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{4 g^2}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac {d^2 \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g^3}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 g}+\frac {c d^2 f \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^4 \sqrt {1-c^2 x^2}}-\frac {c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^5 \sqrt {1-c^2 x^2}}-\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt {1-c^2 x^2}}-\frac {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac {a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac {e^{i \cos ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac {e^{i \cos ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \text {Li}_2\left (-\frac {e^{i \cos ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \text {Li}_2\left (-\frac {e^{i \cos ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [B] time = 20.29, size = 6216, normalized size = 3.80 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a c^{4} d^{2} x^{4} - 2 \, a c^{2} d^{2} x^{2} + a d^{2} + {\left (b c^{4} d^{2} x^{4} - 2 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{g x + f}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.57, size = 4692, normalized size = 2.87 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{f+g\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {acos}{\left (c x \right )}\right )}{f + g x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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