Optimal. Leaf size=1678 \[ -\frac {i b^2 d (e f-d g)^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right ) c^3}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {i b^2 d (e f-d g)^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right ) c^3}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b^2 d (e f-d g)^2 \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right ) c^3}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 d (e f-d g)^2 \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right ) c^3}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b^2 (e f-d g)^2 \log (d+e x) c^2}{e^3 \left (c^2 d^2-e^2\right )}+\frac {b^2 (e f-d g)^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x) c}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a b (e f-d g) \left (4 e^2 g-c^2 d (e f+3 d g)\right ) \tan ^{-1}\left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right ) c}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {4 i b^2 g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right ) c}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {4 i b^2 g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right ) c}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {4 b^2 g (e f-d g) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right ) c}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {4 b^2 g (e f-d g) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right ) c}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {a b (e f-d g)^2 \sqrt {1-c^2 x^2} c}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {i b^2 g^2 \sin ^{-1}(c x)^3}{3 e^3}-\frac {i a b g^2 \sin ^{-1}(c x)^2}{e^3}-\frac {2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}-\frac {b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac {4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}-\frac {a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 a b g^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 a b g^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {a^2 g^2 \log (d+e x)}{e^3}-\frac {2 i a b g^2 \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {2 i b^2 g^2 \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {2 i a b g^2 \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {2 i b^2 g^2 \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 b^2 g^2 \text {Li}_3\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 b^2 g^2 \text {Li}_3\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {2 a^2 g (e f-d g)}{e^3 (d+e x)}-\frac {a^2 (e f-d g)^2}{2 e^3 (d+e x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 3.71, antiderivative size = 1678, normalized size of antiderivative = 1.00, number of steps used = 55, number of rules used = 25, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4759, 43, 4753, 12, 6742, 807, 725, 204, 216, 2404, 4741, 4519, 2190, 2279, 2391, 4743, 4773, 3324, 3323, 2264, 2668, 31, 2531, 2282, 6589} \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 31
Rule 43
Rule 204
Rule 216
Rule 725
Rule 807
Rule 2190
Rule 2264
Rule 2279
Rule 2282
Rule 2391
Rule 2404
Rule 2531
Rule 2668
Rule 3323
Rule 3324
Rule 4519
Rule 4741
Rule 4743
Rule 4753
Rule 4759
Rule 4773
Rule 6589
Rule 6742
Rubi steps
\begin {align*} \int \frac {(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{(d+e x)^3} \, dx &=\int \left (\frac {a^2 (f+g x)^2}{(d+e x)^3}+\frac {2 a b (f+g x)^2 \sin ^{-1}(c x)}{(d+e x)^3}+\frac {b^2 (f+g x)^2 \sin ^{-1}(c x)^2}{(d+e x)^3}\right ) \, dx\\ &=a^2 \int \frac {(f+g x)^2}{(d+e x)^3} \, dx+(2 a b) \int \frac {(f+g x)^2 \sin ^{-1}(c x)}{(d+e x)^3} \, dx+b^2 \int \frac {(f+g x)^2 \sin ^{-1}(c x)^2}{(d+e x)^3} \, dx\\ &=-\frac {a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac {4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}+\frac {2 a b g^2 \sin ^{-1}(c x) \log (d+e x)}{e^3}+a^2 \int \left (\frac {(e f-d g)^2}{e^2 (d+e x)^3}+\frac {2 g (e f-d g)}{e^2 (d+e x)^2}+\frac {g^2}{e^2 (d+e x)}\right ) \, dx+b^2 \int \left (\frac {(e f-d g)^2 \sin ^{-1}(c x)^2}{e^2 (d+e x)^3}+\frac {2 g (e f-d g) \sin ^{-1}(c x)^2}{e^2 (d+e x)^2}+\frac {g^2 \sin ^{-1}(c x)^2}{e^2 (d+e x)}\right ) \, dx-(2 a b c) \int \frac {-(e f-d g) (3 d g+e (f+4 g x))+2 g^2 (d+e x)^2 \log (d+e x)}{2 e^3 (d+e x)^2 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac {2 a^2 g (e f-d g)}{e^3 (d+e x)}-\frac {a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac {4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}+\frac {a^2 g^2 \log (d+e x)}{e^3}+\frac {2 a b g^2 \sin ^{-1}(c x) \log (d+e x)}{e^3}-\frac {(a b c) \int \frac {-(e f-d g) (3 d g+e (f+4 g x))+2 g^2 (d+e x)^2 \log (d+e x)}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{e^3}+\frac {\left (b^2 g^2\right ) \int \frac {\sin ^{-1}(c x)^2}{d+e x} \, dx}{e^2}+\frac {\left (2 b^2 g (e f-d g)\right ) \int \frac {\sin ^{-1}(c x)^2}{(d+e x)^2} \, dx}{e^2}+\frac {\left (b^2 (e f-d g)^2\right ) \int \frac {\sin ^{-1}(c x)^2}{(d+e x)^3} \, dx}{e^2}\\ &=-\frac {a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac {2 a^2 g (e f-d g)}{e^3 (d+e x)}-\frac {a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac {4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}-\frac {b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac {2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}+\frac {a^2 g^2 \log (d+e x)}{e^3}+\frac {2 a b g^2 \sin ^{-1}(c x) \log (d+e x)}{e^3}-\frac {(a b c) \int \left (-\frac {(e f-d g) (e f+3 d g+4 e g x)}{(d+e x)^2 \sqrt {1-c^2 x^2}}+\frac {2 g^2 \log (d+e x)}{\sqrt {1-c^2 x^2}}\right ) \, dx}{e^3}+\frac {\left (b^2 g^2\right ) \operatorname {Subst}\left (\int \frac {x^2 \cos (x)}{c d+e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}+\frac {\left (4 b^2 c g (e f-d g)\right ) \int \frac {\sin ^{-1}(c x)}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{e^3}+\frac {\left (b^2 c (e f-d g)^2\right ) \int \frac {\sin ^{-1}(c x)}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{e^3}\\ &=-\frac {a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac {2 a^2 g (e f-d g)}{e^3 (d+e x)}-\frac {a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac {4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}-\frac {b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac {2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}-\frac {i b^2 g^2 \sin ^{-1}(c x)^3}{3 e^3}+\frac {a^2 g^2 \log (d+e x)}{e^3}+\frac {2 a b g^2 \sin ^{-1}(c x) \log (d+e x)}{e^3}-\frac {\left (2 a b c g^2\right ) \int \frac {\log (d+e x)}{\sqrt {1-c^2 x^2}} \, dx}{e^3}+\frac {\left (b^2 g^2\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x^2}{c d-\sqrt {c^2 d^2-e^2}-i e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}+\frac {\left (b^2 g^2\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x^2}{c d+\sqrt {c^2 d^2-e^2}-i e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}+\frac {(a b c (e f-d g)) \int \frac {e f+3 d g+4 e g x}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{e^3}+\frac {\left (4 b^2 c g (e f-d g)\right ) \operatorname {Subst}\left (\int \frac {x}{c d+e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^3}+\frac {\left (b^2 c^2 (e f-d g)^2\right ) \operatorname {Subst}\left (\int \frac {x}{(c d+e \sin (x))^2} \, dx,x,\sin ^{-1}(c x)\right )}{e^3}\\ &=-\frac {a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac {2 a^2 g (e f-d g)}{e^3 (d+e x)}+\frac {a b c (e f-d g)^2 \sqrt {1-c^2 x^2}}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac {4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}+\frac {b^2 c (e f-d g)^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac {2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}-\frac {i b^2 g^2 \sin ^{-1}(c x)^3}{3 e^3}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {a^2 g^2 \log (d+e x)}{e^3}-\frac {\left (2 b^2 g^2\right ) \operatorname {Subst}\left (\int x \log \left (1-\frac {i e e^{i x}}{c d-\sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3}-\frac {\left (2 b^2 g^2\right ) \operatorname {Subst}\left (\int x \log \left (1-\frac {i e e^{i x}}{c d+\sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3}+\frac {\left (2 a b c g^2\right ) \int \frac {\sin ^{-1}(c x)}{c d+c e x} \, dx}{e^2}+\frac {\left (8 b^2 c g (e f-d g)\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{i e+2 c d e^{i x}-i e e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^3}+\frac {\left (b^2 c^3 d (e f-d g)^2\right ) \operatorname {Subst}\left (\int \frac {x}{c d+e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^3 \left (c^2 d^2-e^2\right )}-\frac {\left (b^2 c^2 (e f-d g)^2\right ) \operatorname {Subst}\left (\int \frac {\cos (x)}{c d+e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^2 \left (c^2 d^2-e^2\right )}-\frac {\left (a b c (e f-d g) \left (4 e^2 g-c^2 d (e f+3 d g)\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{e^3 \left (c^2 d^2-e^2\right )}\\ &=-\frac {a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac {2 a^2 g (e f-d g)}{e^3 (d+e x)}+\frac {a b c (e f-d g)^2 \sqrt {1-c^2 x^2}}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac {4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}+\frac {b^2 c (e f-d g)^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac {2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}-\frac {i b^2 g^2 \sin ^{-1}(c x)^3}{3 e^3}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {a^2 g^2 \log (d+e x)}{e^3}-\frac {2 i b^2 g^2 \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {2 i b^2 g^2 \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {\left (2 i b^2 g^2\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (\frac {i e e^{i x}}{c d-\sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3}+\frac {\left (2 i b^2 g^2\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (\frac {i e e^{i x}}{c d+\sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3}+\frac {\left (2 a b c g^2\right ) \operatorname {Subst}\left (\int \frac {x \cos (x)}{c^2 d+c e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}-\frac {\left (8 i b^2 c g (e f-d g)\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{2 c d-2 \sqrt {c^2 d^2-e^2}-2 i e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {\left (8 i b^2 c g (e f-d g)\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{2 c d+2 \sqrt {c^2 d^2-e^2}-2 i e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {\left (b^2 c^2 (e f-d g)^2\right ) \operatorname {Subst}\left (\int \frac {1}{c d+x} \, dx,x,c e x\right )}{e^3 \left (c^2 d^2-e^2\right )}+\frac {\left (2 b^2 c^3 d (e f-d g)^2\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{i e+2 c d e^{i x}-i e e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^3 \left (c^2 d^2-e^2\right )}+\frac {\left (a b c (e f-d g) \left (4 e^2 g-c^2 d (e f+3 d g)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-c^2 d^2+e^2-x^2} \, dx,x,\frac {e+c^2 d x}{\sqrt {1-c^2 x^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )}\\ &=-\frac {a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac {2 a^2 g (e f-d g)}{e^3 (d+e x)}+\frac {a b c (e f-d g)^2 \sqrt {1-c^2 x^2}}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac {4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}+\frac {b^2 c (e f-d g)^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {i a b g^2 \sin ^{-1}(c x)^2}{e^3}-\frac {b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac {2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}-\frac {i b^2 g^2 \sin ^{-1}(c x)^3}{3 e^3}-\frac {a b c (e f-d g) \left (4 e^2 g-c^2 d (e f+3 d g)\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {4 i b^2 c g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {4 i b^2 c g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {a^2 g^2 \log (d+e x)}{e^3}-\frac {b^2 c^2 (e f-d g)^2 \log (d+e x)}{e^3 \left (c^2 d^2-e^2\right )}-\frac {2 i b^2 g^2 \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {2 i b^2 g^2 \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {\left (2 b^2 g^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i e x}{c d-\sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3}+\frac {\left (2 b^2 g^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i e x}{c d+\sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3}+\frac {\left (2 a b c g^2\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{c^2 d-c \sqrt {c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}+\frac {\left (2 a b c g^2\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{c^2 d+c \sqrt {c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}+\frac {\left (4 i b^2 c g (e f-d g)\right ) \operatorname {Subst}\left (\int \log \left (1-\frac {2 i e e^{i x}}{2 c d-2 \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {\left (4 i b^2 c g (e f-d g)\right ) \operatorname {Subst}\left (\int \log \left (1-\frac {2 i e e^{i x}}{2 c d+2 \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {\left (2 i b^2 c^3 d (e f-d g)^2\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{2 c d-2 \sqrt {c^2 d^2-e^2}-2 i e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {\left (2 i b^2 c^3 d (e f-d g)^2\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{2 c d+2 \sqrt {c^2 d^2-e^2}-2 i e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}\\ &=-\frac {a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac {2 a^2 g (e f-d g)}{e^3 (d+e x)}+\frac {a b c (e f-d g)^2 \sqrt {1-c^2 x^2}}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac {4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}+\frac {b^2 c (e f-d g)^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {i a b g^2 \sin ^{-1}(c x)^2}{e^3}-\frac {b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac {2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}-\frac {i b^2 g^2 \sin ^{-1}(c x)^3}{3 e^3}-\frac {a b c (e f-d g) \left (4 e^2 g-c^2 d (e f+3 d g)\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {2 a b g^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {4 i b^2 c g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {i b^2 c^3 d (e f-d g)^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 a b g^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {4 i b^2 c g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {i b^2 c^3 d (e f-d g)^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {a^2 g^2 \log (d+e x)}{e^3}-\frac {b^2 c^2 (e f-d g)^2 \log (d+e x)}{e^3 \left (c^2 d^2-e^2\right )}-\frac {2 i b^2 g^2 \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {2 i b^2 g^2 \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 b^2 g^2 \text {Li}_3\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 b^2 g^2 \text {Li}_3\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {\left (2 a b g^2\right ) \operatorname {Subst}\left (\int \log \left (1-\frac {i c e e^{i x}}{c^2 d-c \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3}-\frac {\left (2 a b g^2\right ) \operatorname {Subst}\left (\int \log \left (1-\frac {i c e e^{i x}}{c^2 d+c \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3}+\frac {\left (4 b^2 c g (e f-d g)\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {2 i e x}{2 c d-2 \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {\left (4 b^2 c g (e f-d g)\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {2 i e x}{2 c d+2 \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {\left (i b^2 c^3 d (e f-d g)^2\right ) \operatorname {Subst}\left (\int \log \left (1-\frac {2 i e e^{i x}}{2 c d-2 \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {\left (i b^2 c^3 d (e f-d g)^2\right ) \operatorname {Subst}\left (\int \log \left (1-\frac {2 i e e^{i x}}{2 c d+2 \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}\\ &=-\frac {a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac {2 a^2 g (e f-d g)}{e^3 (d+e x)}+\frac {a b c (e f-d g)^2 \sqrt {1-c^2 x^2}}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac {4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}+\frac {b^2 c (e f-d g)^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {i a b g^2 \sin ^{-1}(c x)^2}{e^3}-\frac {b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac {2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}-\frac {i b^2 g^2 \sin ^{-1}(c x)^3}{3 e^3}-\frac {a b c (e f-d g) \left (4 e^2 g-c^2 d (e f+3 d g)\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {2 a b g^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {4 i b^2 c g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {i b^2 c^3 d (e f-d g)^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 a b g^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {4 i b^2 c g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {i b^2 c^3 d (e f-d g)^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {a^2 g^2 \log (d+e x)}{e^3}-\frac {b^2 c^2 (e f-d g)^2 \log (d+e x)}{e^3 \left (c^2 d^2-e^2\right )}-\frac {4 b^2 c g (e f-d g) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {2 i b^2 g^2 \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {4 b^2 c g (e f-d g) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {2 i b^2 g^2 \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 b^2 g^2 \text {Li}_3\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 b^2 g^2 \text {Li}_3\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {\left (2 i a b g^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {i c e x}{c^2 d-c \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3}+\frac {\left (2 i a b g^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {i c e x}{c^2 d+c \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3}+\frac {\left (b^2 c^3 d (e f-d g)^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {2 i e x}{2 c d-2 \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {\left (b^2 c^3 d (e f-d g)^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {2 i e x}{2 c d+2 \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}\\ &=-\frac {a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}-\frac {2 a^2 g (e f-d g)}{e^3 (d+e x)}+\frac {a b c (e f-d g)^2 \sqrt {1-c^2 x^2}}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}-\frac {4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}+\frac {b^2 c (e f-d g)^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {i a b g^2 \sin ^{-1}(c x)^2}{e^3}-\frac {b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac {2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}-\frac {i b^2 g^2 \sin ^{-1}(c x)^3}{3 e^3}-\frac {a b c (e f-d g) \left (4 e^2 g-c^2 d (e f+3 d g)\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {2 a b g^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {4 i b^2 c g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {i b^2 c^3 d (e f-d g)^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 a b g^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {4 i b^2 c g (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {i b^2 c^3 d (e f-d g)^2 \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 g^2 \sin ^{-1}(c x)^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {a^2 g^2 \log (d+e x)}{e^3}-\frac {b^2 c^2 (e f-d g)^2 \log (d+e x)}{e^3 \left (c^2 d^2-e^2\right )}-\frac {2 i a b g^2 \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {4 b^2 c g (e f-d g) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}-\frac {b^2 c^3 d (e f-d g)^2 \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {2 i b^2 g^2 \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {2 i a b g^2 \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {4 b^2 c g (e f-d g) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \sqrt {c^2 d^2-e^2}}+\frac {b^2 c^3 d (e f-d g)^2 \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {2 i b^2 g^2 \sin ^{-1}(c x) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 b^2 g^2 \text {Li}_3\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {2 b^2 g^2 \text {Li}_3\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}\\ \end {align*}
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Mathematica [A] time = 4.98, size = 901, normalized size = 0.54 \[ \frac {-\frac {2 i g^2 \left (a+b \sin ^{-1}(c x)\right )^3}{b}+6 g^2 \log \left (\frac {i e^{i \sin ^{-1}(c x)} e}{\sqrt {c^2 d^2-e^2}-c d}+1\right ) \left (a+b \sin ^{-1}(c x)\right )^2+6 g^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {12 g (d g-e f) \left (a+b \sin ^{-1}(c x)\right )^2}{d+e x}-\frac {3 (e f-d g)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{(d+e x)^2}+\frac {24 b c g (d g-e f) \left (i \left (a+b \sin ^{-1}(c x)\right ) \left (\log \left (\frac {i e^{i \sin ^{-1}(c x)} e}{\sqrt {c^2 d^2-e^2}-c d}+1\right )-\log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )+b \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )-b \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{\sqrt {c^2 d^2-e^2}}+\frac {6 b c^2 (e f-d g)^2 \left (\frac {e \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c d+c e x}-b \log (d+e x)+\frac {c d \left (-i \left (a+b \sin ^{-1}(c x)\right ) \left (\log \left (\frac {i e^{i \sin ^{-1}(c x)} e}{\sqrt {c^2 d^2-e^2}-c d}+1\right )-\log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )-b \text {Li}_2\left (-\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}-c d}\right )+b \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{\sqrt {c^2 d^2-e^2}}\right )}{c^2 d^2-e^2}-12 b g^2 \left (i \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )-b \text {Li}_3\left (-\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}-c d}\right )\right )-12 b g^2 \left (i \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )-b \text {Li}_3\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{6 e^3} \]
Warning: Unable to verify antiderivative.
[In]
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} g^{2} x^{2} + 2 \, a^{2} f g x + a^{2} f^{2} + {\left (b^{2} g^{2} x^{2} + 2 \, b^{2} f g x + b^{2} f^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b g^{2} x^{2} + 2 \, a b f g x + a b f^{2}\right )} \arcsin \left (c x\right )}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 8.68, size = 0, normalized size = 0.00 \[ \int \frac {\left (g x +f \right )^{2} \left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (e x +d \right )^{3}}\, dx \]
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 \[ \int \frac {{\left (f+g\,x\right )}^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (f + g x\right )^{2}}{\left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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