Optimal. Leaf size=738 \[ \frac {f x \left (\frac {3 g^2}{c^2}+f^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {g \left (3 c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {4 i b g \sqrt {1-c^2 x^2} \left (3 c^2 f^2+g^2\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {i f \sqrt {1-c^2 x^2} \left (c^2 f^2+3 g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {2 b f \sqrt {1-c^2 x^2} \left (c^2 f^2+3 g^2\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {f g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 a b g^3 x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 g \sqrt {1-c^2 x^2} \left (3 c^2 f^2+g^2\right ) \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 g \sqrt {1-c^2 x^2} \left (3 c^2 f^2+g^2\right ) \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {i b^2 f \sqrt {1-c^2 x^2} \left (c^2 f^2+3 g^2\right ) \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt {d-c^2 d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.19, antiderivative size = 738, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 15, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {4777, 4775, 4763, 4651, 4675, 3719, 2190, 2279, 2391, 4677, 4657, 4181, 4641, 4619, 261} \[ -\frac {2 i b^2 g \sqrt {1-c^2 x^2} \left (3 c^2 f^2+g^2\right ) \text {PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 g \sqrt {1-c^2 x^2} \left (3 c^2 f^2+g^2\right ) \text {PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {i b^2 f \sqrt {1-c^2 x^2} \left (c^2 f^2+3 g^2\right ) \text {PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g \left (3 c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {f x \left (\frac {3 g^2}{c^2}+f^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}-\frac {i f \sqrt {1-c^2 x^2} \left (c^2 f^2+3 g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {2 b f \sqrt {1-c^2 x^2} \left (c^2 f^2+3 g^2\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {4 i b g \sqrt {1-c^2 x^2} \left (3 c^2 f^2+g^2\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {f g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 a b g^3 x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 261
Rule 2190
Rule 2279
Rule 2391
Rule 3719
Rule 4181
Rule 4619
Rule 4641
Rule 4651
Rule 4657
Rule 4675
Rule 4677
Rule 4763
Rule 4775
Rule 4777
Rubi steps
\begin {align*} \int \frac {(f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {\sqrt {1-c^2 x^2} \int \left (\frac {\left (c^2 f^3+3 f g^2+g \left (3 c^2 f^2+g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 f g^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {1-c^2 x^2}}-\frac {g^3 x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {1-c^2 x^2}}\right ) \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {\sqrt {1-c^2 x^2} \int \frac {\left (c^2 f^3+3 f g^2+g \left (3 c^2 f^2+g^2\right ) x\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (3 f g^2 \sqrt {1-c^2 x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (g^3 \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}\\ &=\frac {g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {f g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} \int \left (\frac {c^2 f^3 \left (1+\frac {3 g^2}{c^2 f^2}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}}+\frac {g \left (3 c^2 f^2+g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}}\right ) \, dx}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b g^3 \sqrt {1-c^2 x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{c^3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {2 a b g^3 x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {f g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 g^3 \sqrt {1-c^2 x^2}\right ) \int \sin ^{-1}(c x) \, dx}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {\left (g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {2 a b g^3 x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g \left (3 c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {f \left (c^2 f^2+3 g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {f g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 g^3 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{c d \sqrt {d-c^2 d x^2}}\\ &=-\frac {2 a b g^3 x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g \left (3 c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {f \left (c^2 f^2+3 g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {f g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {2 a b g^3 x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g \left (3 c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {f \left (c^2 f^2+3 g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {f g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt {d-c^2 d x^2}}+\frac {4 i b g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {\left (4 i b f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{c^3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {2 a b g^3 x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g \left (3 c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {f \left (c^2 f^2+3 g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {f g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt {d-c^2 d x^2}}+\frac {4 i b g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {2 b f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 i b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {2 a b g^3 x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g \left (3 c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {f \left (c^2 f^2+3 g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {f g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt {d-c^2 d x^2}}+\frac {4 i b g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {2 b f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {\left (i b^2 f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {2 a b g^3 x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 g^3 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g \left (3 c^2 f^2+g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {f \left (c^2 f^2+3 g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {f g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c^3 d \sqrt {d-c^2 d x^2}}+\frac {4 i b g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {2 b f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 g \left (3 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {i b^2 f \left (c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 3.44, size = 325, normalized size = 0.44 \[ \frac {\sqrt {1-c^2 x^2} \left (-(c f+g)^3 \left (-\tan \left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i \left (\left (a+b \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)+4 i b \log \left (1+i e^{i \sin ^{-1}(c x)}\right )\right )+4 b^2 \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )\right )\right )+(c f-g)^3 \left (-\cot \left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i \left (\left (a+b \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)-4 i b \log \left (1+i e^{-i \sin ^{-1}(c x)}\right )\right )+4 b^2 \text {Li}_2\left (-i e^{-i \sin ^{-1}(c x)}\right )\right )\right )+2 g^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-4 b g^3 \left (a c x+b \sqrt {1-c^2 x^2}+b c x \sin ^{-1}(c x)\right )-\frac {2 c f g^2 \left (a+b \sin ^{-1}(c x)\right )^3}{b}\right )}{2 c^4 d \sqrt {d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} g^{3} x^{3} + 3 \, a^{2} f g^{2} x^{2} + 3 \, a^{2} f^{2} g x + a^{2} f^{3} + {\left (b^{2} g^{3} x^{3} + 3 \, b^{2} f g^{2} x^{2} + 3 \, b^{2} f^{2} g x + b^{2} f^{3}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b g^{3} x^{3} + 3 \, a b f g^{2} x^{2} + 3 \, a b f^{2} g x + a b f^{3}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 1.39, size = 2663, normalized size = 3.61 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -a^{2} g^{3} {\left (\frac {x^{2}}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} - \frac {2}{\sqrt {-c^{2} d x^{2} + d} c^{4} d}\right )} + 3 \, a^{2} f g^{2} {\left (\frac {x}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} - \frac {\arcsin \left (c x\right )}{c^{3} d^{\frac {3}{2}}}\right )} + \frac {2 \, a b f^{3} x \arcsin \left (c x\right )}{\sqrt {-c^{2} d x^{2} + d} d} + \frac {a^{2} f^{3} x}{\sqrt {-c^{2} d x^{2} + d} d} - \frac {a b f^{3} \log \left (x^{2} - \frac {1}{c^{2}}\right )}{c d^{\frac {3}{2}}} + \frac {3 \, a^{2} f^{2} g}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} - \sqrt {d} \int \frac {{\left (b^{2} g^{3} x^{3} + 3 \, b^{2} f g^{2} x^{2} + 3 \, b^{2} f^{2} g x + b^{2} f^{3}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b g^{3} x^{3} + 3 \, a b f g^{2} x^{2} + 3 \, a b f^{2} g x\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{{\left (c^{2} d^{2} x^{2} - d^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (f+g\,x\right )}^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [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.
[In]
[Out]
________________________________________________________________________________________