Optimal. Leaf size=752 \[ \frac {5 i b c^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b c^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {26 i b c^2 \sqrt {1-c^2 x^2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 c^2 \sqrt {1-c^2 x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {13 i b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {13 i b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_3\left (-e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_3\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 c^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{d^2 \sqrt {d-c^2 d x^2}} \]
[Out]
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Rubi [A] time = 1.26, antiderivative size = 752, normalized size of antiderivative = 1.00, number of steps used = 39, number of rules used = 18, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.621, Rules used = {4701, 4705, 4713, 4709, 4183, 2531, 2282, 6589, 4657, 4181, 2279, 2391, 4655, 261, 266, 51, 63, 208} \[ \frac {5 i b c^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b c^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {13 i b^2 c^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {13 i b^2 c^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 b^2 c^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (3,-e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 b^2 c^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (3,e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {26 i b c^2 \sqrt {1-c^2 x^2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 c^2 \sqrt {1-c^2 x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {b^2 c^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{d^2 \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 208
Rule 261
Rule 266
Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 4181
Rule 4183
Rule 4655
Rule 4657
Rule 4701
Rule 4705
Rule 4709
Rule 4713
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx &=-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {1}{2} \left (5 c^2\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx+\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x^2 \left (1-c^2 x^2\right )^2} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {\left (5 c^2\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx}{2 d}+\frac {\left (b^2 c^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \left (1-c^2 x^2\right )^{3/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^2} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 c^2\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \sqrt {d-c^2 d x^2}} \, dx}{2 d^2}+\frac {\left (b^2 c^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 b^2 c^4 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (3 b^2 c^4 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 c^2 \sqrt {1-c^2 x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \sqrt {1-c^2 x^2}} \, dx}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b c^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 b c^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b c^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 c^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {26 i b c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 c^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 b^2 c^2 \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 )}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b^2 c^2 \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 )}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (3 b^2 c^2 \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 )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 b^2 c^2 \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 )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 b^2 c^2 \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 )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b^2 c^2 \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 )}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {26 i b c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b c^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 b c^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 i b^2 c^2 \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 )}{6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 i b^2 c^2 \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 )}{6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 i b^2 c^2 \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 )}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (3 i b^2 c^2 \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 )}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 i b^2 c^2 \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 )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 i b^2 c^2 \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 )}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {26 i b c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {13 i b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {13 i b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 i b^2 c^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 i b^2 c^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {26 i b c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {13 i b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {13 i b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b^2 c^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 b^2 c^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {26 i b c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {13 i b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {13 i b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {5 b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_3\left (-e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_3\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 11.32, size = 1090, normalized size = 1.45 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}\right )}}{c^{6} d^{3} x^{9} - 3 \, c^{4} d^{3} x^{7} + 3 \, c^{2} d^{3} x^{5} - d^{3} x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.74, size = 1876, normalized size = 2.49 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{6} \, a^{2} {\left (\frac {15 \, c^{2} \log \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {d}}{{\left | x \right |}} + \frac {2 \, d}{{\left | x \right |}}\right )}{d^{\frac {5}{2}}} - \frac {15 \, c^{2}}{\sqrt {-c^{2} d x^{2} + d} d^{2}} - \frac {5 \, c^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d} + \frac {3}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x^{2}}\right )} - \sqrt {d} \int \frac {{\left (b^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, a b \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}{c^{6} d^{3} x^{9} - 3 \, c^{4} d^{3} x^{7} + 3 \, c^{2} d^{3} x^{5} - d^{3} x^{3}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^3\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{3} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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