Optimal. Leaf size=590 \[ -\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}+\frac {3 a b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {b c^3 d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {3 b^2 c^2 d \sqrt {d-c^2 d x^2} \text {Li}_3\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {3 b^2 c^2 d \sqrt {d-c^2 d x^2} \text {Li}_3\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+2 b^2 c^2 d \sqrt {d-c^2 d x^2}-\frac {b^2 c^2 d \sqrt {d-c^2 d x^2} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{\sqrt {1-c^2 x^2}}+\frac {3 b^2 c^3 d x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \]
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Rubi [A] time = 0.61, antiderivative size = 590, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 15, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {4695, 4697, 4709, 4183, 2531, 2282, 6589, 4619, 261, 14, 4687, 446, 80, 63, 208} \[ -\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {3 b^2 c^2 d \sqrt {d-c^2 d x^2} \text {PolyLog}\left (3,-e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {3 b^2 c^2 d \sqrt {d-c^2 d x^2} \text {PolyLog}\left (3,e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {3 a b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {b c^3 d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}+2 b^2 c^2 d \sqrt {d-c^2 d x^2}+\frac {3 b^2 c^3 d x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}-\frac {b^2 c^2 d \sqrt {d-c^2 d x^2} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{\sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 63
Rule 80
Rule 208
Rule 261
Rule 446
Rule 2282
Rule 2531
Rule 4183
Rule 4619
Rule 4687
Rule 4695
Rule 4697
Rule 4709
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x^3} \, dx &=-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}-\frac {1}{2} \left (3 c^2 d\right ) \int \frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{x^2} \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {b c d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x \sqrt {1-c^2 x^2}}-\frac {b c^3 d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}-\frac {\left (3 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {-1-c^2 x^2}{x \sqrt {1-c^2 x^2}} \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (3 b c^3 d \sqrt {d-c^2 d x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {3 a b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}-\frac {b c d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x \sqrt {1-c^2 x^2}}-\frac {b c^3 d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}-\frac {\left (3 c^2 d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {-1-c^2 x}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{2 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 c^3 d \sqrt {d-c^2 d x^2}\right ) \int \sin ^{-1}(c x) \, dx}{\sqrt {1-c^2 x^2}}\\ &=-b^2 c^2 d \sqrt {d-c^2 d x^2}+\frac {3 a b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}+\frac {3 b^2 c^3 d x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}-\frac {b c d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x \sqrt {1-c^2 x^2}}-\frac {b c^3 d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (3 b c^2 d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (3 b c^2 d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{2 \sqrt {1-c^2 x^2}}-\frac {\left (3 b^2 c^4 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {1-c^2 x^2}}\\ &=2 b^2 c^2 d \sqrt {d-c^2 d x^2}+\frac {3 a b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}+\frac {3 b^2 c^3 d x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}-\frac {b c d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x \sqrt {1-c^2 x^2}}-\frac {b c^3 d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (b^2 d \sqrt {d-c^2 d 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 )}{\sqrt {1-c^2 x^2}}+\frac {\left (3 i b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (3 i b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}\\ &=2 b^2 c^2 d \sqrt {d-c^2 d x^2}+\frac {3 a b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}+\frac {3 b^2 c^3 d x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}-\frac {b c d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x \sqrt {1-c^2 x^2}}-\frac {b c^3 d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {b^2 c^2 d \sqrt {d-c^2 d x^2} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{\sqrt {1-c^2 x^2}}-\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (3 b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}\\ &=2 b^2 c^2 d \sqrt {d-c^2 d x^2}+\frac {3 a b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}+\frac {3 b^2 c^3 d x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}-\frac {b c d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x \sqrt {1-c^2 x^2}}-\frac {b c^3 d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {b^2 c^2 d \sqrt {d-c^2 d x^2} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{\sqrt {1-c^2 x^2}}-\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {3 b^2 c^2 d \sqrt {d-c^2 d x^2} \text {Li}_3\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {3 b^2 c^2 d \sqrt {d-c^2 d x^2} \text {Li}_3\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 7.19, size = 854, normalized size = 1.45 \[ -\frac {3}{2} a^2 d^{3/2} \log (x) c^2+\frac {3}{2} a^2 d^{3/2} \log \left (d+\sqrt {-d \left (c^2 x^2-1\right )} \sqrt {d}\right ) c^2-2 a b d \sqrt {d \left (1-c^2 x^2\right )} \left (-\frac {c x}{\sqrt {1-c^2 x^2}}+\sin ^{-1}(c x)+\frac {\sin ^{-1}(c x) \left (\log \left (1-e^{i \sin ^{-1}(c x)}\right )-\log \left (1+e^{i \sin ^{-1}(c x)}\right )\right )}{\sqrt {1-c^2 x^2}}+\frac {i \left (\text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )-\text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )\right )}{\sqrt {1-c^2 x^2}}\right ) c^2-b^2 d \sqrt {d \left (1-c^2 x^2\right )} \left (\frac {\left (\log \left (1-e^{i \sin ^{-1}(c x)}\right )-\log \left (1+e^{i \sin ^{-1}(c x)}\right )\right ) \sin ^{-1}(c x)^2}{\sqrt {1-c^2 x^2}}+\sin ^{-1}(c x)^2+\frac {2 i \left (\text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )-\text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )\right ) \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}-\frac {2 c x \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}+\frac {2 \left (\text {Li}_3\left (e^{i \sin ^{-1}(c x)}\right )-\text {Li}_3\left (-e^{i \sin ^{-1}(c x)}\right )\right )}{\sqrt {1-c^2 x^2}}-2\right ) c^2+\frac {a b d^2 \sqrt {1-c^2 x^2} \left (-\sin ^{-1}(c x) \csc ^2\left (\frac {1}{2} \sin ^{-1}(c x)\right )+\sin ^{-1}(c x) \sec ^2\left (\frac {1}{2} \sin ^{-1}(c x)\right )-2 \cot \left (\frac {1}{2} \sin ^{-1}(c x)\right )-4 \sin ^{-1}(c x) \log \left (1-e^{i \sin ^{-1}(c x)}\right )+4 \sin ^{-1}(c x) \log \left (1+e^{i \sin ^{-1}(c x)}\right )-4 i \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )+4 i \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )-2 \tan \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right ) c^2}{4 \sqrt {d \left (1-c^2 x^2\right )}}+\frac {b^2 d^2 \sqrt {1-c^2 x^2} \left (-\csc ^2\left (\frac {1}{2} \sin ^{-1}(c x)\right ) \sin ^{-1}(c x)^2+\sec ^2\left (\frac {1}{2} \sin ^{-1}(c x)\right ) \sin ^{-1}(c x)^2-4 \log \left (1-e^{i \sin ^{-1}(c x)}\right ) \sin ^{-1}(c x)^2+4 \log \left (1+e^{i \sin ^{-1}(c x)}\right ) \sin ^{-1}(c x)^2-4 \cot \left (\frac {1}{2} \sin ^{-1}(c x)\right ) \sin ^{-1}(c x)-8 i \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right ) \sin ^{-1}(c x)+8 i \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right ) \sin ^{-1}(c x)-4 \tan \left (\frac {1}{2} \sin ^{-1}(c x)\right ) \sin ^{-1}(c x)+8 \log \left (\tan \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )+8 \text {Li}_3\left (-e^{i \sin ^{-1}(c x)}\right )-8 \text {Li}_3\left (e^{i \sin ^{-1}(c x)}\right )\right ) c^2}{8 \sqrt {d \left (1-c^2 x^2\right )}}+\left (-c^2 d a^2-\frac {d a^2}{2 x^2}\right ) \sqrt {-d \left (c^2 x^2-1\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a^{2} c^{2} d x^{2} - a^{2} d + {\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{2} d x^{2} - a b d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{x^{3}}, x\right ) \]
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 = 0.60, size = 1372, normalized size = 2.33 \[ \text {result too large to display} \]
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 \[ \frac {1}{2} \, {\left (3 \, c^{2} d^{\frac {3}{2}} \log \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {d}}{{\left | x \right |}} + \frac {2 \, d}{{\left | x \right |}}\right ) - {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} - 3 \, \sqrt {-c^{2} d x^{2} + d} c^{2} d - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{2}}\right )} a^{2} - \sqrt {d} \int \frac {{\left ({\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b c^{2} d x^{2} - a b d\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}{x^{3}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^3} \,d x \]
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 \[ \int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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