Optimal. Leaf size=412 \[ \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d^2}+\frac {4 i b \sqrt {1-c^2 x^2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {4 a b x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {2 b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {4 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.45, antiderivative size = 412, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {4703, 4677, 4619, 261, 4715, 4657, 4181, 2279, 2391} \[ -\frac {2 i b^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d^2}-\frac {4 a b x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {2 b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {4 i b \sqrt {1-c^2 x^2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {4 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 261
Rule 2279
Rule 2391
Rule 4181
Rule 4619
Rule 4657
Rule 4677
Rule 4703
Rule 4715
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx &=\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}} \, dx}{c^2 d}-\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {x^2 \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 b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d^2}-\frac {\left (2 b \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 (4 b \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 {\left (2 b^2 \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 {4 a b x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {2 b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d^2}-\frac {\left (2 b \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 (4 b^2 \sqrt {1-c^2 x^2}\right ) \int \sin ^{-1}(c x) \, dx}{c^3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {4 a b x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {4 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {2 b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d^2}+\frac {4 i b \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 \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 \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 b^2 \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 {4 a b x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {4 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {2 b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d^2}+\frac {4 i b \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 i b^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 )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^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 )}{c^4 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {4 a b x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {4 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {2 b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^4 d^2}+\frac {4 i b \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 i b^2 \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 \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}}\\ \end {align*}
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Mathematica [A] time = 0.60, size = 369, normalized size = 0.90 \[ \frac {-2 a^2 c^2 x^2+4 a^2+4 a b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )-4 a b \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )+6 a b \sin ^{-1}(c x)-2 a b \sin \left (2 \sin ^{-1}(c x)\right )+2 a b \sin ^{-1}(c x) \cos \left (2 \sin ^{-1}(c x)\right )-4 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )+4 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )-4 b^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )+4 b^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log \left (1+i e^{i \sin ^{-1}(c x)}\right )+3 b^2 \sin ^{-1}(c x)^2-2 b^2 \sin ^{-1}(c x) \sin \left (2 \sin ^{-1}(c x)\right )-2 b^2 \cos \left (2 \sin ^{-1}(c x)\right )+b^2 \sin ^{-1}(c x)^2 \cos \left (2 \sin ^{-1}(c x)\right )-2 b^2}{2 c^4 d \sqrt {d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{3} \arcsin \left (c x\right )^{2} + 2 \, a b x^{3} \arcsin \left (c x\right ) + a^{2} x^{3}\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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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 = 0.54, size = 830, normalized size = 2.01 \[ -\frac {a^{2} x^{2}}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 a^{2}}{d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x}{c^{3} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )^{2} x^{2}}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{2}}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )^{2}}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 i b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \dilog \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 i b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \dilog \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, x}{c^{3} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x^{2}}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {4 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )} \]
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 \[ -a b c {\left (\frac {2 \, x}{c^{4} d^{\frac {3}{2}}} + \frac {\log \left (c x + 1\right )}{c^{5} d^{\frac {3}{2}}} - \frac {\log \left (c x - 1\right )}{c^{5} d^{\frac {3}{2}}}\right )} - 2 \, a b {\left (\frac {x^{2}}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} - \frac {2}{\sqrt {-c^{2} d x^{2} + d} c^{4} d}\right )} \arcsin \left (c x\right ) - a^{2} {\left (\frac {x^{2}}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} - \frac {2}{\sqrt {-c^{2} d x^{2} + d} c^{4} d}\right )} + \frac {\frac {1}{3} \, {\left (3 \, {\left (c^{2} x^{2} - 2\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \sqrt {d} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} - \frac {{\left (c^{6} d^{2} x^{2} - c^{4} d^{2}\right )} {\left (c^{4} d^{2} \int \frac {2 \, {\left (c^{3} x^{3} - 3 \, c x\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (-c x + 1\right )\right )} + 3 \, e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (-c x + 1\right )\right )} \log \left (c x + 1\right ) - 3 \, e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (-c x + 1\right )\right )} \log \left (-c x + 1\right )}{c^{7} d^{2} x^{4} - c^{5} d^{2} x^{2} - {\left (c^{5} d^{2} x^{2} - c^{3} d^{2}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )}}\,{d x} + 2 \, {\left (c^{3} x^{3} - 3 \, c x\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + 3 \, \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) \log \left (c x + 1\right ) - 3 \, \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) \log \left (-c x + 1\right )\right )}}{c^{4} d^{\frac {3}{2}}}\right )} b^{2}}{c^{6} d^{2} x^{2} - c^{4} d^{2}} \]
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 \[ \int \frac {x^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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