Optimal. Leaf size=169 \[ \frac {3 b^2 \text {Li}_3\left (e^{2 i \sin ^{-1}(c+d x)}\right ) \left (a+b \sin ^{-1}(c+d x)\right )}{2 d e}-\frac {3 i b \text {Li}_2\left (e^{2 i \sin ^{-1}(c+d x)}\right ) \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d e}-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )^4}{4 b d e}+\frac {\log \left (1-e^{2 i \sin ^{-1}(c+d x)}\right ) \left (a+b \sin ^{-1}(c+d x)\right )^3}{d e}+\frac {3 i b^3 \text {Li}_4\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{4 d e} \]
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Rubi [A] time = 0.20, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {4805, 12, 4625, 3717, 2190, 2531, 6609, 2282, 6589} \[ \frac {3 b^2 \text {PolyLog}\left (3,e^{2 i \sin ^{-1}(c+d x)}\right ) \left (a+b \sin ^{-1}(c+d x)\right )}{2 d e}-\frac {3 i b \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(c+d x)}\right ) \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d e}+\frac {3 i b^3 \text {PolyLog}\left (4,e^{2 i \sin ^{-1}(c+d x)}\right )}{4 d e}-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )^4}{4 b d e}+\frac {\log \left (1-e^{2 i \sin ^{-1}(c+d x)}\right ) \left (a+b \sin ^{-1}(c+d x)\right )^3}{d e} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2190
Rule 2282
Rule 2531
Rule 3717
Rule 4625
Rule 4805
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c+d x)\right )^3}{c e+d e x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^3}{e x} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^3}{x} \, dx,x,c+d x\right )}{d e}\\ &=\frac {\operatorname {Subst}\left (\int (a+b x)^3 \cot (x) \, dx,x,\sin ^{-1}(c+d x)\right )}{d e}\\ &=-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )^4}{4 b d e}-\frac {(2 i) \operatorname {Subst}\left (\int \frac {e^{2 i x} (a+b x)^3}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{d e}\\ &=-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )^4}{4 b d e}+\frac {\left (a+b \sin ^{-1}(c+d x)\right )^3 \log \left (1-e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}-\frac {(3 b) \operatorname {Subst}\left (\int (a+b x)^2 \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{d e}\\ &=-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )^4}{4 b d e}+\frac {\left (a+b \sin ^{-1}(c+d x)\right )^3 \log \left (1-e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}-\frac {3 i b \left (a+b \sin ^{-1}(c+d x)\right )^2 \text {Li}_2\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{2 d e}+\frac {\left (3 i b^2\right ) \operatorname {Subst}\left (\int (a+b x) \text {Li}_2\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{d e}\\ &=-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )^4}{4 b d e}+\frac {\left (a+b \sin ^{-1}(c+d x)\right )^3 \log \left (1-e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}-\frac {3 i b \left (a+b \sin ^{-1}(c+d x)\right )^2 \text {Li}_2\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{2 d e}+\frac {3 b^2 \left (a+b \sin ^{-1}(c+d x)\right ) \text {Li}_3\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{2 d e}-\frac {\left (3 b^3\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{2 d e}\\ &=-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )^4}{4 b d e}+\frac {\left (a+b \sin ^{-1}(c+d x)\right )^3 \log \left (1-e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}-\frac {3 i b \left (a+b \sin ^{-1}(c+d x)\right )^2 \text {Li}_2\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{2 d e}+\frac {3 b^2 \left (a+b \sin ^{-1}(c+d x)\right ) \text {Li}_3\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{2 d e}+\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c+d x)}\right )}{4 d e}\\ &=-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )^4}{4 b d e}+\frac {\left (a+b \sin ^{-1}(c+d x)\right )^3 \log \left (1-e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}-\frac {3 i b \left (a+b \sin ^{-1}(c+d x)\right )^2 \text {Li}_2\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{2 d e}+\frac {3 b^2 \left (a+b \sin ^{-1}(c+d x)\right ) \text {Li}_3\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{2 d e}+\frac {3 i b^3 \text {Li}_4\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{4 d e}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 304, normalized size = 1.80 \[ -\frac {i \left (64 i a^3 \log (c+d x)+96 a^2 b \text {Li}_2\left (e^{2 i \sin ^{-1}(c+d x)}\right )+96 a^2 b \sin ^{-1}(c+d x)^2+192 i a^2 b \sin ^{-1}(c+d x) \log \left (1-e^{2 i \sin ^{-1}(c+d x)}\right )-96 b^2 \sin ^{-1}(c+d x) \text {Li}_2\left (e^{-2 i \sin ^{-1}(c+d x)}\right ) \left (2 a+b \sin ^{-1}(c+d x)\right )+96 i a b^2 \text {Li}_3\left (e^{-2 i \sin ^{-1}(c+d x)}\right )-64 a b^2 \sin ^{-1}(c+d x)^3+192 i a b^2 \sin ^{-1}(c+d x)^2 \log \left (1-e^{-2 i \sin ^{-1}(c+d x)}\right )+8 \pi ^3 a b^2+96 i b^3 \sin ^{-1}(c+d x) \text {Li}_3\left (e^{-2 i \sin ^{-1}(c+d x)}\right )+48 b^3 \text {Li}_4\left (e^{-2 i \sin ^{-1}(c+d x)}\right )-16 b^3 \sin ^{-1}(c+d x)^4+64 i b^3 \sin ^{-1}(c+d x)^3 \log \left (1-e^{-2 i \sin ^{-1}(c+d x)}\right )+\pi ^4 b^3\right )}{64 d e} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \arcsin \left (d x + c\right )^{3} + 3 \, a b^{2} \arcsin \left (d x + c\right )^{2} + 3 \, a^{2} b \arcsin \left (d x + c\right ) + a^{3}}{d e x + c e}, x\right ) \]
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 \[ \int \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{3}}{d e x + c e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 828, normalized size = 4.90 \[ \frac {a^{3} \ln \left (d x +c \right )}{d e}-\frac {3 i b^{3} \arcsin \left (d x +c \right )^{2} \polylog \left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{d e}+\frac {b^{3} \arcsin \left (d x +c \right )^{3} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d e}-\frac {6 i a \,b^{2} \arcsin \left (d x +c \right ) \polylog \left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{d e}+\frac {6 b^{3} \arcsin \left (d x +c \right ) \polylog \left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{d e}-\frac {3 i a^{2} b \polylog \left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d e}+\frac {b^{3} \arcsin \left (d x +c \right )^{3} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{d e}-\frac {i a \,b^{2} \arcsin \left (d x +c \right )^{3}}{d e}+\frac {6 b^{3} \arcsin \left (d x +c \right ) \polylog \left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d e}-\frac {3 i a^{2} b \arcsin \left (d x +c \right )^{2}}{2 d e}-\frac {3 i a^{2} b \polylog \left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{d e}+\frac {3 a \,b^{2} \arcsin \left (d x +c \right )^{2} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d e}+\frac {6 i b^{3} \polylog \left (4, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d e}+\frac {6 a \,b^{2} \polylog \left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{d e}+\frac {3 a \,b^{2} \arcsin \left (d x +c \right )^{2} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{d e}+\frac {6 i b^{3} \polylog \left (4, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{d e}+\frac {6 a \,b^{2} \polylog \left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d e}-\frac {6 i a \,b^{2} \arcsin \left (d x +c \right ) \polylog \left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d e}+\frac {3 a^{2} b \arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d e}+\frac {3 a^{2} b \arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{d e}-\frac {i b^{3} \arcsin \left (d x +c \right )^{4}}{4 d e}-\frac {3 i b^{3} \arcsin \left (d x +c \right )^{2} \polylog \left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d e} \]
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 \[ \frac {a^{3} \log \left (d e x + c e\right )}{d e} + \int \frac {b^{3} \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{3} + 3 \, a b^{2} \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{2} + 3 \, a^{2} b \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )}{d e x + c e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3}{c\,e+d\,e\,x} \,d x \]
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 \[ \frac {\int \frac {a^{3}}{c + d x}\, dx + \int \frac {b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {3 a b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {3 a^{2} b \operatorname {asin}{\left (c + d x \right )}}{c + d x}\, dx}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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