Optimal. Leaf size=169 \[ -\frac {i (a+b \text {ArcSin}(c+d x))^4}{4 b d e}+\frac {(a+b \text {ArcSin}(c+d x))^3 \log \left (1-e^{2 i \text {ArcSin}(c+d x)}\right )}{d e}-\frac {3 i b (a+b \text {ArcSin}(c+d x))^2 \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c+d x)}\right )}{2 d e}+\frac {3 b^2 (a+b \text {ArcSin}(c+d x)) \text {PolyLog}\left (3,e^{2 i \text {ArcSin}(c+d x)}\right )}{2 d e}+\frac {3 i b^3 \text {PolyLog}\left (4,e^{2 i \text {ArcSin}(c+d x)}\right )}{4 d e} \]
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Rubi [A]
time = 0.15, 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 = {4889, 12, 4721,
3798, 2221, 2611, 6744, 2320, 6724} \begin {gather*} \frac {3 b^2 \text {Li}_3\left (e^{2 i \text {ArcSin}(c+d x)}\right ) (a+b \text {ArcSin}(c+d x))}{2 d e}-\frac {3 i b \text {Li}_2\left (e^{2 i \text {ArcSin}(c+d x)}\right ) (a+b \text {ArcSin}(c+d x))^2}{2 d e}-\frac {i (a+b \text {ArcSin}(c+d x))^4}{4 b d e}+\frac {\log \left (1-e^{2 i \text {ArcSin}(c+d x)}\right ) (a+b \text {ArcSin}(c+d x))^3}{d e}+\frac {3 i b^3 \text {Li}_4\left (e^{2 i \text {ArcSin}(c+d x)}\right )}{4 d e} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2221
Rule 2320
Rule 2611
Rule 3798
Rule 4721
Rule 4889
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c+d x)\right )^3}{c e+d e x} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^3}{e x} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^3}{x} \, dx,x,c+d x\right )}{d e}\\ &=\frac {\text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 = 0.12, size = 304, normalized size = 1.80 \begin {gather*} -\frac {i \left (8 a b^2 \pi ^3+b^3 \pi ^4+96 a^2 b \text {ArcSin}(c+d x)^2-64 a b^2 \text {ArcSin}(c+d x)^3-16 b^3 \text {ArcSin}(c+d x)^4+192 i a b^2 \text {ArcSin}(c+d x)^2 \log \left (1-e^{-2 i \text {ArcSin}(c+d x)}\right )+64 i b^3 \text {ArcSin}(c+d x)^3 \log \left (1-e^{-2 i \text {ArcSin}(c+d x)}\right )+192 i a^2 b \text {ArcSin}(c+d x) \log \left (1-e^{2 i \text {ArcSin}(c+d x)}\right )+64 i a^3 \log (c+d x)-96 b^2 \text {ArcSin}(c+d x) (2 a+b \text {ArcSin}(c+d x)) \text {PolyLog}\left (2,e^{-2 i \text {ArcSin}(c+d x)}\right )+96 a^2 b \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c+d x)}\right )+96 i a b^2 \text {PolyLog}\left (3,e^{-2 i \text {ArcSin}(c+d x)}\right )+96 i b^3 \text {ArcSin}(c+d x) \text {PolyLog}\left (3,e^{-2 i \text {ArcSin}(c+d x)}\right )+48 b^3 \text {PolyLog}\left (4,e^{-2 i \text {ArcSin}(c+d x)}\right )\right )}{64 d e} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 765 vs. \(2 (206 ) = 412\).
time = 0.22, size = 766, normalized size = 4.53
method | result | size |
derivativedivides | \(\frac {\frac {a^{3} \ln \left (d x +c \right )}{e}-\frac {i b^{3} \arcsin \left (d x +c \right )^{4}}{4 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 )}{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 )}{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 )}{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 )}{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 )}{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 )}{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 )}{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 )}{e}-\frac {i a \,b^{2} \arcsin \left (d x +c \right )^{3}}{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 )}{e}+\frac {6 i b^{3} \polylog \left (4, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{e}+\frac {6 a \,b^{2} \polylog \left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{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 )}{e}-\frac {3 i a^{2} b \arcsin \left (d x +c \right )^{2}}{2 e}+\frac {6 a \,b^{2} \polylog \left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{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 )}{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 )}{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 )}{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 )}{e}+\frac {6 i b^{3} \polylog \left (4, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{e}}{d}\) | \(766\) |
default | \(\frac {\frac {a^{3} \ln \left (d x +c \right )}{e}-\frac {i b^{3} \arcsin \left (d x +c \right )^{4}}{4 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 )}{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 )}{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 )}{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 )}{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 )}{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 )}{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 )}{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 )}{e}-\frac {i a \,b^{2} \arcsin \left (d x +c \right )^{3}}{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 )}{e}+\frac {6 i b^{3} \polylog \left (4, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{e}+\frac {6 a \,b^{2} \polylog \left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{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 )}{e}-\frac {3 i a^{2} b \arcsin \left (d x +c \right )^{2}}{2 e}+\frac {6 a \,b^{2} \polylog \left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{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 )}{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 )}{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 )}{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 )}{e}+\frac {6 i b^{3} \polylog \left (4, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{e}}{d}\) | \(766\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3}{c\,e+d\,e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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