Optimal. Leaf size=126 \[ -\frac {i (a+b \text {ArcSin}(c+d x))^3}{3 b d e}+\frac {(a+b \text {ArcSin}(c+d x))^2 \log \left (1-e^{2 i \text {ArcSin}(c+d x)}\right )}{d e}-\frac {i b (a+b \text {ArcSin}(c+d x)) \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c+d x)}\right )}{d e}+\frac {b^2 \text {PolyLog}\left (3,e^{2 i \text {ArcSin}(c+d x)}\right )}{2 d e} \]
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Rubi [A]
time = 0.13, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {4889, 12, 4721,
3798, 2221, 2611, 2320, 6724} \begin {gather*} -\frac {i b \text {Li}_2\left (e^{2 i \text {ArcSin}(c+d x)}\right ) (a+b \text {ArcSin}(c+d x))}{d e}-\frac {i (a+b \text {ArcSin}(c+d x))^3}{3 b d e}+\frac {\log \left (1-e^{2 i \text {ArcSin}(c+d x)}\right ) (a+b \text {ArcSin}(c+d x))^2}{d e}+\frac {b^2 \text {Li}_3\left (e^{2 i \text {ArcSin}(c+d x)}\right )}{2 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
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c+d x)\right )^2}{c e+d e x} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^2}{e x} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^2}{x} \, dx,x,c+d x\right )}{d e}\\ &=\frac {\text {Subst}\left (\int (a+b x)^2 \cot (x) \, dx,x,\sin ^{-1}(c+d x)\right )}{d e}\\ &=-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )^3}{3 b d e}-\frac {(2 i) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)^2}{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 )^3}{3 b d e}+\frac {\left (a+b \sin ^{-1}(c+d x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}-\frac {(2 b) \text {Subst}\left (\int (a+b x) \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 )^3}{3 b d e}+\frac {\left (a+b \sin ^{-1}(c+d x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}-\frac {i b \left (a+b \sin ^{-1}(c+d x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \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 )^3}{3 b d e}+\frac {\left (a+b \sin ^{-1}(c+d x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}-\frac {i b \left (a+b \sin ^{-1}(c+d x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}+\frac {b^2 \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c+d x)}\right )}{2 d e}\\ &=-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )^3}{3 b d e}+\frac {\left (a+b \sin ^{-1}(c+d x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}-\frac {i b \left (a+b \sin ^{-1}(c+d x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}+\frac {b^2 \text {Li}_3\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{2 d e}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 170, normalized size = 1.35 \begin {gather*} \frac {2 a b \text {ArcSin}(c+d x) \log \left (1-e^{2 i \text {ArcSin}(c+d x)}\right )+a^2 \log (c+d x)-i a b \left (\text {ArcSin}(c+d x)^2+\text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c+d x)}\right )\right )+b^2 \left (-\frac {i \pi ^3}{24}+\frac {1}{3} i \text {ArcSin}(c+d x)^3+\text {ArcSin}(c+d x)^2 \log \left (1-e^{-2 i \text {ArcSin}(c+d x)}\right )+i \text {ArcSin}(c+d x) \text {PolyLog}\left (2,e^{-2 i \text {ArcSin}(c+d x)}\right )+\frac {1}{2} \text {PolyLog}\left (3,e^{-2 i \text {ArcSin}(c+d x)}\right )\right )}{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. 419 vs. \(2 (156 ) = 312\).
time = 0.20, size = 420, normalized size = 3.33
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} \ln \left (d x +c \right )}{e}-\frac {i b^{2} \arcsin \left (d x +c \right )^{3}}{3 e}+\frac {b^{2} \arcsin \left (d x +c \right )^{2} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{e}-\frac {2 i b^{2} \arcsin \left (d x +c \right ) \polylog \left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{e}+\frac {2 b^{2} \polylog \left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{e}+\frac {b^{2} \arcsin \left (d x +c \right )^{2} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{e}-\frac {2 i b^{2} \arcsin \left (d x +c \right ) \polylog \left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{e}+\frac {2 b^{2} \polylog \left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{e}-\frac {i a b \arcsin \left (d x +c \right )^{2}}{e}+\frac {2 a b \arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{e}+\frac {2 a b \arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{e}-\frac {2 i a b \polylog \left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{e}-\frac {2 i a b \polylog \left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{e}}{d}\) | \(420\) |
default | \(\frac {\frac {a^{2} \ln \left (d x +c \right )}{e}-\frac {i b^{2} \arcsin \left (d x +c \right )^{3}}{3 e}+\frac {b^{2} \arcsin \left (d x +c \right )^{2} \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{e}-\frac {2 i b^{2} \arcsin \left (d x +c \right ) \polylog \left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{e}+\frac {2 b^{2} \polylog \left (3, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{e}+\frac {b^{2} \arcsin \left (d x +c \right )^{2} \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{e}-\frac {2 i b^{2} \arcsin \left (d x +c \right ) \polylog \left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{e}+\frac {2 b^{2} \polylog \left (3, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{e}-\frac {i a b \arcsin \left (d x +c \right )^{2}}{e}+\frac {2 a b \arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{e}+\frac {2 a b \arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{e}-\frac {2 i a b \polylog \left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{e}-\frac {2 i a b \polylog \left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )}{e}}{d}\) | \(420\) |
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^{2}}{c + d x}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {2 a 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 )}^2}{c\,e+d\,e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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