Optimal. Leaf size=202 \[ -\frac {i (a+b \text {ArcSin}(c+d x))^5}{5 b d e}+\frac {(a+b \text {ArcSin}(c+d x))^4 \log \left (1-e^{2 i \text {ArcSin}(c+d x)}\right )}{d e}-\frac {2 i b (a+b \text {ArcSin}(c+d x))^3 \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c+d x)}\right )}{d e}+\frac {3 b^2 (a+b \text {ArcSin}(c+d x))^2 \text {PolyLog}\left (3,e^{2 i \text {ArcSin}(c+d x)}\right )}{d e}+\frac {3 i b^3 (a+b \text {ArcSin}(c+d x)) \text {PolyLog}\left (4,e^{2 i \text {ArcSin}(c+d x)}\right )}{d e}-\frac {3 b^4 \text {PolyLog}\left (5,e^{2 i \text {ArcSin}(c+d x)}\right )}{2 d e} \]
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Rubi [A]
time = 0.18, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps
used = 10, 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 i b^3 \text {Li}_4\left (e^{2 i \text {ArcSin}(c+d x)}\right ) (a+b \text {ArcSin}(c+d x))}{d e}+\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 {2 i b \text {Li}_2\left (e^{2 i \text {ArcSin}(c+d x)}\right ) (a+b \text {ArcSin}(c+d x))^3}{d e}-\frac {i (a+b \text {ArcSin}(c+d x))^5}{5 b d e}+\frac {\log \left (1-e^{2 i \text {ArcSin}(c+d x)}\right ) (a+b \text {ArcSin}(c+d x))^4}{d e}-\frac {3 b^4 \text {Li}_5\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
Rule 6744
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c+d x)\right )^4}{c e+d e x} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^4}{e x} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^4}{x} \, dx,x,c+d x\right )}{d e}\\ &=\frac {\text {Subst}\left (\int (a+b x)^4 \cot (x) \, dx,x,\sin ^{-1}(c+d x)\right )}{d e}\\ &=-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )^5}{5 b d e}-\frac {(2 i) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)^4}{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 )^5}{5 b d e}+\frac {\left (a+b \sin ^{-1}(c+d x)\right )^4 \log \left (1-e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}-\frac {(4 b) \text {Subst}\left (\int (a+b x)^3 \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 )^5}{5 b d e}+\frac {\left (a+b \sin ^{-1}(c+d x)\right )^4 \log \left (1-e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}-\frac {2 i b \left (a+b \sin ^{-1}(c+d x)\right )^3 \text {Li}_2\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}+\frac {\left (6 i b^2\right ) \text {Subst}\left (\int (a+b x)^2 \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 )^5}{5 b d e}+\frac {\left (a+b \sin ^{-1}(c+d x)\right )^4 \log \left (1-e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}-\frac {2 i b \left (a+b \sin ^{-1}(c+d x)\right )^3 \text {Li}_2\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}+\frac {3 b^2 \left (a+b \sin ^{-1}(c+d x)\right )^2 \text {Li}_3\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}-\frac {\left (6 b^3\right ) \text {Subst}\left (\int (a+b x) \text {Li}_3\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 )^5}{5 b d e}+\frac {\left (a+b \sin ^{-1}(c+d x)\right )^4 \log \left (1-e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}-\frac {2 i b \left (a+b \sin ^{-1}(c+d x)\right )^3 \text {Li}_2\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}+\frac {3 b^2 \left (a+b \sin ^{-1}(c+d x)\right )^2 \text {Li}_3\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}+\frac {3 i b^3 \left (a+b \sin ^{-1}(c+d x)\right ) \text {Li}_4\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}-\frac {\left (3 i b^4\right ) \text {Subst}\left (\int \text {Li}_4\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 )^5}{5 b d e}+\frac {\left (a+b \sin ^{-1}(c+d x)\right )^4 \log \left (1-e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}-\frac {2 i b \left (a+b \sin ^{-1}(c+d x)\right )^3 \text {Li}_2\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}+\frac {3 b^2 \left (a+b \sin ^{-1}(c+d x)\right )^2 \text {Li}_3\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}+\frac {3 i b^3 \left (a+b \sin ^{-1}(c+d x)\right ) \text {Li}_4\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}-\frac {\left (3 b^4\right ) \text {Subst}\left (\int \frac {\text {Li}_4(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 )^5}{5 b d e}+\frac {\left (a+b \sin ^{-1}(c+d x)\right )^4 \log \left (1-e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}-\frac {2 i b \left (a+b \sin ^{-1}(c+d x)\right )^3 \text {Li}_2\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}+\frac {3 b^2 \left (a+b \sin ^{-1}(c+d x)\right )^2 \text {Li}_3\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}+\frac {3 i b^3 \left (a+b \sin ^{-1}(c+d x)\right ) \text {Li}_4\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}-\frac {3 b^4 \text {Li}_5\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{2 d e}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(439\) vs. \(2(202)=404\).
time = 0.25, size = 439, normalized size = 2.17 \begin {gather*} \frac {16 a^4 \log (c+d x)+64 a^3 b \left (\text {ArcSin}(c+d x) \log \left (1-e^{2 i \text {ArcSin}(c+d x)}\right )-\frac {1}{2} i \left (\text {ArcSin}(c+d x)^2+\text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c+d x)}\right )\right )\right )+4 a^2 b^2 \left (-i \pi ^3+8 i \text {ArcSin}(c+d x)^3+24 \text {ArcSin}(c+d x)^2 \log \left (1-e^{-2 i \text {ArcSin}(c+d x)}\right )+24 i \text {ArcSin}(c+d x) \text {PolyLog}\left (2,e^{-2 i \text {ArcSin}(c+d x)}\right )+12 \text {PolyLog}\left (3,e^{-2 i \text {ArcSin}(c+d x)}\right )\right )-i a b^3 \left (\pi ^4-16 \text {ArcSin}(c+d x)^4+64 i \text {ArcSin}(c+d x)^3 \log \left (1-e^{-2 i \text {ArcSin}(c+d x)}\right )-96 \text {ArcSin}(c+d x)^2 \text {PolyLog}\left (2,e^{-2 i \text {ArcSin}(c+d x)}\right )+96 i \text {ArcSin}(c+d x) \text {PolyLog}\left (3,e^{-2 i \text {ArcSin}(c+d x)}\right )+48 \text {PolyLog}\left (4,e^{-2 i \text {ArcSin}(c+d x)}\right )\right )+16 b^4 \left (-\frac {i \pi ^5}{160}+\frac {1}{5} i \text {ArcSin}(c+d x)^5+\text {ArcSin}(c+d x)^4 \log \left (1-e^{-2 i \text {ArcSin}(c+d x)}\right )+2 i \text {ArcSin}(c+d x)^3 \text {PolyLog}\left (2,e^{-2 i \text {ArcSin}(c+d x)}\right )+3 \text {ArcSin}(c+d x)^2 \text {PolyLog}\left (3,e^{-2 i \text {ArcSin}(c+d x)}\right )-3 i \text {ArcSin}(c+d x) \text {PolyLog}\left (4,e^{-2 i \text {ArcSin}(c+d x)}\right )-\frac {3}{2} \text {PolyLog}\left (5,e^{-2 i \text {ArcSin}(c+d x)}\right )\right )}{16 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. 1199 vs. \(2 (255 ) = 510\).
time = 0.19, size = 1200, normalized size = 5.94
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1200\) |
default | \(\text {Expression too large to display}\) | \(1200\) |
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^{4}}{c + d x}\, dx + \int \frac {b^{4} \operatorname {asin}^{4}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {4 a b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {6 a^{2} b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {4 a^{3} 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 = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^4}{c\,e+d\,e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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