Optimal. Leaf size=155 \[ \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 b d e}+\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3 \log \left (1-e^{-2 \sinh ^{-1}(c+d x)}\right )}{d e}-\frac {3 b \left (a+b \sinh ^{-1}(c+d x)\right )^2 \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c+d x)}\right )}{2 d e}-\frac {3 b^2 \left (a+b \sinh ^{-1}(c+d x)\right ) \text {PolyLog}\left (3,e^{-2 \sinh ^{-1}(c+d x)}\right )}{2 d e}-\frac {3 b^3 \text {PolyLog}\left (4,e^{-2 \sinh ^{-1}(c+d x)}\right )}{4 d e} \]
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Rubi [A]
time = 0.21, antiderivative size = 155, 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 = {5859, 12, 5775,
3797, 2221, 2611, 6744, 2320, 6724} \begin {gather*} -\frac {3 b^2 \text {Li}_3\left (e^{-2 \sinh ^{-1}(c+d x)}\right ) \left (a+b \sinh ^{-1}(c+d x)\right )}{2 d e}-\frac {3 b \text {Li}_2\left (e^{-2 \sinh ^{-1}(c+d x)}\right ) \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e}+\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 b d e}+\frac {\log \left (1-e^{-2 \sinh ^{-1}(c+d x)}\right ) \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d e}-\frac {3 b^3 \text {Li}_4\left (e^{-2 \sinh ^{-1}(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 3797
Rule 5775
Rule 5859
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{c e+d e x} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^3}{e x} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^3}{x} \, dx,x,c+d x\right )}{d e}\\ &=\frac {\text {Subst}\left (\int (a+b x)^3 \coth (x) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 b d e}-\frac {2 \text {Subst}\left (\int \frac {e^{2 x} (a+b x)^3}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 b d e}+\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3 \log \left (1-e^{2 \sinh ^{-1}(c+d x)}\right )}{d e}-\frac {(3 b) \text {Subst}\left (\int (a+b x)^2 \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 b d e}+\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3 \log \left (1-e^{2 \sinh ^{-1}(c+d x)}\right )}{d e}+\frac {3 b \left (a+b \sinh ^{-1}(c+d x)\right )^2 \text {Li}_2\left (e^{2 \sinh ^{-1}(c+d x)}\right )}{2 d e}-\frac {\left (3 b^2\right ) \text {Subst}\left (\int (a+b x) \text {Li}_2\left (e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 b d e}+\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3 \log \left (1-e^{2 \sinh ^{-1}(c+d x)}\right )}{d e}+\frac {3 b \left (a+b \sinh ^{-1}(c+d x)\right )^2 \text {Li}_2\left (e^{2 \sinh ^{-1}(c+d x)}\right )}{2 d e}-\frac {3 b^2 \left (a+b \sinh ^{-1}(c+d x)\right ) \text {Li}_3\left (e^{2 \sinh ^{-1}(c+d x)}\right )}{2 d e}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int \text {Li}_3\left (e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{2 d e}\\ &=-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 b d e}+\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3 \log \left (1-e^{2 \sinh ^{-1}(c+d x)}\right )}{d e}+\frac {3 b \left (a+b \sinh ^{-1}(c+d x)\right )^2 \text {Li}_2\left (e^{2 \sinh ^{-1}(c+d x)}\right )}{2 d e}-\frac {3 b^2 \left (a+b \sinh ^{-1}(c+d x)\right ) \text {Li}_3\left (e^{2 \sinh ^{-1}(c+d x)}\right )}{2 d e}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c+d x)}\right )}{4 d e}\\ &=-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 b d e}+\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3 \log \left (1-e^{2 \sinh ^{-1}(c+d x)}\right )}{d e}+\frac {3 b \left (a+b \sinh ^{-1}(c+d x)\right )^2 \text {Li}_2\left (e^{2 \sinh ^{-1}(c+d x)}\right )}{2 d e}-\frac {3 b^2 \left (a+b \sinh ^{-1}(c+d x)\right ) \text {Li}_3\left (e^{2 \sinh ^{-1}(c+d x)}\right )}{2 d e}+\frac {3 b^3 \text {Li}_4\left (e^{2 \sinh ^{-1}(c+d x)}\right )}{4 d e}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 128, normalized size = 0.83 \begin {gather*} \frac {-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^4}{b}+4 \left (a+b \sinh ^{-1}(c+d x)\right )^3 \log \left (1-e^{2 \sinh ^{-1}(c+d x)}\right )+6 b \left (a+b \sinh ^{-1}(c+d x)\right )^2 \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c+d x)}\right )-6 b^2 \left (a+b \sinh ^{-1}(c+d x)\right ) \text {PolyLog}\left (3,e^{2 \sinh ^{-1}(c+d x)}\right )+3 b^3 \text {PolyLog}\left (4,e^{2 \sinh ^{-1}(c+d x)}\right )}{4 d e} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(673\) vs.
\(2(179)=358\).
time = 2.67, size = 674, normalized size = 4.35
method | result | size |
derivativedivides | \(\frac {\frac {a^{3} \ln \left (d x +c \right )}{e}-\frac {b^{3} \arcsinh \left (d x +c \right )^{4}}{4 e}+\frac {b^{3} \arcsinh \left (d x +c \right )^{3} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {3 b^{3} \arcsinh \left (d x +c \right )^{2} \polylog \left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}-\frac {6 b^{3} \arcsinh \left (d x +c \right ) \polylog \left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {6 b^{3} \polylog \left (4, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {b^{3} \arcsinh \left (d x +c \right )^{3} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {3 b^{3} \arcsinh \left (d x +c \right )^{2} \polylog \left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}-\frac {6 b^{3} \arcsinh \left (d x +c \right ) \polylog \left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {6 b^{3} \polylog \left (4, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}-\frac {a \,b^{2} \arcsinh \left (d x +c \right )^{3}}{e}+\frac {3 a \,b^{2} \arcsinh \left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {6 a \,b^{2} \arcsinh \left (d x +c \right ) \polylog \left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}-\frac {6 a \,b^{2} \polylog \left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {3 a \,b^{2} \arcsinh \left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {6 a \,b^{2} \arcsinh \left (d x +c \right ) \polylog \left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}-\frac {6 a \,b^{2} \polylog \left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}-\frac {3 a^{2} b \arcsinh \left (d x +c \right )^{2}}{2 e}+\frac {3 a^{2} b \arcsinh \left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {3 a^{2} b \polylog \left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {3 a^{2} b \arcsinh \left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {3 a^{2} b \polylog \left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}}{d}\) | \(674\) |
default | \(\frac {\frac {a^{3} \ln \left (d x +c \right )}{e}-\frac {b^{3} \arcsinh \left (d x +c \right )^{4}}{4 e}+\frac {b^{3} \arcsinh \left (d x +c \right )^{3} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {3 b^{3} \arcsinh \left (d x +c \right )^{2} \polylog \left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}-\frac {6 b^{3} \arcsinh \left (d x +c \right ) \polylog \left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {6 b^{3} \polylog \left (4, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {b^{3} \arcsinh \left (d x +c \right )^{3} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {3 b^{3} \arcsinh \left (d x +c \right )^{2} \polylog \left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}-\frac {6 b^{3} \arcsinh \left (d x +c \right ) \polylog \left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {6 b^{3} \polylog \left (4, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}-\frac {a \,b^{2} \arcsinh \left (d x +c \right )^{3}}{e}+\frac {3 a \,b^{2} \arcsinh \left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {6 a \,b^{2} \arcsinh \left (d x +c \right ) \polylog \left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}-\frac {6 a \,b^{2} \polylog \left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {3 a \,b^{2} \arcsinh \left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {6 a \,b^{2} \arcsinh \left (d x +c \right ) \polylog \left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}-\frac {6 a \,b^{2} \polylog \left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}-\frac {3 a^{2} b \arcsinh \left (d x +c \right )^{2}}{2 e}+\frac {3 a^{2} b \arcsinh \left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {3 a^{2} b \polylog \left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {3 a^{2} b \arcsinh \left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {3 a^{2} b \polylog \left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}}{d}\) | \(674\) |
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 {asinh}^{3}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {3 a b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {3 a^{2} b \operatorname {asinh}{\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 {asinh}\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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