Optimal. Leaf size=157 \[ \frac {3 b \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^3}-\frac {3 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac {3 b^2 \left (a+b \sinh ^{-1}(c+d x)\right ) \log \left (1-e^{-2 \sinh ^{-1}(c+d x)}\right )}{d e^3}-\frac {3 b^3 \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c+d x)}\right )}{2 d e^3} \]
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Rubi [A]
time = 0.21, antiderivative size = 157, 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, 5776,
5800, 5775, 3797, 2221, 2317, 2438} \begin {gather*} \frac {3 b^2 \log \left (1-e^{-2 \sinh ^{-1}(c+d x)}\right ) \left (a+b \sinh ^{-1}(c+d x)\right )}{d e^3}-\frac {3 b \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}+\frac {3 b \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^3}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}-\frac {3 b^3 \text {Li}_2\left (e^{-2 \sinh ^{-1}(c+d x)}\right )}{2 d e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 5775
Rule 5776
Rule 5800
Rule 5859
Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{(c e+d e x)^3} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^3}{e^3 x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^3}{x^3} \, dx,x,c+d x\right )}{d e^3}\\ &=-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac {(3 b) \text {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^2}{x^2 \sqrt {1+x^2}} \, dx,x,c+d x\right )}{2 d e^3}\\ &=-\frac {3 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{x} \, dx,x,c+d x\right )}{d e^3}\\ &=-\frac {3 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d e^3}\\ &=-\frac {3 b \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^3}-\frac {3 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}-\frac {\left (6 b^2\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{d e^3}\\ &=-\frac {3 b \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^3}-\frac {3 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac {3 b^2 \left (a+b \sinh ^{-1}(c+d x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c+d x)}\right )}{d e^3}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d e^3}\\ &=-\frac {3 b \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^3}-\frac {3 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac {3 b^2 \left (a+b \sinh ^{-1}(c+d x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c+d x)}\right )}{d e^3}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c+d x)}\right )}{2 d e^3}\\ &=-\frac {3 b \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^3}-\frac {3 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac {3 b^2 \left (a+b \sinh ^{-1}(c+d x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c+d x)}\right )}{d e^3}+\frac {3 b^3 \text {Li}_2\left (e^{2 \sinh ^{-1}(c+d x)}\right )}{2 d e^3}\\ \end {align*}
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Mathematica [A]
time = 0.56, size = 229, normalized size = 1.46 \begin {gather*} -\frac {3 b^2 \left (a+b (c+d x) \left (-c-d x+\sqrt {1+c^2+2 c d x+d^2 x^2}\right )\right ) \sinh ^{-1}(c+d x)^2+b^3 \sinh ^{-1}(c+d x)^3+3 b \sinh ^{-1}(c+d x) \left (a \left (a+2 b (c+d x) \sqrt {1+c^2+2 c d x+d^2 x^2}\right )-2 b^2 (c+d x)^2 \log \left (1-e^{-2 \sinh ^{-1}(c+d x)}\right )\right )+a \left (a \left (a+3 b (c+d x) \sqrt {1+c^2+2 c d x+d^2 x^2}\right )-6 b^2 (c+d x)^2 \log (c+d x)\right )+3 b^3 (c+d x)^2 \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c+d x)}\right )}{2 d e^3 (c+d x)^2} \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. \(366\) vs.
\(2(163)=326\).
time = 4.58, size = 367, normalized size = 2.34
method | result | size |
derivativedivides | \(\frac {-\frac {a^{3}}{2 e^{3} \left (d x +c \right )^{2}}-\frac {3 b^{3} \arcsinh \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{2 e^{3} \left (d x +c \right )}-\frac {3 b^{3} \arcsinh \left (d x +c \right )^{2}}{2 e^{3}}-\frac {b^{3} \arcsinh \left (d x +c \right )^{3}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {3 b^{3} \arcsinh \left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{3}}+\frac {3 b^{3} \polylog \left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{3}}+\frac {3 b^{3} \arcsinh \left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{3}}+\frac {3 b^{3} \polylog \left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{3}}-\frac {3 a \,b^{2} \arcsinh \left (d x +c \right )}{e^{3}}-\frac {3 a \,b^{2} \arcsinh \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{e^{3} \left (d x +c \right )}-\frac {3 a \,b^{2} \arcsinh \left (d x +c \right )^{2}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {3 a \,b^{2} \ln \left (\left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )^{2}-1\right )}{e^{3}}+\frac {3 a^{2} b \left (-\frac {\arcsinh \left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{2 \left (d x +c \right )}\right )}{e^{3}}}{d}\) | \(367\) |
default | \(\frac {-\frac {a^{3}}{2 e^{3} \left (d x +c \right )^{2}}-\frac {3 b^{3} \arcsinh \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{2 e^{3} \left (d x +c \right )}-\frac {3 b^{3} \arcsinh \left (d x +c \right )^{2}}{2 e^{3}}-\frac {b^{3} \arcsinh \left (d x +c \right )^{3}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {3 b^{3} \arcsinh \left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{3}}+\frac {3 b^{3} \polylog \left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{3}}+\frac {3 b^{3} \arcsinh \left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{3}}+\frac {3 b^{3} \polylog \left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{3}}-\frac {3 a \,b^{2} \arcsinh \left (d x +c \right )}{e^{3}}-\frac {3 a \,b^{2} \arcsinh \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{e^{3} \left (d x +c \right )}-\frac {3 a \,b^{2} \arcsinh \left (d x +c \right )^{2}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {3 a \,b^{2} \ln \left (\left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )^{2}-1\right )}{e^{3}}+\frac {3 a^{2} b \left (-\frac {\arcsinh \left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{2 \left (d x +c \right )}\right )}{e^{3}}}{d}\) | \(367\) |
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^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {3 a b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {3 a^{2} b \operatorname {asinh}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx}{e^{3}} \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}{{\left (c\,e+d\,e\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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