Optimal. Leaf size=164 \[ -\frac {3 b \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3}+\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}-\frac {3 b^2 \left (a+b \cosh ^{-1}(c+d x)\right ) \log \left (1+e^{-2 \cosh ^{-1}(c+d x)}\right )}{d e^3}+\frac {3 b^3 \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c+d x)}\right )}{2 d e^3} \]
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Rubi [A]
time = 0.26, antiderivative size = 164, 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 = {5996, 12, 5883,
5918, 5882, 3799, 2221, 2317, 2438} \begin {gather*} -\frac {3 b^2 \log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^3}+\frac {3 b \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}-\frac {3 b \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac {3 b^3 \text {Li}_2\left (-e^{-2 \cosh ^{-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 3799
Rule 5882
Rule 5883
Rule 5918
Rule 5996
Rubi steps
\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^3}{(c e+d e x)^3} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^3}{e^3 x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^3}{x^3} \, dx,x,c+d x\right )}{d e^3}\\ &=-\frac {\left (a+b \cosh ^{-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 \cosh ^{-1}(x)\right )^2}{\sqrt {-1+x} x^2 \sqrt {1+x}} \, dx,x,c+d x\right )}{2 d e^3}\\ &=\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}-\frac {\left (a+b \cosh ^{-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 \cosh ^{-1}(x)}{x} \, dx,x,c+d x\right )}{d e^3}\\ &=\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}-\frac {\left (a+b \cosh ^{-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) \tanh (x) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^3}\\ &=\frac {3 b \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3}+\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}-\frac {\left (a+b \cosh ^{-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,\cosh ^{-1}(c+d x)\right )}{d e^3}\\ &=\frac {3 b \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3}+\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}-\frac {3 b^2 \left (a+b \cosh ^{-1}(c+d x)\right ) \log \left (1+e^{2 \cosh ^{-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,\cosh ^{-1}(c+d x)\right )}{d e^3}\\ &=\frac {3 b \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3}+\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}-\frac {3 b^2 \left (a+b \cosh ^{-1}(c+d x)\right ) \log \left (1+e^{2 \cosh ^{-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 \cosh ^{-1}(c+d x)}\right )}{2 d e^3}\\ &=\frac {3 b \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3}+\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}-\frac {3 b^2 \left (a+b \cosh ^{-1}(c+d x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e^3}-\frac {3 b^3 \text {Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{2 d e^3}\\ \end {align*}
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Mathematica [A]
time = 0.78, size = 266, normalized size = 1.62 \begin {gather*} \frac {-\frac {a^3}{(c+d x)^2}+\frac {3 a^2 b \left (\sqrt {\frac {-1+c+d x}{1+c+d x}} \left (c+c^2+2 c d x+d x (1+d x)\right )-\cosh ^{-1}(c+d x)\right )}{(c+d x)^2}-\frac {b^3 \cosh ^{-1}(c+d x)^3}{(c+d x)^2}+6 a b^2 \left (\frac {\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \cosh ^{-1}(c+d x)}{c+d x}-\frac {\cosh ^{-1}(c+d x)^2}{2 (c+d x)^2}-\log (c+d x)\right )+3 b^3 \left (\cosh ^{-1}(c+d x) \left (-\cosh ^{-1}(c+d x)+\frac {\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \cosh ^{-1}(c+d x)}{c+d x}-2 \log \left (1+e^{-2 \cosh ^{-1}(c+d x)}\right )\right )+\text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c+d x)}\right )\right )}{2 d e^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 42.89, size = 339, normalized size = 2.07
method | result | size |
derivativedivides | \(\frac {-\frac {a^{3}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {3 b^{3} \mathrm {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c +1}\, \sqrt {d x +c -1}}{2 e^{3} \left (d x +c \right )}+\frac {3 b^{3} \mathrm {arccosh}\left (d x +c \right )^{2}}{2 e^{3}}-\frac {b^{3} \mathrm {arccosh}\left (d x +c \right )^{3}}{2 e^{3} \left (d x +c \right )^{2}}-\frac {3 b^{3} \mathrm {arccosh}\left (d x +c \right ) \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e^{3}}-\frac {3 b^{3} \polylog \left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2 e^{3}}+\frac {3 a \,b^{2} \mathrm {arccosh}\left (d x +c \right )}{e^{3}}+\frac {3 a \,b^{2} \mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c +1}\, \sqrt {d x +c -1}}{e^{3} \left (d x +c \right )}-\frac {3 a \,b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}}{2 e^{3} \left (d x +c \right )^{2}}-\frac {3 a \,b^{2} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e^{3}}+\frac {3 a^{2} b \left (-\frac {\mathrm {arccosh}\left (d x +c \right )}{2 \left (d x +c \right )^{2}}+\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2 d x +2 c}\right )}{e^{3}}}{d}\) | \(339\) |
default | \(\frac {-\frac {a^{3}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {3 b^{3} \mathrm {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c +1}\, \sqrt {d x +c -1}}{2 e^{3} \left (d x +c \right )}+\frac {3 b^{3} \mathrm {arccosh}\left (d x +c \right )^{2}}{2 e^{3}}-\frac {b^{3} \mathrm {arccosh}\left (d x +c \right )^{3}}{2 e^{3} \left (d x +c \right )^{2}}-\frac {3 b^{3} \mathrm {arccosh}\left (d x +c \right ) \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e^{3}}-\frac {3 b^{3} \polylog \left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2 e^{3}}+\frac {3 a \,b^{2} \mathrm {arccosh}\left (d x +c \right )}{e^{3}}+\frac {3 a \,b^{2} \mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c +1}\, \sqrt {d x +c -1}}{e^{3} \left (d x +c \right )}-\frac {3 a \,b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}}{2 e^{3} \left (d x +c \right )^{2}}-\frac {3 a \,b^{2} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e^{3}}+\frac {3 a^{2} b \left (-\frac {\mathrm {arccosh}\left (d x +c \right )}{2 \left (d x +c \right )^{2}}+\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2 d x +2 c}\right )}{e^{3}}}{d}\) | \(339\) |
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 {acosh}^{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 {acosh}^{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 {acosh}{\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 {acosh}\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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