Optimal. Leaf size=164 \[ -\frac {3 b^2 \log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^3}+\frac {3 b \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}-\frac {3 b \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac {3 b^3 \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c+d x)}\right )}{2 d e^3} \]
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Rubi [A] time = 0.37, 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 = {5866, 12, 5662, 5724, 5660, 3718, 2190, 2279, 2391} \[ -\frac {3 b^3 \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(c+d x)}\right )}{2 d e^3}-\frac {3 b^2 \log \left (e^{2 \cosh ^{-1}(c+d x)}+1\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^3}+\frac {3 b \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}+\frac {3 b \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2} \]
Warning: Unable to verify antiderivative.
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Rule 12
Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 5660
Rule 5662
Rule 5724
Rule 5866
Rubi steps
\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^3}{(c e+d e x)^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^3}{e^3 x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 1.11, size = 266, normalized size = 1.62 \[ \frac {-\frac {a^3}{(c+d x)^2}+\frac {3 a^2 b \left (\sqrt {\frac {c+d x-1}{c+d x+1}} \left (c^2+2 c d x+c+d x (d x+1)\right )-\cosh ^{-1}(c+d x)\right )}{(c+d x)^2}+6 a b^2 \left (-\log (c+d x)-\frac {\cosh ^{-1}(c+d x)^2}{2 (c+d x)^2}+\frac {\sqrt {\frac {c+d x-1}{c+d x+1}} (c+d x+1) \cosh ^{-1}(c+d x)}{c+d x}\right )+3 b^3 \left (\text {Li}_2\left (-e^{-2 \cosh ^{-1}(c+d x)}\right )+\cosh ^{-1}(c+d x) \left (\frac {\sqrt {\frac {c+d x-1}{c+d x+1}} (c+d x+1) \cosh ^{-1}(c+d x)}{c+d x}-\cosh ^{-1}(c+d x)-2 \log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right )\right )\right )-\frac {b^3 \cosh ^{-1}(c+d x)^3}{(c+d x)^2}}{2 d e^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \operatorname {arcosh}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname {arcosh}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname {arcosh}\left (d x + c\right ) + a^{3}}{d^{3} e^{3} x^{3} + 3 \, c d^{2} e^{3} x^{2} + 3 \, c^{2} d e^{3} x + c^{3} e^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.29, size = 375, normalized size = 2.29 \[ -\frac {a^{3}}{2 d \,e^{3} \left (d x +c \right )^{2}}+\frac {3 b^{3} \mathrm {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c +1}\, \sqrt {d x +c -1}}{2 d \,e^{3} \left (d x +c \right )}+\frac {3 b^{3} \mathrm {arccosh}\left (d x +c \right )^{2}}{2 d \,e^{3}}-\frac {b^{3} \mathrm {arccosh}\left (d x +c \right )^{3}}{2 d \,e^{3} \left (d x +c \right )^{2}}-\frac {3 b^{3} \mathrm {arccosh}\left (d x +c \right ) \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{d \,e^{3}}-\frac {3 b^{3} \polylog \left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2 d \,e^{3}}+\frac {3 a \,b^{2} \mathrm {arccosh}\left (d x +c \right )}{d \,e^{3}}+\frac {3 a \,b^{2} \mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c +1}\, \sqrt {d x +c -1}}{d \,e^{3} \left (d x +c \right )}-\frac {3 a \,b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}}{2 d \,e^{3} \left (d x +c \right )^{2}}-\frac {3 a \,b^{2} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{d \,e^{3}}-\frac {3 a^{2} b \,\mathrm {arccosh}\left (d x +c \right )}{2 d \,e^{3} \left (d x +c \right )^{2}}+\frac {3 a^{2} b \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2 d \,e^{3} \left (d x +c \right )} \]
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 \[ 3 \, {\left (\frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d \operatorname {arcosh}\left (d x + c\right )}{d^{3} e^{3} x + c d^{2} e^{3}} - \frac {\log \left (d x + c\right )}{d e^{3}}\right )} a b^{2} - \frac {1}{2} \, {\left (\frac {\log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{3}}{d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}} - 2 \, \int \frac {3 \, {\left (d^{2} x^{2} + 2 \, c d x + \sqrt {d x + c + 1} {\left (d x + c\right )} \sqrt {d x + c - 1} + c^{2} - 1\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{2}}{2 \, {\left (d^{5} e^{3} x^{5} + 5 \, c d^{4} e^{3} x^{4} + c^{5} e^{3} - c^{3} e^{3} + {\left (10 \, c^{2} d^{3} e^{3} - d^{3} e^{3}\right )} x^{3} + {\left (10 \, c^{3} d^{2} e^{3} - 3 \, c d^{2} e^{3}\right )} x^{2} + {\left (d^{4} e^{3} x^{4} + 4 \, c d^{3} e^{3} x^{3} + c^{4} e^{3} - c^{2} e^{3} + {\left (6 \, c^{2} d^{2} e^{3} - d^{2} e^{3}\right )} x^{2} + 2 \, {\left (2 \, c^{3} d e^{3} - c d e^{3}\right )} x\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + {\left (5 \, c^{4} d e^{3} - 3 \, c^{2} d e^{3}\right )} x\right )}}\,{d x}\right )} b^{3} + \frac {3}{2} \, a^{2} b {\left (\frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d}{d^{3} e^{3} x + c d^{2} e^{3}} - \frac {\operatorname {arcosh}\left (d x + c\right )}{d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}}\right )} - \frac {3 \, a b^{2} \operatorname {arcosh}\left (d x + c\right )^{2}}{2 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} - \frac {a^{3}}{2 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^3}{{\left (c\,e+d\,e\,x\right )}^3} \,d x \]
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 \[ \frac {\int \frac {a^{3}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {3 a b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {3 a^{2} b \operatorname {acosh}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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