Optimal. Leaf size=288 \[ \frac {\left (a^2-b^2\right ) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {\left (a^2-b^2\right ) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {x \left (a^2-b^2\right ) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b^3 d}+\frac {x \left (a^2-b^2\right ) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b^3 d}-\frac {x^2 \left (a^2-b^2\right )}{2 b^3}+\frac {a \sinh (c+d x)}{b^2 d^2}-\frac {a x \cosh (c+d x)}{b^2 d}-\frac {\sinh (c+d x) \cosh (c+d x)}{4 b d^2}+\frac {x \sinh ^2(c+d x)}{2 b d}+\frac {x}{4 b d} \]
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Rubi [A] time = 0.34, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {5566, 3296, 2637, 5372, 2635, 8, 5562, 2190, 2279, 2391} \[ \frac {\left (a^2-b^2\right ) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {\left (a^2-b^2\right ) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}\right )}{b^3 d^2}+\frac {x \left (a^2-b^2\right ) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b^3 d}+\frac {x \left (a^2-b^2\right ) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b^3 d}-\frac {x^2 \left (a^2-b^2\right )}{2 b^3}+\frac {a \sinh (c+d x)}{b^2 d^2}-\frac {a x \cosh (c+d x)}{b^2 d}-\frac {\sinh (c+d x) \cosh (c+d x)}{4 b d^2}+\frac {x \sinh ^2(c+d x)}{2 b d}+\frac {x}{4 b d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2190
Rule 2279
Rule 2391
Rule 2635
Rule 2637
Rule 3296
Rule 5372
Rule 5562
Rule 5566
Rubi steps
\begin {align*} \int \frac {x \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx &=-\frac {a \int x \sinh (c+d x) \, dx}{b^2}+\frac {\int x \cosh (c+d x) \sinh (c+d x) \, dx}{b}+\frac {\left (a^2-b^2\right ) \int \frac {x \sinh (c+d x)}{a+b \cosh (c+d x)} \, dx}{b^2}\\ &=-\frac {\left (a^2-b^2\right ) x^2}{2 b^3}-\frac {a x \cosh (c+d x)}{b^2 d}+\frac {x \sinh ^2(c+d x)}{2 b d}+\frac {\left (a^2-b^2\right ) \int \frac {e^{c+d x} x}{a-\sqrt {a^2-b^2}+b e^{c+d x}} \, dx}{b^2}+\frac {\left (a^2-b^2\right ) \int \frac {e^{c+d x} x}{a+\sqrt {a^2-b^2}+b e^{c+d x}} \, dx}{b^2}+\frac {a \int \cosh (c+d x) \, dx}{b^2 d}-\frac {\int \sinh ^2(c+d x) \, dx}{2 b d}\\ &=-\frac {\left (a^2-b^2\right ) x^2}{2 b^3}-\frac {a x \cosh (c+d x)}{b^2 d}+\frac {\left (a^2-b^2\right ) x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {\left (a^2-b^2\right ) x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {a \sinh (c+d x)}{b^2 d^2}-\frac {\cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {x \sinh ^2(c+d x)}{2 b d}+\frac {\int 1 \, dx}{4 b d}-\frac {\left (a^2-b^2\right ) \int \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{b^3 d}-\frac {\left (a^2-b^2\right ) \int \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{b^3 d}\\ &=\frac {x}{4 b d}-\frac {\left (a^2-b^2\right ) x^2}{2 b^3}-\frac {a x \cosh (c+d x)}{b^2 d}+\frac {\left (a^2-b^2\right ) x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {\left (a^2-b^2\right ) x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {a \sinh (c+d x)}{b^2 d^2}-\frac {\cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {x \sinh ^2(c+d x)}{2 b d}-\frac {\left (a^2-b^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a-\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^3 d^2}-\frac {\left (a^2-b^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^3 d^2}\\ &=\frac {x}{4 b d}-\frac {\left (a^2-b^2\right ) x^2}{2 b^3}-\frac {a x \cosh (c+d x)}{b^2 d}+\frac {\left (a^2-b^2\right ) x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {\left (a^2-b^2\right ) x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {\left (a^2-b^2\right ) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {\left (a^2-b^2\right ) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {a \sinh (c+d x)}{b^2 d^2}-\frac {\cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {x \sinh ^2(c+d x)}{2 b d}\\ \end {align*}
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Mathematica [A] time = 2.96, size = 414, normalized size = 1.44 \[ \frac {4 \left (a^2-b^2\right ) \left (2 \text {Li}_2\left (\frac {b (\cosh (c+d x)+\sinh (c+d x))}{\sqrt {a^2-b^2}-a}\right )+2 \text {Li}_2\left (-\frac {b (\cosh (c+d x)+\sinh (c+d x))}{a+\sqrt {a^2-b^2}}\right )+2 (c+d x) \log \left (\frac {b (\sinh (c+d x)+\cosh (c+d x))}{a-\sqrt {a^2-b^2}}+1\right )+2 (c+d x) \log \left (\frac {b (\sinh (c+d x)+\cosh (c+d x))}{\sqrt {a^2-b^2}+a}+1\right )+\frac {4 a c \sqrt {-\left (a^2-b^2\right )^2} \tan ^{-1}\left (\frac {a+b \sinh (c+d x)+b \cosh (c+d x)}{\sqrt {b^2-a^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+\frac {4 a c \sqrt {-\left (a^2-b^2\right )^2} \tanh ^{-1}\left (\frac {a+b \sinh (c+d x)+b \cosh (c+d x)}{\sqrt {a^2-b^2}}\right )}{\left (b^2-a^2\right )^{3/2}}-2 c \log (2 (\sinh (c+d x)+\cosh (c+d x)) (a+b \cosh (c+d x)))-(c+d x)^2+2 c (c+d x)\right )+8 a b \sinh (c+d x)-8 a b d x \cosh (c+d x)-b^2 \sinh (2 (c+d x))+2 b^2 d x \cosh (2 (c+d x))}{8 b^3 d^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 1196, normalized size = 4.15 \[ \text {result too large to display} \]
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 \[ \int \frac {x \sinh \left (d x + c\right )^{3}}{b \cosh \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.30, size = 860, normalized size = 2.99 \[ -\frac {\ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right ) x}{d b}-\frac {\ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right ) c}{d^{2} b}+\frac {c \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}+b \right )}{d^{2} b}-\frac {2 c \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} b}+\frac {2 c x}{d b}-\frac {\ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right ) x}{d b}-\frac {\ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right ) c}{d^{2} b}-\frac {a^{2} c^{2}}{d^{2} b^{3}}-\frac {a \left (d x +1\right ) {\mathrm e}^{-d x -c}}{2 b^{2} d^{2}}-\frac {a \left (d x -1\right ) {\mathrm e}^{d x +c}}{2 b^{2} d^{2}}-\frac {x^{2} a^{2}}{2 b^{3}}+\frac {\left (2 d x +1\right ) {\mathrm e}^{-2 d x -2 c}}{16 b \,d^{2}}-\frac {\dilog \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right )}{d^{2} b}-\frac {\dilog \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right )}{d^{2} b}+\frac {a^{2} \dilog \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right )}{d^{2} b^{3}}+\frac {a^{2} \dilog \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right )}{d^{2} b^{3}}-\frac {2 a^{2} c x}{d \,b^{3}}+\frac {\ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right ) a^{2} x}{d \,b^{3}}+\frac {\ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right ) a^{2} c}{d^{2} b^{3}}+\frac {\ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right ) a^{2} x}{d \,b^{3}}+\frac {\ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right ) a^{2} c}{d^{2} b^{3}}+\frac {2 c \,a^{2} \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} b^{3}}-\frac {c \,a^{2} \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}+b \right )}{d^{2} b^{3}}+\frac {c^{2}}{d^{2} b}+\frac {\left (2 d x -1\right ) {\mathrm e}^{2 d x +2 c}}{16 b \,d^{2}}+\frac {x^{2}}{2 b} \]
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 \[ \frac {{\left (8 \, {\left (a^{2} d^{2} e^{\left (2 \, c\right )} - b^{2} d^{2} e^{\left (2 \, c\right )}\right )} x^{2} + {\left (2 \, b^{2} d x e^{\left (4 \, c\right )} - b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 8 \, {\left (a b d x e^{\left (3 \, c\right )} - a b e^{\left (3 \, c\right )}\right )} e^{\left (d x\right )} - 8 \, {\left (a b d x e^{c} + a b e^{c}\right )} e^{\left (-d x\right )} + {\left (2 \, b^{2} d x + b^{2}\right )} e^{\left (-2 \, d x\right )}\right )} e^{\left (-2 \, c\right )}}{16 \, b^{3} d^{2}} - \frac {1}{8} \, \int \frac {16 \, {\left ({\left (a^{3} e^{c} - a b^{2} e^{c}\right )} x e^{\left (d x\right )} + {\left (a^{2} b - b^{3}\right )} x\right )}}{b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b^{3} e^{\left (d x + c\right )} + b^{4}}\,{d x} \]
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 \[ \int \frac {x\,{\mathrm {sinh}\left (c+d\,x\right )}^3}{a+b\,\mathrm {cosh}\left (c+d\,x\right )} \,d x \]
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 \[ \int \frac {x \sinh ^{3}{\left (c + d x \right )}}{a + b \cosh {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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