Optimal. Leaf size=182 \[ -\frac {a^3 d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^5}-\frac {a^3 d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^5}+\frac {a^3 \cosh (c+d x)}{b^4 (a+b x)}+\frac {3 a^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^4}+\frac {3 a^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^4}-\frac {2 a \sinh (c+d x)}{b^3 d}-\frac {\cosh (c+d x)}{b^2 d^2}+\frac {x \sinh (c+d x)}{b^2 d} \]
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Rubi [A] time = 0.42, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {6742, 2637, 3296, 2638, 3297, 3303, 3298, 3301} \[ -\frac {a^3 d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^5}+\frac {3 a^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^4}+\frac {3 a^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^4}-\frac {a^3 d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^5}+\frac {a^3 \cosh (c+d x)}{b^4 (a+b x)}-\frac {2 a \sinh (c+d x)}{b^3 d}-\frac {\cosh (c+d x)}{b^2 d^2}+\frac {x \sinh (c+d x)}{b^2 d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rule 3296
Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 6742
Rubi steps
\begin {align*} \int \frac {x^3 \cosh (c+d x)}{(a+b x)^2} \, dx &=\int \left (-\frac {2 a \cosh (c+d x)}{b^3}+\frac {x \cosh (c+d x)}{b^2}-\frac {a^3 \cosh (c+d x)}{b^3 (a+b x)^2}+\frac {3 a^2 \cosh (c+d x)}{b^3 (a+b x)}\right ) \, dx\\ &=-\frac {(2 a) \int \cosh (c+d x) \, dx}{b^3}+\frac {\left (3 a^2\right ) \int \frac {\cosh (c+d x)}{a+b x} \, dx}{b^3}-\frac {a^3 \int \frac {\cosh (c+d x)}{(a+b x)^2} \, dx}{b^3}+\frac {\int x \cosh (c+d x) \, dx}{b^2}\\ &=\frac {a^3 \cosh (c+d x)}{b^4 (a+b x)}-\frac {2 a \sinh (c+d x)}{b^3 d}+\frac {x \sinh (c+d x)}{b^2 d}-\frac {\int \sinh (c+d x) \, dx}{b^2 d}-\frac {\left (a^3 d\right ) \int \frac {\sinh (c+d x)}{a+b x} \, dx}{b^4}+\frac {\left (3 a^2 \cosh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cosh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^3}+\frac {\left (3 a^2 \sinh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sinh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^3}\\ &=-\frac {\cosh (c+d x)}{b^2 d^2}+\frac {a^3 \cosh (c+d x)}{b^4 (a+b x)}+\frac {3 a^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^4}-\frac {2 a \sinh (c+d x)}{b^3 d}+\frac {x \sinh (c+d x)}{b^2 d}+\frac {3 a^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^4}-\frac {\left (a^3 d \cosh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sinh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^4}-\frac {\left (a^3 d \sinh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cosh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^4}\\ &=-\frac {\cosh (c+d x)}{b^2 d^2}+\frac {a^3 \cosh (c+d x)}{b^4 (a+b x)}+\frac {3 a^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^4}-\frac {a^3 d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^5}-\frac {2 a \sinh (c+d x)}{b^3 d}+\frac {x \sinh (c+d x)}{b^2 d}-\frac {a^3 d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^5}+\frac {3 a^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^4}\\ \end {align*}
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Mathematica [A] time = 0.96, size = 156, normalized size = 0.86 \[ \frac {a^2 \text {Chi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (3 b \cosh \left (c-\frac {a d}{b}\right )-a d \sinh \left (c-\frac {a d}{b}\right )\right )-a^2 \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (a d \cosh \left (c-\frac {a d}{b}\right )-3 b \sinh \left (c-\frac {a d}{b}\right )\right )+\frac {b \left (\left (a^3 d^2-a b^2-b^3 x\right ) \cosh (c+d x)+b d \left (-2 a^2-a b x+b^2 x^2\right ) \sinh (c+d x)\right )}{d^2 (a+b x)}}{b^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 333, normalized size = 1.83 \[ \frac {2 \, {\left (a^{3} b d^{2} - b^{4} x - a b^{3}\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{4} d^{3} - 3 \, a^{3} b d^{2} + {\left (a^{3} b d^{3} - 3 \, a^{2} b^{2} d^{2}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\left (a^{4} d^{3} + 3 \, a^{3} b d^{2} + {\left (a^{3} b d^{3} + 3 \, a^{2} b^{2} d^{2}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) + 2 \, {\left (b^{4} d x^{2} - a b^{3} d x - 2 \, a^{2} b^{2} d\right )} \sinh \left (d x + c\right ) + {\left ({\left (a^{4} d^{3} - 3 \, a^{3} b d^{2} + {\left (a^{3} b d^{3} - 3 \, a^{2} b^{2} d^{2}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{4} d^{3} + 3 \, a^{3} b d^{2} + {\left (a^{3} b d^{3} + 3 \, a^{2} b^{2} d^{2}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{2 \, {\left (b^{6} d^{2} x + a b^{5} d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 1991, normalized size = 10.94 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 325, normalized size = 1.79 \[ -\frac {{\mathrm e}^{-d x -c} x}{2 d \,b^{2}}+\frac {{\mathrm e}^{-d x -c} a}{d \,b^{3}}-\frac {{\mathrm e}^{-d x -c}}{2 d^{2} b^{2}}-\frac {d \,{\mathrm e}^{\frac {d a -c b}{b}} \Ei \left (1, d x +c +\frac {d a -c b}{b}\right ) a^{3}}{2 b^{5}}-\frac {3 \,{\mathrm e}^{\frac {d a -c b}{b}} \Ei \left (1, d x +c +\frac {d a -c b}{b}\right ) a^{2}}{2 b^{4}}+\frac {d \,{\mathrm e}^{-d x -c} a^{3}}{2 b^{4} \left (b d x +d a \right )}+\frac {{\mathrm e}^{d x +c} x}{2 d \,b^{2}}-\frac {3 \,{\mathrm e}^{-\frac {d a -c b}{b}} \Ei \left (1, -d x -c -\frac {d a -c b}{b}\right ) a^{2}}{2 b^{4}}-\frac {a \,{\mathrm e}^{d x +c}}{d \,b^{3}}-\frac {{\mathrm e}^{d x +c}}{2 d^{2} b^{2}}+\frac {d \,{\mathrm e}^{d x +c} a^{3}}{2 b^{5} \left (\frac {a d}{b}+d x \right )}+\frac {d \,{\mathrm e}^{-\frac {d a -c b}{b}} \Ei \left (1, -d x -c -\frac {d a -c b}{b}\right ) a^{3}}{2 b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 312, normalized size = 1.71 \[ -\frac {1}{4} \, {\left (2 \, a^{3} {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b^{5}} - \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b^{5}}\right )} + \frac {6 \, a^{2} {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b} + \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{b^{3} d} - \frac {4 \, a {\left (\frac {{\left (d x e^{c} - e^{c}\right )} e^{\left (d x\right )}}{d^{2}} + \frac {{\left (d x + 1\right )} e^{\left (-d x - c\right )}}{d^{2}}\right )}}{b^{3}} + \frac {\frac {{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} e^{\left (d x\right )}}{d^{3}} + \frac {{\left (d^{2} x^{2} + 2 \, d x + 2\right )} e^{\left (-d x - c\right )}}{d^{3}}}{b^{2}} + \frac {12 \, a^{2} \cosh \left (d x + c\right ) \log \left (b x + a\right )}{b^{4} d}\right )} d + \frac {1}{2} \, {\left (\frac {2 \, a^{3}}{b^{5} x + a b^{4}} + \frac {6 \, a^{2} \log \left (b x + a\right )}{b^{4}} + \frac {b x^{2} - 4 \, a x}{b^{3}}\right )} \cosh \left (d x + c\right ) \]
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 \[ \int \frac {x^3\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (a+b\,x\right )}^2} \,d x \]
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 \[ \int \frac {x^{3} \cosh {\left (c + d x \right )}}{\left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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