Integrand size = 15, antiderivative size = 178 \[ \int \frac {x \cosh (c+d x)}{(a+b x)^3} \, dx=\frac {a \cosh (c+d x)}{2 b^2 (a+b x)^2}-\frac {\cosh (c+d x)}{b^2 (a+b x)}-\frac {a d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{2 b^4}+\frac {d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^3}+\frac {a d \sinh (c+d x)}{2 b^3 (a+b x)}+\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^3}-\frac {a d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{2 b^4} \]
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Time = 0.26 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6874, 3378, 3384, 3379, 3382} \[ \int \frac {x \cosh (c+d x)}{(a+b x)^3} \, dx=-\frac {a d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{2 b^4}-\frac {a d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{2 b^4}+\frac {d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {a d \sinh (c+d x)}{2 b^3 (a+b x)}-\frac {\cosh (c+d x)}{b^2 (a+b x)}+\frac {a \cosh (c+d x)}{2 b^2 (a+b x)^2} \]
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Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a \cosh (c+d x)}{b (a+b x)^3}+\frac {\cosh (c+d x)}{b (a+b x)^2}\right ) \, dx \\ & = \frac {\int \frac {\cosh (c+d x)}{(a+b x)^2} \, dx}{b}-\frac {a \int \frac {\cosh (c+d x)}{(a+b x)^3} \, dx}{b} \\ & = \frac {a \cosh (c+d x)}{2 b^2 (a+b x)^2}-\frac {\cosh (c+d x)}{b^2 (a+b x)}+\frac {d \int \frac {\sinh (c+d x)}{a+b x} \, dx}{b^2}-\frac {(a d) \int \frac {\sinh (c+d x)}{(a+b x)^2} \, dx}{2 b^2} \\ & = \frac {a \cosh (c+d x)}{2 b^2 (a+b x)^2}-\frac {\cosh (c+d x)}{b^2 (a+b x)}+\frac {a d \sinh (c+d x)}{2 b^3 (a+b x)}-\frac {\left (a d^2\right ) \int \frac {\cosh (c+d x)}{a+b x} \, dx}{2 b^3}+\frac {\left (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^2}+\frac {\left (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^2} \\ & = \frac {a \cosh (c+d x)}{2 b^2 (a+b x)^2}-\frac {\cosh (c+d x)}{b^2 (a+b x)}+\frac {d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^3}+\frac {a d \sinh (c+d x)}{2 b^3 (a+b x)}+\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^3}-\frac {\left (a d^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}{2 b^3}-\frac {\left (a d^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}{2 b^3} \\ & = \frac {a \cosh (c+d x)}{2 b^2 (a+b x)^2}-\frac {\cosh (c+d x)}{b^2 (a+b x)}-\frac {a d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{2 b^4}+\frac {d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^3}+\frac {a d \sinh (c+d x)}{2 b^3 (a+b x)}+\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^3}-\frac {a d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{2 b^4} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.89 \[ \int \frac {x \cosh (c+d x)}{(a+b x)^3} \, dx=-\frac {b \cosh (d x) (b (a+2 b x) \cosh (c)-a d (a+b x) \sinh (c))-b (a d (a+b x) \cosh (c)-b (a+2 b x) \sinh (c)) \sinh (d x)+d (a+b x)^2 \left (\text {Chi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (a d \cosh \left (c-\frac {a d}{b}\right )-2 b \sinh \left (c-\frac {a d}{b}\right )\right )+\left (-2 b \cosh \left (c-\frac {a d}{b}\right )+a d \sinh \left (c-\frac {a d}{b}\right )\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )\right )}{2 b^4 (a+b x)^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(434\) vs. \(2(175)=350\).
Time = 0.26 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.44
method | result | size |
risch | \(-\frac {d^{3} {\mathrm e}^{-d x -c} a x}{4 b^{2} \left (x^{2} d^{2} b^{2}+2 a b \,d^{2} x +a^{2} d^{2}\right )}-\frac {d^{3} {\mathrm e}^{-d x -c} a^{2}}{4 b^{3} \left (x^{2} d^{2} b^{2}+2 a b \,d^{2} x +a^{2} d^{2}\right )}-\frac {d^{2} {\mathrm e}^{-d x -c} x}{2 b \left (x^{2} d^{2} b^{2}+2 a b \,d^{2} x +a^{2} d^{2}\right )}-\frac {d^{2} {\mathrm e}^{-d x -c} a}{4 b^{2} \left (x^{2} d^{2} b^{2}+2 a b \,d^{2} x +a^{2} d^{2}\right )}+\frac {d^{2} {\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right ) a}{4 b^{4}}+\frac {d \,{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right )}{2 b^{3}}+\frac {d^{2} {\mathrm e}^{d x +c} a}{4 b^{4} \left (\frac {d a}{b}+d x \right )^{2}}+\frac {d^{2} {\mathrm e}^{d x +c} a}{4 b^{4} \left (\frac {d a}{b}+d x \right )}+\frac {d^{2} {\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right ) a}{4 b^{4}}-\frac {d \,{\mathrm e}^{d x +c}}{2 b^{3} \left (\frac {d a}{b}+d x \right )}-\frac {d \,{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right )}{2 b^{3}}\) | \(435\) |
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Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (175) = 350\).
Time = 0.25 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.10 \[ \int \frac {x \cosh (c+d x)}{(a+b x)^3} \, dx=-\frac {2 \, {\left (2 \, b^{3} x + a b^{2}\right )} \cosh \left (d x + c\right ) + {\left ({\left (a^{3} d^{2} - 2 \, a^{2} b d + {\left (a b^{2} d^{2} - 2 \, b^{3} d\right )} x^{2} + 2 \, {\left (a^{2} b d^{2} - 2 \, a b^{2} d\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{3} d^{2} + 2 \, a^{2} b d + {\left (a b^{2} d^{2} + 2 \, b^{3} d\right )} x^{2} + 2 \, {\left (a^{2} b d^{2} + 2 \, a b^{2} d\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) - 2 \, {\left (a b^{2} d x + a^{2} b d\right )} \sinh \left (d x + c\right ) - {\left ({\left (a^{3} d^{2} - 2 \, a^{2} b d + {\left (a b^{2} d^{2} - 2 \, b^{3} d\right )} x^{2} + 2 \, {\left (a^{2} b d^{2} - 2 \, a b^{2} d\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\left (a^{3} d^{2} + 2 \, a^{2} b d + {\left (a b^{2} d^{2} + 2 \, b^{3} d\right )} x^{2} + 2 \, {\left (a^{2} b d^{2} + 2 \, a b^{2} d\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{4 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} \]
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Timed out. \[ \int \frac {x \cosh (c+d x)}{(a+b x)^3} \, dx=\text {Timed out} \]
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\[ \int \frac {x \cosh (c+d x)}{(a+b x)^3} \, dx=\int { \frac {x \cosh \left (d x + c\right )}{{\left (b x + a\right )}^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 529 vs. \(2 (175) = 350\).
Time = 0.28 (sec) , antiderivative size = 529, normalized size of antiderivative = 2.97 \[ \int \frac {x \cosh (c+d x)}{(a+b x)^3} \, dx=-\frac {a b^{2} d^{2} x^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + a b^{2} d^{2} x^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 2 \, a^{2} b d^{2} x {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} - 2 \, b^{3} d x^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 2 \, a^{2} b d^{2} x {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 2 \, b^{3} d x^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + a^{3} d^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} - 4 \, a b^{2} d x {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + a^{3} d^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 4 \, a b^{2} d x {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} - a b^{2} d x e^{\left (d x + c\right )} + a b^{2} d x e^{\left (-d x - c\right )} - 2 \, a^{2} b d {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 2 \, a^{2} b d {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} - a^{2} b d e^{\left (d x + c\right )} + 2 \, b^{3} x e^{\left (d x + c\right )} + a^{2} b d e^{\left (-d x - c\right )} + 2 \, b^{3} x e^{\left (-d x - c\right )} + a b^{2} e^{\left (d x + c\right )} + a b^{2} e^{\left (-d x - c\right )}}{4 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} \]
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Timed out. \[ \int \frac {x \cosh (c+d x)}{(a+b x)^3} \, dx=\int \frac {x\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (a+b\,x\right )}^3} \,d x \]
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