Integrand size = 18, antiderivative size = 123 \[ \int \frac {a+b \sinh (e+f x)}{(c+d x)^3} \, dx=-\frac {a}{2 d (c+d x)^2}-\frac {b f \cosh (e+f x)}{2 d^2 (c+d x)}+\frac {b f^2 \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{2 d^3}-\frac {b \sinh (e+f x)}{2 d (c+d x)^2}+\frac {b f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{2 d^3} \]
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Time = 0.15 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3398, 3378, 3384, 3379, 3382} \[ \int \frac {a+b \sinh (e+f x)}{(c+d x)^3} \, dx=-\frac {a}{2 d (c+d x)^2}+\frac {b f^2 \text {Chi}\left (x f+\frac {c f}{d}\right ) \sinh \left (e-\frac {c f}{d}\right )}{2 d^3}+\frac {b f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{2 d^3}-\frac {b f \cosh (e+f x)}{2 d^2 (c+d x)}-\frac {b \sinh (e+f x)}{2 d (c+d x)^2} \]
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Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 3398
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a}{(c+d x)^3}+\frac {b \sinh (e+f x)}{(c+d x)^3}\right ) \, dx \\ & = -\frac {a}{2 d (c+d x)^2}+b \int \frac {\sinh (e+f x)}{(c+d x)^3} \, dx \\ & = -\frac {a}{2 d (c+d x)^2}-\frac {b \sinh (e+f x)}{2 d (c+d x)^2}+\frac {(b f) \int \frac {\cosh (e+f x)}{(c+d x)^2} \, dx}{2 d} \\ & = -\frac {a}{2 d (c+d x)^2}-\frac {b f \cosh (e+f x)}{2 d^2 (c+d x)}-\frac {b \sinh (e+f x)}{2 d (c+d x)^2}+\frac {\left (b f^2\right ) \int \frac {\sinh (e+f x)}{c+d x} \, dx}{2 d^2} \\ & = -\frac {a}{2 d (c+d x)^2}-\frac {b f \cosh (e+f x)}{2 d^2 (c+d x)}-\frac {b \sinh (e+f x)}{2 d (c+d x)^2}+\frac {\left (b f^2 \cosh \left (e-\frac {c f}{d}\right )\right ) \int \frac {\sinh \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx}{2 d^2}+\frac {\left (b f^2 \sinh \left (e-\frac {c f}{d}\right )\right ) \int \frac {\cosh \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx}{2 d^2} \\ & = -\frac {a}{2 d (c+d x)^2}-\frac {b f \cosh (e+f x)}{2 d^2 (c+d x)}+\frac {b f^2 \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{2 d^3}-\frac {b \sinh (e+f x)}{2 d (c+d x)^2}+\frac {b f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{2 d^3} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.77 \[ \int \frac {a+b \sinh (e+f x)}{(c+d x)^3} \, dx=\frac {b f^2 \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right ) \sinh \left (e-\frac {c f}{d}\right )-\frac {d (b f (c+d x) \cosh (e+f x)+d (a+b \sinh (e+f x)))}{(c+d x)^2}+b f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )}{2 d^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(295\) vs. \(2(115)=230\).
Time = 1.08 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.41
method | result | size |
risch | \(-\frac {a}{2 d \left (d x +c \right )^{2}}-\frac {f^{3} b \,{\mathrm e}^{-f x -e} x}{4 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{3} b \,{\mathrm e}^{-f x -e} c}{4 d^{2} \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{2} b \,{\mathrm e}^{-f x -e}}{4 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{2} b \,{\mathrm e}^{\frac {c f -d e}{d}} \operatorname {Ei}_{1}\left (f x +e +\frac {c f -d e}{d}\right )}{4 d^{3}}-\frac {f^{2} b \,{\mathrm e}^{f x +e}}{4 d^{3} \left (\frac {c f}{d}+f x \right )^{2}}-\frac {f^{2} b \,{\mathrm e}^{f x +e}}{4 d^{3} \left (\frac {c f}{d}+f x \right )}-\frac {f^{2} b \,{\mathrm e}^{-\frac {c f -d e}{d}} \operatorname {Ei}_{1}\left (-f x -e -\frac {c f -d e}{d}\right )}{4 d^{3}}\) | \(296\) |
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Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (115) = 230\).
Time = 0.25 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.23 \[ \int \frac {a+b \sinh (e+f x)}{(c+d x)^3} \, dx=-\frac {2 \, b d^{2} \sinh \left (f x + e\right ) + 2 \, a d^{2} + 2 \, {\left (b d^{2} f x + b c d f\right )} \cosh \left (f x + e\right ) - {\left ({\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + b c^{2} f^{2}\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) - {\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + b c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \cosh \left (-\frac {d e - c f}{d}\right ) + {\left ({\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + b c^{2} f^{2}\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) + {\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + b c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \sinh \left (-\frac {d e - c f}{d}\right )}{4 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]
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Timed out. \[ \int \frac {a+b \sinh (e+f x)}{(c+d x)^3} \, dx=\text {Timed out} \]
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Time = 0.23 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.80 \[ \int \frac {a+b \sinh (e+f x)}{(c+d x)^3} \, dx=\frac {1}{2} \, b {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{3}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d} - \frac {e^{\left (e - \frac {c f}{d}\right )} E_{3}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d}\right )} - \frac {a}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 319 vs. \(2 (115) = 230\).
Time = 0.27 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.59 \[ \int \frac {a+b \sinh (e+f x)}{(c+d x)^3} \, dx=\frac {b d^{2} f^{2} x^{2} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (e - \frac {c f}{d}\right )} - b d^{2} f^{2} x^{2} {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (-e + \frac {c f}{d}\right )} + 2 \, b c d f^{2} x {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (e - \frac {c f}{d}\right )} - 2 \, b c d f^{2} x {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (-e + \frac {c f}{d}\right )} + b c^{2} f^{2} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (e - \frac {c f}{d}\right )} - b c^{2} f^{2} {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (-e + \frac {c f}{d}\right )} - b d^{2} f x e^{\left (f x + e\right )} - b d^{2} f x e^{\left (-f x - e\right )} - b c d f e^{\left (f x + e\right )} - b c d f e^{\left (-f x - e\right )} - b d^{2} e^{\left (f x + e\right )} + b d^{2} e^{\left (-f x - e\right )} - 2 \, a d^{2}}{4 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]
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Timed out. \[ \int \frac {a+b \sinh (e+f x)}{(c+d x)^3} \, dx=\int \frac {a+b\,\mathrm {sinh}\left (e+f\,x\right )}{{\left (c+d\,x\right )}^3} \,d x \]
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