Integrand size = 20, antiderivative size = 242 \[ \int \frac {(a+b \sinh (e+f x))^2}{(c+d x)^3} \, dx=-\frac {a^2}{2 d (c+d x)^2}-\frac {a b f \cosh (e+f x)}{d^2 (c+d x)}+\frac {b^2 f^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{d^3}+\frac {a b f^2 \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^3}-\frac {a b \sinh (e+f x)}{d (c+d x)^2}-\frac {b^2 f \cosh (e+f x) \sinh (e+f x)}{d^2 (c+d x)}-\frac {b^2 \sinh ^2(e+f x)}{2 d (c+d x)^2}+\frac {a b f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^3}+\frac {b^2 f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{d^3} \]
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Time = 0.30 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3398, 3378, 3384, 3379, 3382, 3395, 31, 3393} \[ \int \frac {(a+b \sinh (e+f x))^2}{(c+d x)^3} \, dx=-\frac {a^2}{2 d (c+d x)^2}+\frac {a b f^2 \text {Chi}\left (x f+\frac {c f}{d}\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^3}+\frac {a b f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d^3}-\frac {a b f \cosh (e+f x)}{d^2 (c+d x)}-\frac {a b \sinh (e+f x)}{d (c+d x)^2}+\frac {b^2 f^2 \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{d^3}+\frac {b^2 f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{d^3}-\frac {b^2 f \sinh (e+f x) \cosh (e+f x)}{d^2 (c+d x)}-\frac {b^2 \sinh ^2(e+f x)}{2 d (c+d x)^2} \]
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Rule 31
Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 3393
Rule 3395
Rule 3398
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2}{(c+d x)^3}+\frac {2 a b \sinh (e+f x)}{(c+d x)^3}+\frac {b^2 \sinh ^2(e+f x)}{(c+d x)^3}\right ) \, dx \\ & = -\frac {a^2}{2 d (c+d x)^2}+(2 a b) \int \frac {\sinh (e+f x)}{(c+d x)^3} \, dx+b^2 \int \frac {\sinh ^2(e+f x)}{(c+d x)^3} \, dx \\ & = -\frac {a^2}{2 d (c+d x)^2}-\frac {a b \sinh (e+f x)}{d (c+d x)^2}-\frac {b^2 f \cosh (e+f x) \sinh (e+f x)}{d^2 (c+d x)}-\frac {b^2 \sinh ^2(e+f x)}{2 d (c+d x)^2}+\frac {(a b f) \int \frac {\cosh (e+f x)}{(c+d x)^2} \, dx}{d}+\frac {\left (b^2 f^2\right ) \int \frac {1}{c+d x} \, dx}{d^2}+\frac {\left (2 b^2 f^2\right ) \int \frac {\sinh ^2(e+f x)}{c+d x} \, dx}{d^2} \\ & = -\frac {a^2}{2 d (c+d x)^2}-\frac {a b f \cosh (e+f x)}{d^2 (c+d x)}+\frac {b^2 f^2 \log (c+d x)}{d^3}-\frac {a b \sinh (e+f x)}{d (c+d x)^2}-\frac {b^2 f \cosh (e+f x) \sinh (e+f x)}{d^2 (c+d x)}-\frac {b^2 \sinh ^2(e+f x)}{2 d (c+d x)^2}+\frac {\left (a b f^2\right ) \int \frac {\sinh (e+f x)}{c+d x} \, dx}{d^2}-\frac {\left (2 b^2 f^2\right ) \int \left (\frac {1}{2 (c+d x)}-\frac {\cosh (2 e+2 f x)}{2 (c+d x)}\right ) \, dx}{d^2} \\ & = -\frac {a^2}{2 d (c+d x)^2}-\frac {a b f \cosh (e+f x)}{d^2 (c+d x)}-\frac {a b \sinh (e+f x)}{d (c+d x)^2}-\frac {b^2 f \cosh (e+f x) \sinh (e+f x)}{d^2 (c+d x)}-\frac {b^2 \sinh ^2(e+f x)}{2 d (c+d x)^2}+\frac {\left (b^2 f^2\right ) \int \frac {\cosh (2 e+2 f x)}{c+d x} \, dx}{d^2}+\frac {\left (a 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}{d^2}+\frac {\left (a 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}{d^2} \\ & = -\frac {a^2}{2 d (c+d x)^2}-\frac {a b f \cosh (e+f x)}{d^2 (c+d x)}+\frac {a b f^2 \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^3}-\frac {a b \sinh (e+f x)}{d (c+d x)^2}-\frac {b^2 f \cosh (e+f x) \sinh (e+f x)}{d^2 (c+d x)}-\frac {b^2 \sinh ^2(e+f x)}{2 d (c+d x)^2}+\frac {a b f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^3}+\frac {\left (b^2 f^2 \cosh \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\cosh \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{d^2}+\frac {\left (b^2 f^2 \sinh \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\sinh \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{d^2} \\ & = -\frac {a^2}{2 d (c+d x)^2}-\frac {a b f \cosh (e+f x)}{d^2 (c+d x)}+\frac {b^2 f^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{d^3}+\frac {a b f^2 \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^3}-\frac {a b \sinh (e+f x)}{d (c+d x)^2}-\frac {b^2 f \cosh (e+f x) \sinh (e+f x)}{d^2 (c+d x)}-\frac {b^2 \sinh ^2(e+f x)}{2 d (c+d x)^2}+\frac {a b f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^3}+\frac {b^2 f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{d^3} \\ \end{align*}
Time = 0.61 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.63 \[ \int \frac {(a+b \sinh (e+f x))^2}{(c+d x)^3} \, dx=\frac {-2 a^2 d^2+b^2 d^2-4 a b c d f \cosh (e+f x)-4 a b d^2 f x \cosh (e+f x)-b^2 d^2 \cosh (2 (e+f x))+4 b^2 f^2 (c+d x)^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 f (c+d x)}{d}\right )+4 a b f^2 (c+d x)^2 \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right ) \sinh \left (e-\frac {c f}{d}\right )-4 a b d^2 \sinh (e+f x)-2 b^2 c d f \sinh (2 (e+f x))-2 b^2 d^2 f x \sinh (2 (e+f x))+4 a b c^2 f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )+8 a b c d f^2 x \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )+4 a b d^2 f^2 x^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )+4 b^2 c^2 f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )+8 b^2 c d f^2 x \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )+4 b^2 d^2 f^2 x^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )}{4 d^3 (c+d x)^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(625\) vs. \(2(242)=484\).
Time = 4.36 (sec) , antiderivative size = 626, normalized size of antiderivative = 2.59
method | result | size |
risch | \(-\frac {f^{2} b a \,{\mathrm e}^{f x +e}}{2 d^{3} \left (\frac {c f}{d}+f x \right )^{2}}-\frac {f^{2} b a \,{\mathrm e}^{f x +e}}{2 d^{3} \left (\frac {c f}{d}+f x \right )}-\frac {f^{2} b a \,{\mathrm e}^{-\frac {c f -d e}{d}} \operatorname {Ei}_{1}\left (-f x -e -\frac {c f -d e}{d}\right )}{2 d^{3}}-\frac {a^{2}}{2 d \left (d x +c \right )^{2}}+\frac {b^{2}}{4 \left (d x +c \right )^{2} d}+\frac {f^{3} b^{2} {\mathrm e}^{-2 f x -2 e} x}{4 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{3} b^{2} {\mathrm e}^{-2 f x -2 e} c}{4 d^{2} \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{2} b^{2} {\mathrm e}^{-2 f x -2 e}}{8 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{2} b^{2} {\mathrm e}^{\frac {2 c f -2 d e}{d}} \operatorname {Ei}_{1}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{2 d^{3}}-\frac {f^{2} b^{2} {\mathrm e}^{2 f x +2 e}}{8 d^{3} \left (\frac {c f}{d}+f x \right )^{2}}-\frac {f^{2} b^{2} {\mathrm e}^{2 f x +2 e}}{4 d^{3} \left (\frac {c f}{d}+f x \right )}-\frac {f^{2} b^{2} {\mathrm e}^{-\frac {2 \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (-2 f x -2 e -\frac {2 \left (c f -d e \right )}{d}\right )}{2 d^{3}}-\frac {f^{3} a b \,{\mathrm e}^{-f x -e} x}{2 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{3} a b \,{\mathrm e}^{-f x -e} c}{2 d^{2} \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{2} a b \,{\mathrm e}^{-f x -e}}{2 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{2} a b \,{\mathrm e}^{\frac {c f -d e}{d}} \operatorname {Ei}_{1}\left (f x +e +\frac {c f -d e}{d}\right )}{2 d^{3}}\) | \(626\) |
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Leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (242) = 484\).
Time = 0.25 (sec) , antiderivative size = 590, normalized size of antiderivative = 2.44 \[ \int \frac {(a+b \sinh (e+f x))^2}{(c+d x)^3} \, dx=-\frac {b^{2} d^{2} \cosh \left (f x + e\right )^{2} + b^{2} d^{2} \sinh \left (f x + e\right )^{2} + {\left (2 \, a^{2} - b^{2}\right )} d^{2} + 4 \, {\left (a b d^{2} f x + a b c d f\right )} \cosh \left (f x + e\right ) - 2 \, {\left ({\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2}\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) - {\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \cosh \left (-\frac {d e - c f}{d}\right ) - 2 \, {\left ({\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + {\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right )\right )} \cosh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + 4 \, {\left (a b d^{2} + {\left (b^{2} d^{2} f x + b^{2} c d f\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + 2 \, {\left ({\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2}\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) + {\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \sinh \left (-\frac {d e - c f}{d}\right ) + 2 \, {\left ({\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - {\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right )\right )} \sinh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )}{4 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]
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\[ \int \frac {(a+b \sinh (e+f x))^2}{(c+d x)^3} \, dx=\int \frac {\left (a + b \sinh {\left (e + f x \right )}\right )^{2}}{\left (c + d x\right )^{3}}\, dx \]
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Time = 0.26 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b \sinh (e+f x))^2}{(c+d x)^3} \, dx=\frac {1}{4} \, b^{2} {\left (\frac {1}{d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d} - \frac {e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{3}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d} - \frac {e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} E_{3}\left (-\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d}\right )} + a b {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{3}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d} - \frac {e^{\left (e - \frac {c f}{d}\right )} E_{3}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d}\right )} - \frac {a^{2}}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 678 vs. \(2 (242) = 484\).
Time = 0.28 (sec) , antiderivative size = 678, normalized size of antiderivative = 2.80 \[ \int \frac {(a+b \sinh (e+f x))^2}{(c+d x)^3} \, dx=\frac {4 \, b^{2} d^{2} f^{2} x^{2} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} + 4 \, a 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 )} - 4 \, a 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 )} + 4 \, b^{2} d^{2} f^{2} x^{2} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} + 8 \, b^{2} c d f^{2} x {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} + 8 \, a 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 )} - 8 \, a 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 )} + 8 \, b^{2} c d f^{2} x {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} + 4 \, b^{2} c^{2} f^{2} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} + 4 \, a b c^{2} f^{2} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (e - \frac {c f}{d}\right )} - 4 \, a b c^{2} f^{2} {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (-e + \frac {c f}{d}\right )} + 4 \, b^{2} c^{2} f^{2} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} - 2 \, b^{2} d^{2} f x e^{\left (2 \, f x + 2 \, e\right )} - 4 \, a b d^{2} f x e^{\left (f x + e\right )} - 4 \, a b d^{2} f x e^{\left (-f x - e\right )} + 2 \, b^{2} d^{2} f x e^{\left (-2 \, f x - 2 \, e\right )} - 2 \, b^{2} c d f e^{\left (2 \, f x + 2 \, e\right )} - 4 \, a b c d f e^{\left (f x + e\right )} - 4 \, a b c d f e^{\left (-f x - e\right )} + 2 \, b^{2} c d f e^{\left (-2 \, f x - 2 \, e\right )} - b^{2} d^{2} e^{\left (2 \, f x + 2 \, e\right )} - 4 \, a b d^{2} e^{\left (f x + e\right )} + 4 \, a b d^{2} e^{\left (-f x - e\right )} - b^{2} d^{2} e^{\left (-2 \, f x - 2 \, e\right )} - 4 \, a^{2} d^{2} + 2 \, b^{2} d^{2}}{8 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]
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Timed out. \[ \int \frac {(a+b \sinh (e+f x))^2}{(c+d x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {sinh}\left (e+f\,x\right )\right )}^2}{{\left (c+d\,x\right )}^3} \,d x \]
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