Optimal. Leaf size=123 \[ -\frac {a}{2 d (c+d x)^2}-\frac {b f \cosh (e+f x)}{2 d^2 (c+d x)}+\frac {b f^2 \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{2 d^3}-\frac {b \sinh (e+f x)}{2 d (c+d x)^2}+\frac {b f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{2 d^3} \]
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Rubi [A]
time = 0.16, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3398, 3378,
3384, 3379, 3382} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 3398
Rubi steps
\begin {align*} \int \frac {a+b \sinh (e+f x)}{(c+d x)^3} \, dx &=\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*}
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Mathematica [A]
time = 0.39, size = 95, normalized size = 0.77 \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(295\) vs.
\(2(115)=230\).
time = 0.45, size = 296, normalized size = 2.41
method | result | size |
risch | \(-\frac {a}{2 d \left (d x +c \right )^{2}}-\frac {f^{3} b \,{\mathrm e}^{-f x -e} x}{4 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{3} b \,{\mathrm e}^{-f x -e} c}{4 d^{2} \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{2} b \,{\mathrm e}^{-f x -e}}{4 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{2} b \,{\mathrm e}^{\frac {c f -d e}{d}} \expIntegral \left (1, f x +e +\frac {c f -d e}{d}\right )}{4 d^{3}}-\frac {b \,f^{2} {\mathrm e}^{f x +e}}{4 d^{3} \left (\frac {c f}{d}+f x \right )^{2}}-\frac {b \,f^{2} {\mathrm e}^{f x +e}}{4 d^{3} \left (\frac {c f}{d}+f x \right )}-\frac {b \,f^{2} {\mathrm e}^{-\frac {c f -d e}{d}} \expIntegral \left (1, -f x -e -\frac {c f -d e}{d}\right )}{4 d^{3}}\) | \(296\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 101, normalized size = 0.82 \begin {gather*} \frac {1}{2} \, b {\left (\frac {e^{\left (\frac {c f}{d} - e\right )} E_{3}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d} - \frac {e^{\left (-\frac {c f}{d} + e\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 293 vs.
\(2 (119) = 238\).
time = 0.36, size = 293, normalized size = 2.38 \begin {gather*} -\frac {2 \, b d^{2} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + 2 \, a d^{2} + 2 \, {\left (b d^{2} f x + b c d f\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\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 {c f - d \cosh \left (1\right ) - d \sinh \left (1\right )}{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 {c f - d \cosh \left (1\right ) - d \sinh \left (1\right )}{d}\right )}{4 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 319 vs.
\(2 (115) = 230\).
time = 0.43, size = 319, normalized size = 2.59 \begin {gather*} \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 )}} \end {gather*}
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 \begin {gather*} \int \frac {a+b\,\mathrm {sinh}\left (e+f\,x\right )}{{\left (c+d\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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