Optimal. Leaf size=180 \[ -\frac {\cosh (c+d x)}{a f (e+f x)}-\frac {i d \cosh \left (2 c-\frac {2 d e}{f}\right ) \text {Chi}\left (\frac {2 d e}{f}+2 d x\right )}{a f^2}+\frac {d \text {Chi}\left (\frac {d e}{f}+d x\right ) \sinh \left (c-\frac {d e}{f}\right )}{a f^2}+\frac {i \sinh (2 c+2 d x)}{2 a f (e+f x)}+\frac {d \cosh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{a f^2}-\frac {i d \sinh \left (2 c-\frac {2 d e}{f}\right ) \text {Shi}\left (\frac {2 d e}{f}+2 d x\right )}{a f^2} \]
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Rubi [A]
time = 0.27, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {5682, 3378,
3384, 3379, 3382, 5556, 12} \begin {gather*} \frac {d \sinh \left (c-\frac {d e}{f}\right ) \text {Chi}\left (\frac {d e}{f}+d x\right )}{a f^2}-\frac {i d \cosh \left (2 c-\frac {2 d e}{f}\right ) \text {Chi}\left (\frac {2 d e}{f}+2 d x\right )}{a f^2}-\frac {i d \sinh \left (2 c-\frac {2 d e}{f}\right ) \text {Shi}\left (\frac {2 d e}{f}+2 d x\right )}{a f^2}+\frac {d \cosh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{a f^2}+\frac {i \sinh (2 c+2 d x)}{2 a f (e+f x)}-\frac {\cosh (c+d x)}{a f (e+f x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 5682
Rubi steps
\begin {align*} \int \frac {\cosh ^3(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx &=-\frac {i \int \frac {\cosh (c+d x) \sinh (c+d x)}{(e+f x)^2} \, dx}{a}+\frac {\int \frac {\cosh (c+d x)}{(e+f x)^2} \, dx}{a}\\ &=-\frac {\cosh (c+d x)}{a f (e+f x)}-\frac {i \int \frac {\sinh (2 c+2 d x)}{2 (e+f x)^2} \, dx}{a}+\frac {d \int \frac {\sinh (c+d x)}{e+f x} \, dx}{a f}\\ &=-\frac {\cosh (c+d x)}{a f (e+f x)}-\frac {i \int \frac {\sinh (2 c+2 d x)}{(e+f x)^2} \, dx}{2 a}+\frac {\left (d \cosh \left (c-\frac {d e}{f}\right )\right ) \int \frac {\sinh \left (\frac {d e}{f}+d x\right )}{e+f x} \, dx}{a f}+\frac {\left (d \sinh \left (c-\frac {d e}{f}\right )\right ) \int \frac {\cosh \left (\frac {d e}{f}+d x\right )}{e+f x} \, dx}{a f}\\ &=-\frac {\cosh (c+d x)}{a f (e+f x)}+\frac {d \text {Chi}\left (\frac {d e}{f}+d x\right ) \sinh \left (c-\frac {d e}{f}\right )}{a f^2}+\frac {i \sinh (2 c+2 d x)}{2 a f (e+f x)}+\frac {d \cosh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{a f^2}-\frac {(i d) \int \frac {\cosh (2 c+2 d x)}{e+f x} \, dx}{a f}\\ &=-\frac {\cosh (c+d x)}{a f (e+f x)}+\frac {d \text {Chi}\left (\frac {d e}{f}+d x\right ) \sinh \left (c-\frac {d e}{f}\right )}{a f^2}+\frac {i \sinh (2 c+2 d x)}{2 a f (e+f x)}+\frac {d \cosh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{a f^2}-\frac {\left (i d \cosh \left (2 c-\frac {2 d e}{f}\right )\right ) \int \frac {\cosh \left (\frac {2 d e}{f}+2 d x\right )}{e+f x} \, dx}{a f}-\frac {\left (i d \sinh \left (2 c-\frac {2 d e}{f}\right )\right ) \int \frac {\sinh \left (\frac {2 d e}{f}+2 d x\right )}{e+f x} \, dx}{a f}\\ &=-\frac {\cosh (c+d x)}{a f (e+f x)}-\frac {i d \cosh \left (2 c-\frac {2 d e}{f}\right ) \text {Chi}\left (\frac {2 d e}{f}+2 d x\right )}{a f^2}+\frac {d \text {Chi}\left (\frac {d e}{f}+d x\right ) \sinh \left (c-\frac {d e}{f}\right )}{a f^2}+\frac {i \sinh (2 c+2 d x)}{2 a f (e+f x)}+\frac {d \cosh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{a f^2}-\frac {i d \sinh \left (2 c-\frac {2 d e}{f}\right ) \text {Shi}\left (\frac {2 d e}{f}+2 d x\right )}{a f^2}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 212, normalized size = 1.18 \begin {gather*} \frac {-2 f \cosh (c+d x)-2 i d (e+f x) \cosh \left (2 c-\frac {2 d e}{f}\right ) \text {Chi}\left (\frac {2 d (e+f x)}{f}\right )+2 d (e+f x) \text {Chi}\left (d \left (\frac {e}{f}+x\right )\right ) \sinh \left (c-\frac {d e}{f}\right )+i f \sinh (2 (c+d x))+2 d e \cosh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (d \left (\frac {e}{f}+x\right )\right )+2 d f x \cosh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (d \left (\frac {e}{f}+x\right )\right )-2 i d e \sinh \left (2 c-\frac {2 d e}{f}\right ) \text {Shi}\left (\frac {2 d (e+f x)}{f}\right )-2 i d f x \sinh \left (2 c-\frac {2 d e}{f}\right ) \text {Shi}\left (\frac {2 d (e+f x)}{f}\right )}{2 a f^2 (e+f x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.47, size = 299, normalized size = 1.66
method | result | size |
risch | \(-\frac {d \,{\mathrm e}^{-d x -c}}{2 a f \left (d x f +d e \right )}+\frac {d \,{\mathrm e}^{-\frac {c f -d e}{f}} \expIntegral \left (1, d x +c -\frac {c f -d e}{f}\right )}{2 a \,f^{2}}-\frac {d \,{\mathrm e}^{d x +c}}{2 a \,f^{2} \left (\frac {d e}{f}+d x \right )}-\frac {d \,{\mathrm e}^{\frac {c f -d e}{f}} \expIntegral \left (1, -d x -c -\frac {-c f +d e}{f}\right )}{2 a \,f^{2}}+\frac {i d \,{\mathrm e}^{2 d x +2 c}}{4 a \,f^{2} \left (\frac {d e}{f}+d x \right )}+\frac {i d \,{\mathrm e}^{\frac {2 c f -2 d e}{f}} \expIntegral \left (1, -2 d x -2 c -\frac {2 \left (-c f +d e \right )}{f}\right )}{2 a \,f^{2}}-\frac {i d \,{\mathrm e}^{-2 d x -2 c}}{4 a f \left (d x f +d e \right )}+\frac {i d \,{\mathrm e}^{-\frac {2 \left (c f -d e \right )}{f}} \expIntegral \left (1, 2 d x +2 c -\frac {2 \left (c f -d e \right )}{f}\right )}{2 a \,f^{2}}\) | \(299\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 257, normalized size = 1.43 \begin {gather*} -\frac {{\left (2 \, {\left (d f x + d e\right )} {\rm Ei}\left (-\frac {d f x + d e}{f}\right ) e^{\left (2 \, d x + 2 \, c - \frac {c f - d e}{f}\right )} + 2 \, {\left (i \, d f x + i \, d e\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + d e\right )}}{f}\right ) e^{\left (2 \, d x + 2 \, c - \frac {2 \, {\left (c f - d e\right )}}{f}\right )} - i \, f e^{\left (4 \, d x + 4 \, c\right )} + 2 \, f e^{\left (3 \, d x + 3 \, c\right )} + 2 \, {\left ({\left (i \, d f x + i \, d e\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + d e\right )}}{f}\right ) e^{\left (\frac {2 \, {\left (c f - d e\right )}}{f}\right )} - {\left (d f x + d e\right )} {\rm Ei}\left (\frac {d f x + d e}{f}\right ) e^{\left (\frac {c f - d e}{f}\right )}\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, f e^{\left (d x + c\right )} + i \, f\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{4 \, {\left (a f^{3} x + a f^{2} e\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] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1080 vs. \(2 (172) = 344\).
time = 0.55, size = 1080, normalized size = 6.00 \begin {gather*} -\frac {{\left (2 i \, {\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} d^{2} {\rm Ei}\left (-\frac {2 \, {\left ({\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} + d e - c f\right )}}{f}\right ) e^{\left (\frac {2 \, {\left (d e - c f\right )}}{f}\right )} + 2 i \, d^{3} e {\rm Ei}\left (-\frac {2 \, {\left ({\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} + d e - c f\right )}}{f}\right ) e^{\left (\frac {2 \, {\left (d e - c f\right )}}{f}\right )} - 2 i \, c d^{2} f {\rm Ei}\left (-\frac {2 \, {\left ({\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} + d e - c f\right )}}{f}\right ) e^{\left (\frac {2 \, {\left (d e - c f\right )}}{f}\right )} + 2 \, {\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} d^{2} {\rm Ei}\left (-\frac {{\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} + d e - c f}{f}\right ) e^{\left (\frac {d e - c f}{f}\right )} + 2 \, d^{3} e {\rm Ei}\left (-\frac {{\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} + d e - c f}{f}\right ) e^{\left (\frac {d e - c f}{f}\right )} - 2 \, c d^{2} f {\rm Ei}\left (-\frac {{\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} + d e - c f}{f}\right ) e^{\left (\frac {d e - c f}{f}\right )} - 2 \, {\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} d^{2} {\rm Ei}\left (\frac {{\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} + d e - c f}{f}\right ) e^{\left (-\frac {d e - c f}{f}\right )} - 2 \, d^{3} e {\rm Ei}\left (\frac {{\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} + d e - c f}{f}\right ) e^{\left (-\frac {d e - c f}{f}\right )} + 2 \, c d^{2} f {\rm Ei}\left (\frac {{\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} + d e - c f}{f}\right ) e^{\left (-\frac {d e - c f}{f}\right )} + 2 i \, {\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} d^{2} {\rm Ei}\left (\frac {2 \, {\left ({\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} + d e - c f\right )}}{f}\right ) e^{\left (-\frac {2 \, {\left (d e - c f\right )}}{f}\right )} + 2 i \, d^{3} e {\rm Ei}\left (\frac {2 \, {\left ({\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} + d e - c f\right )}}{f}\right ) e^{\left (-\frac {2 \, {\left (d e - c f\right )}}{f}\right )} - 2 i \, c d^{2} f {\rm Ei}\left (\frac {2 \, {\left ({\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} + d e - c f\right )}}{f}\right ) e^{\left (-\frac {2 \, {\left (d e - c f\right )}}{f}\right )} - i \, d^{2} f e^{\left (\frac {2 \, {\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )}}{f}\right )} + 2 \, d^{2} f e^{\left (\frac {{\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )}}{f}\right )} + 2 \, d^{2} f e^{\left (-\frac {{\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )}}{f}\right )} + i \, d^{2} f e^{\left (-\frac {2 \, {\left (f x + e\right )} {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )}}{f}\right )}\right )} f^{2}}{4 \, {\left ({\left (f x + e\right )} a {\left (d - \frac {d e}{f x + e} + \frac {c f}{f x + e}\right )} f^{4} + a d e f^{4} - a c f^{5}\right )} d} \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 {{\mathrm {cosh}\left (c+d\,x\right )}^3}{{\left (e+f\,x\right )}^2\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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