Optimal. Leaf size=420 \[ -\frac {1}{4 a^2 d (c+d x)}+\frac {\cosh (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac {\cosh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}+\frac {f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{a^2 d^2}-\frac {f \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Chi}\left (\frac {4 c f}{d}+4 f x\right )}{a^2 d^2}+\frac {f \text {Chi}\left (\frac {4 c f}{d}+4 f x\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{a^2 d^2}-\frac {f \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{a^2 d^2}-\frac {\sinh (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac {\sinh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}+\frac {\sinh (4 e+4 f x)}{4 a^2 d (c+d x)}-\frac {f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a^2 d^2}+\frac {f \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a^2 d^2}+\frac {f \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{a^2 d^2}-\frac {f \sinh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{a^2 d^2} \]
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Rubi [A]
time = 0.53, antiderivative size = 420, normalized size of antiderivative = 1.00, number of steps
used = 24, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {3809, 3378,
3384, 3379, 3382, 3394, 12} \begin {gather*} \frac {f \text {Chi}\left (4 x f+\frac {4 c f}{d}\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{a^2 d^2}-\frac {f \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{a^2 d^2}+\frac {f \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{a^2 d^2}-\frac {f \text {Chi}\left (4 x f+\frac {4 c f}{d}\right ) \cosh \left (4 e-\frac {4 c f}{d}\right )}{a^2 d^2}+\frac {f \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{a^2 d^2}-\frac {f \sinh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (4 x f+\frac {4 c f}{d}\right )}{a^2 d^2}-\frac {f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{a^2 d^2}+\frac {f \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (4 x f+\frac {4 c f}{d}\right )}{a^2 d^2}-\frac {\sinh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}-\frac {\sinh (2 e+2 f x)}{2 a^2 d (c+d x)}+\frac {\sinh (4 e+4 f x)}{4 a^2 d (c+d x)}-\frac {\cosh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}+\frac {\cosh (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac {1}{4 a^2 d (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 3394
Rule 3809
Rubi steps
\begin {align*} \int \frac {1}{(c+d x)^2 (a+a \coth (e+f x))^2} \, dx &=\int \left (\frac {1}{4 a^2 (c+d x)^2}-\frac {\cosh (2 e+2 f x)}{2 a^2 (c+d x)^2}+\frac {\cosh ^2(2 e+2 f x)}{4 a^2 (c+d x)^2}+\frac {\sinh (2 e+2 f x)}{2 a^2 (c+d x)^2}+\frac {\sinh ^2(2 e+2 f x)}{4 a^2 (c+d x)^2}-\frac {\sinh (4 e+4 f x)}{4 a^2 (c+d x)^2}\right ) \, dx\\ &=-\frac {1}{4 a^2 d (c+d x)}+\frac {\int \frac {\cosh ^2(2 e+2 f x)}{(c+d x)^2} \, dx}{4 a^2}+\frac {\int \frac {\sinh ^2(2 e+2 f x)}{(c+d x)^2} \, dx}{4 a^2}-\frac {\int \frac {\sinh (4 e+4 f x)}{(c+d x)^2} \, dx}{4 a^2}-\frac {\int \frac {\cosh (2 e+2 f x)}{(c+d x)^2} \, dx}{2 a^2}+\frac {\int \frac {\sinh (2 e+2 f x)}{(c+d x)^2} \, dx}{2 a^2}\\ &=-\frac {1}{4 a^2 d (c+d x)}+\frac {\cosh (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac {\cosh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}-\frac {\sinh (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac {\sinh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}+\frac {\sinh (4 e+4 f x)}{4 a^2 d (c+d x)}+\frac {(i f) \int -\frac {i \sinh (4 e+4 f x)}{2 (c+d x)} \, dx}{a^2 d}-\frac {(i f) \int \frac {i \sinh (4 e+4 f x)}{2 (c+d x)} \, dx}{a^2 d}+\frac {f \int \frac {\cosh (2 e+2 f x)}{c+d x} \, dx}{a^2 d}-\frac {f \int \frac {\cosh (4 e+4 f x)}{c+d x} \, dx}{a^2 d}-\frac {f \int \frac {\sinh (2 e+2 f x)}{c+d x} \, dx}{a^2 d}\\ &=-\frac {1}{4 a^2 d (c+d x)}+\frac {\cosh (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac {\cosh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}-\frac {\sinh (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac {\sinh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}+\frac {\sinh (4 e+4 f x)}{4 a^2 d (c+d x)}+2 \frac {f \int \frac {\sinh (4 e+4 f x)}{c+d x} \, dx}{2 a^2 d}-\frac {\left (f \cosh \left (4 e-\frac {4 c f}{d}\right )\right ) \int \frac {\cosh \left (\frac {4 c f}{d}+4 f x\right )}{c+d x} \, dx}{a^2 d}+\frac {\left (f \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}{a^2 d}-\frac {\left (f \cosh \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}{a^2 d}-\frac {\left (f \sinh \left (4 e-\frac {4 c f}{d}\right )\right ) \int \frac {\sinh \left (\frac {4 c f}{d}+4 f x\right )}{c+d x} \, dx}{a^2 d}-\frac {\left (f \sinh \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}{a^2 d}+\frac {\left (f \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}{a^2 d}\\ &=-\frac {1}{4 a^2 d (c+d x)}+\frac {\cosh (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac {\cosh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}+\frac {f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{a^2 d^2}-\frac {f \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Chi}\left (\frac {4 c f}{d}+4 f x\right )}{a^2 d^2}-\frac {f \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{a^2 d^2}-\frac {\sinh (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac {\sinh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}+\frac {\sinh (4 e+4 f x)}{4 a^2 d (c+d x)}-\frac {f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a^2 d^2}+\frac {f \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a^2 d^2}-\frac {f \sinh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{a^2 d^2}+2 \left (\frac {\left (f \cosh \left (4 e-\frac {4 c f}{d}\right )\right ) \int \frac {\sinh \left (\frac {4 c f}{d}+4 f x\right )}{c+d x} \, dx}{2 a^2 d}+\frac {\left (f \sinh \left (4 e-\frac {4 c f}{d}\right )\right ) \int \frac {\cosh \left (\frac {4 c f}{d}+4 f x\right )}{c+d x} \, dx}{2 a^2 d}\right )\\ &=-\frac {1}{4 a^2 d (c+d x)}+\frac {\cosh (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac {\cosh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}+\frac {f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{a^2 d^2}-\frac {f \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Chi}\left (\frac {4 c f}{d}+4 f x\right )}{a^2 d^2}-\frac {f \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{a^2 d^2}-\frac {\sinh (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac {\sinh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}+\frac {\sinh (4 e+4 f x)}{4 a^2 d (c+d x)}-\frac {f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a^2 d^2}+\frac {f \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a^2 d^2}-\frac {f \sinh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{a^2 d^2}+2 \left (\frac {f \text {Chi}\left (\frac {4 c f}{d}+4 f x\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{2 a^2 d^2}+\frac {f \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{2 a^2 d^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.99, size = 442, normalized size = 1.05 \begin {gather*} \frac {\left (-\cosh \left (2 \left (e+f \left (-\frac {c}{d}+x\right )\right )\right )+\sinh \left (2 \left (e+f \left (-\frac {c}{d}+x\right )\right )\right )\right ) \left (-2 d \cosh \left (\frac {2 c f}{d}\right )+d \cosh \left (2 \left (e+f \left (-\frac {c}{d}+x\right )\right )\right )+d \cosh \left (2 \left (e+f \left (\frac {c}{d}+x\right )\right )\right )+2 d \sinh \left (\frac {2 c f}{d}\right )-4 f (c+d x) \text {Chi}\left (\frac {2 f (c+d x)}{d}\right ) (\cosh (2 f x)+\sinh (2 f x))+d \sinh \left (2 \left (e+f \left (-\frac {c}{d}+x\right )\right )\right )-d \sinh \left (2 \left (e+f \left (\frac {c}{d}+x\right )\right )\right )+4 f (c+d x) \text {Chi}\left (\frac {4 f (c+d x)}{d}\right ) \left (\cosh \left (2 e-\frac {2 f (c+d x)}{d}\right )-\sinh \left (2 e-\frac {2 f (c+d x)}{d}\right )\right )+4 c f \cosh (2 f x) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )+4 d f x \cosh (2 f x) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )+4 c f \sinh (2 f x) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )+4 d f x \sinh (2 f x) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )-4 c f \cosh \left (2 e-\frac {2 f (c+d x)}{d}\right ) \text {Shi}\left (\frac {4 f (c+d x)}{d}\right )-4 d f x \cosh \left (2 e-\frac {2 f (c+d x)}{d}\right ) \text {Shi}\left (\frac {4 f (c+d x)}{d}\right )+4 c f \sinh \left (2 e-\frac {2 f (c+d x)}{d}\right ) \text {Shi}\left (\frac {4 f (c+d x)}{d}\right )+4 d f x \sinh \left (2 e-\frac {2 f (c+d x)}{d}\right ) \text {Shi}\left (\frac {4 f (c+d x)}{d}\right )\right )}{4 a^2 d^2 (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 9.35, size = 164, normalized size = 0.39
method | result | size |
risch | \(-\frac {1}{4 a^{2} d \left (d x +c \right )}-\frac {f \,{\mathrm e}^{-4 f x -4 e}}{4 a^{2} d \left (d x f +c f \right )}+\frac {f \,{\mathrm e}^{\frac {4 c f -4 d e}{d}} \expIntegral \left (1, 4 f x +4 e +\frac {4 c f -4 d e}{d}\right )}{a^{2} d^{2}}+\frac {f \,{\mathrm e}^{-2 f x -2 e}}{2 a^{2} d \left (d x f +c f \right )}-\frac {f \,{\mathrm e}^{\frac {2 c f -2 d e}{d}} \expIntegral \left (1, 2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{a^{2} d^{2}}\) | \(164\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.25, size = 102, normalized size = 0.24 \begin {gather*} -\frac {1}{4 \, {\left (a^{2} d^{2} x + a^{2} c d\right )}} - \frac {e^{\left (\frac {4 \, c f}{d} - 4 \, e\right )} E_{2}\left (\frac {4 \, {\left (d x + c\right )} f}{d}\right )}{4 \, {\left (d x + c\right )} a^{2} d} + \frac {e^{\left (\frac {2 \, c f}{d} - 2 \, e\right )} E_{2}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{2 \, {\left (d x + c\right )} a^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 713, normalized size = 1.70 \begin {gather*} \frac {2 \, {\left (d f x + c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} \cosh \left (-\frac {2 \, {\left (c f - d \cosh \left (1\right ) - d \sinh \left (1\right )\right )}}{d}\right ) - 2 \, {\left (d f x + c f\right )} {\rm Ei}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} \cosh \left (-\frac {4 \, {\left (c f - d \cosh \left (1\right ) - d \sinh \left (1\right )\right )}}{d}\right ) - d \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} + {\left (2 \, {\left (d f x + c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (-\frac {2 \, {\left (c f - d \cosh \left (1\right ) - d \sinh \left (1\right )\right )}}{d}\right ) - 2 \, {\left (d f x + c f\right )} {\rm Ei}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (-\frac {4 \, {\left (c f - d \cosh \left (1\right ) - d \sinh \left (1\right )\right )}}{d}\right ) - d\right )} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} + 4 \, {\left ({\left (d f x + c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) \cosh \left (-\frac {2 \, {\left (c f - d \cosh \left (1\right ) - d \sinh \left (1\right )\right )}}{d}\right ) - {\left (d f x + c f\right )} {\rm Ei}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) \cosh \left (-\frac {4 \, {\left (c f - d \cosh \left (1\right ) - d \sinh \left (1\right )\right )}}{d}\right )\right )} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - 2 \, {\left ({\left (d f x + c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} + 2 \, {\left (d f x + c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + {\left (d f x + c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2}\right )} \sinh \left (-\frac {2 \, {\left (c f - d \cosh \left (1\right ) - d \sinh \left (1\right )\right )}}{d}\right ) + 2 \, {\left ({\left (d f x + c f\right )} {\rm Ei}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} + 2 \, {\left (d f x + c f\right )} {\rm Ei}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + {\left (d f x + c f\right )} {\rm Ei}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2}\right )} \sinh \left (-\frac {4 \, {\left (c f - d \cosh \left (1\right ) - d \sinh \left (1\right )\right )}}{d}\right ) + d}{2 \, {\left ({\left (a^{2} d^{3} x + a^{2} c d^{2}\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} + 2 \, {\left (a^{2} d^{3} x + a^{2} c d^{2}\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + {\left (a^{2} d^{3} x + a^{2} c d^{2}\right )} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{c^{2} \coth ^{2}{\left (e + f x \right )} + 2 c^{2} \coth {\left (e + f x \right )} + c^{2} + 2 c d x \coth ^{2}{\left (e + f x \right )} + 4 c d x \coth {\left (e + f x \right )} + 2 c d x + d^{2} x^{2} \coth ^{2}{\left (e + f x \right )} + 2 d^{2} x^{2} \coth {\left (e + f x \right )} + d^{2} x^{2}}\, dx}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 585, normalized size = 1.39 \begin {gather*} \frac {{\left (4 \, {\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} f^{2} {\rm Ei}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )} - 4 \, d e f^{2} {\rm Ei}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )} + 4 \, c f^{3} {\rm Ei}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )} - 4 \, {\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} f^{2} {\rm Ei}\left (-\frac {4 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (-\frac {4 \, {\left (d e - c f\right )}}{d}\right )} + 4 \, d e f^{2} {\rm Ei}\left (-\frac {4 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (-\frac {4 \, {\left (d e - c f\right )}}{d}\right )} - 4 \, c f^{3} {\rm Ei}\left (-\frac {4 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (-\frac {4 \, {\left (d e - c f\right )}}{d}\right )} + 2 \, d f^{2} e^{\left (-\frac {2 \, {\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )}}{d}\right )} - d f^{2} e^{\left (-\frac {4 \, {\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )}}{d}\right )} - d f^{2}\right )} d^{2}}{4 \, {\left ({\left (d x + c\right )} a^{2} d^{4} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - a^{2} d^{5} e + a^{2} c d^{4} f\right )} f} \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 = -0.00 \begin {gather*} \int \frac {1}{{\left (a+a\,\mathrm {coth}\left (e+f\,x\right )\right )}^2\,{\left (c+d\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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