Optimal. Leaf size=297 \[ -\frac {\cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}+\frac {\cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Chi}\left (\frac {4 c f}{d}+4 f x\right )}{4 a^2 d}+\frac {\log (c+d x)}{4 a^2 d}-\frac {\text {Chi}\left (\frac {4 c f}{d}+4 f x\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{4 a^2 d}+\frac {\text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{2 a^2 d}+\frac {\cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}-\frac {\sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}-\frac {\cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{4 a^2 d}+\frac {\sinh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{4 a^2 d} \]
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Rubi [A]
time = 0.52, antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps
used = 21, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3809, 3384,
3379, 3382, 3393} \begin {gather*} \frac {\text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{2 a^2 d}-\frac {\text {Chi}\left (4 x f+\frac {4 c f}{d}\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{4 a^2 d}-\frac {\text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{2 a^2 d}+\frac {\text {Chi}\left (4 x f+\frac {4 c f}{d}\right ) \cosh \left (4 e-\frac {4 c f}{d}\right )}{4 a^2 d}-\frac {\sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{2 a^2 d}+\frac {\sinh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (4 x f+\frac {4 c f}{d}\right )}{4 a^2 d}+\frac {\cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{2 a^2 d}-\frac {\cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (4 x f+\frac {4 c f}{d}\right )}{4 a^2 d}+\frac {\log (c+d x)}{4 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 3382
Rule 3384
Rule 3393
Rule 3809
Rubi steps
\begin {align*} \int \frac {1}{(c+d x) (a+a \coth (e+f x))^2} \, dx &=\int \left (\frac {1}{4 a^2 (c+d x)}-\frac {\cosh (2 e+2 f x)}{2 a^2 (c+d x)}+\frac {\cosh ^2(2 e+2 f x)}{4 a^2 (c+d x)}+\frac {\sinh (2 e+2 f x)}{2 a^2 (c+d x)}+\frac {\sinh ^2(2 e+2 f x)}{4 a^2 (c+d x)}-\frac {\sinh (4 e+4 f x)}{4 a^2 (c+d x)}\right ) \, dx\\ &=\frac {\log (c+d x)}{4 a^2 d}+\frac {\int \frac {\cosh ^2(2 e+2 f x)}{c+d x} \, dx}{4 a^2}+\frac {\int \frac {\sinh ^2(2 e+2 f x)}{c+d x} \, dx}{4 a^2}-\frac {\int \frac {\sinh (4 e+4 f x)}{c+d x} \, dx}{4 a^2}-\frac {\int \frac {\cosh (2 e+2 f x)}{c+d x} \, dx}{2 a^2}+\frac {\int \frac {\sinh (2 e+2 f x)}{c+d x} \, dx}{2 a^2}\\ &=\frac {\log (c+d x)}{4 a^2 d}-\frac {\int \left (\frac {1}{2 (c+d x)}-\frac {\cosh (4 e+4 f x)}{2 (c+d x)}\right ) \, dx}{4 a^2}+\frac {\int \left (\frac {1}{2 (c+d x)}+\frac {\cosh (4 e+4 f x)}{2 (c+d x)}\right ) \, dx}{4 a^2}-\frac {\cosh \left (4 e-\frac {4 c f}{d}\right ) \int \frac {\sinh \left (\frac {4 c f}{d}+4 f x\right )}{c+d x} \, dx}{4 a^2}-\frac {\cosh \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\cosh \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{2 a^2}+\frac {\cosh \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sinh \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{2 a^2}-\frac {\sinh \left (4 e-\frac {4 c f}{d}\right ) \int \frac {\cosh \left (\frac {4 c f}{d}+4 f x\right )}{c+d x} \, dx}{4 a^2}+\frac {\sinh \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\cosh \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{2 a^2}-\frac {\sinh \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sinh \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{2 a^2}\\ &=-\frac {\cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}+\frac {\log (c+d x)}{4 a^2 d}-\frac {\text {Chi}\left (\frac {4 c f}{d}+4 f x\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{4 a^2 d}+\frac {\text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{2 a^2 d}+\frac {\cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}-\frac {\sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}-\frac {\cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{4 a^2 d}+2 \frac {\int \frac {\cosh (4 e+4 f x)}{c+d x} \, dx}{8 a^2}\\ &=-\frac {\cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}+\frac {\log (c+d x)}{4 a^2 d}-\frac {\text {Chi}\left (\frac {4 c f}{d}+4 f x\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{4 a^2 d}+\frac {\text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{2 a^2 d}+\frac {\cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}-\frac {\sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}-\frac {\cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{4 a^2 d}+2 \left (\frac {\cosh \left (4 e-\frac {4 c f}{d}\right ) \int \frac {\cosh \left (\frac {4 c f}{d}+4 f x\right )}{c+d x} \, dx}{8 a^2}+\frac {\sinh \left (4 e-\frac {4 c f}{d}\right ) \int \frac {\sinh \left (\frac {4 c f}{d}+4 f x\right )}{c+d x} \, dx}{8 a^2}\right )\\ &=-\frac {\cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}+\frac {\log (c+d x)}{4 a^2 d}-\frac {\text {Chi}\left (\frac {4 c f}{d}+4 f x\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{4 a^2 d}+\frac {\text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{2 a^2 d}+\frac {\cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}-\frac {\sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}-\frac {\cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{4 a^2 d}+2 \left (\frac {\cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Chi}\left (\frac {4 c f}{d}+4 f x\right )}{8 a^2 d}+\frac {\sinh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{8 a^2 d}\right )\\ \end {align*}
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Mathematica [A]
time = 0.35, size = 199, normalized size = 0.67 \begin {gather*} \frac {\left (\cosh \left (2 e-\frac {2 c f}{d}\right )-\sinh \left (2 e-\frac {2 c f}{d}\right )\right ) \left (-2 \text {Chi}\left (\frac {2 f (c+d x)}{d}\right )+\cosh \left (2 e-\frac {2 c f}{d}\right ) \log (f (c+d x))+\text {Chi}\left (\frac {4 f (c+d x)}{d}\right ) \left (\cosh \left (2 e-\frac {2 c f}{d}\right )-\sinh \left (2 e-\frac {2 c f}{d}\right )\right )+\log (f (c+d x)) \sinh \left (2 e-\frac {2 c f}{d}\right )+2 \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )-\cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {4 f (c+d x)}{d}\right )+\sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {4 f (c+d x)}{d}\right )\right )}{4 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 10.40, size = 106, normalized size = 0.36
method | result | size |
risch | \(\frac {\ln \left (d x +c \right )}{4 a^{2} d}-\frac {{\mathrm e}^{\frac {4 c f -4 d e}{d}} \expIntegral \left (1, 4 f x +4 e +\frac {4 c f -4 d e}{d}\right )}{4 a^{2} d}+\frac {{\mathrm e}^{\frac {2 c f -2 d e}{d}} \expIntegral \left (1, 2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{2 a^{2} d}\) | \(106\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.74, size = 83, normalized size = 0.28 \begin {gather*} -\frac {e^{\left (\frac {4 \, c f}{d} - 4 \, e\right )} E_{1}\left (\frac {4 \, {\left (d x + c\right )} f}{d}\right )}{4 \, a^{2} d} + \frac {e^{\left (\frac {2 \, c f}{d} - 2 \, e\right )} E_{1}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{2 \, a^{2} d} + \frac {\log \left (d x + c\right )}{4 \, a^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 160, normalized size = 0.54 \begin {gather*} -\frac {2 \, {\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 ) - {\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 ) - 2 \, {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (-\frac {2 \, {\left (c f - d \cosh \left (1\right ) - d \sinh \left (1\right )\right )}}{d}\right ) + {\rm Ei}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (-\frac {4 \, {\left (c f - d \cosh \left (1\right ) - d \sinh \left (1\right )\right )}}{d}\right ) - \log \left (d x + c\right )}{4 \, a^{2} d} \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 \coth ^{2}{\left (e + f x \right )} + 2 c \coth {\left (e + f x \right )} + c + d x \coth ^{2}{\left (e + f x \right )} + 2 d x \coth {\left (e + f x \right )} + d x}\, dx}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 77, normalized size = 0.26 \begin {gather*} -\frac {{\left (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 )} - {\rm Ei}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac {4 \, c f}{d}\right )} - e^{\left (4 \, e\right )} \log \left (d x + c\right )\right )} e^{\left (-4 \, e\right )}}{4 \, a^{2} 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 = -0.00 \begin {gather*} \int \frac {1}{{\left (a+a\,\mathrm {coth}\left (e+f\,x\right )\right )}^2\,\left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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