Optimal. Leaf size=159 \[ \frac {f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^2}-\frac {1}{d (c+d x) (a+a \coth (e+f x))}-\frac {f \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{a d^2}-\frac {f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^2}+\frac {f \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^2} \]
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Rubi [A]
time = 0.17, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3805, 3384,
3379, 3382} \begin {gather*} -\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 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 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 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 d^2}-\frac {1}{d (c+d x) (a \coth (e+f x)+a)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 3382
Rule 3384
Rule 3805
Rubi steps
\begin {align*} \int \frac {1}{(c+d x)^2 (a+a \coth (e+f x))} \, dx &=-\frac {1}{d (c+d x) (a+a \coth (e+f x))}-\frac {(i f) \int \frac {\sin \left (2 \left (i e+\frac {\pi }{2}\right )+2 i f x\right )}{c+d x} \, dx}{a d}-\frac {f \int \frac {\cos \left (2 \left (i e+\frac {\pi }{2}\right )+2 i f x\right )}{c+d x} \, dx}{a d}\\ &=-\frac {1}{d (c+d x) (a+a \coth (e+f x))}+\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 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 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 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 d}\\ &=\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 d^2}-\frac {1}{d (c+d x) (a+a \coth (e+f x))}-\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 d^2}-\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 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 d^2}\\ \end {align*}
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Mathematica [A]
time = 0.55, size = 206, normalized size = 1.30 \begin {gather*} -\frac {\text {csch}(e+f x) \left (\cosh \left (\frac {c f}{d}\right )+\sinh \left (\frac {c f}{d}\right )\right ) \left (d \left (\cosh \left (e+f \left (-\frac {c}{d}+x\right )\right )-\cosh \left (e+f \left (\frac {c}{d}+x\right )\right )+\sinh \left (e+f \left (-\frac {c}{d}+x\right )\right )+\sinh \left (e+f \left (\frac {c}{d}+x\right )\right )\right )+2 f (c+d x) \text {Chi}\left (\frac {2 f (c+d x)}{d}\right ) \left (-\cosh \left (e-\frac {f (c+d x)}{d}\right )+\sinh \left (e-\frac {f (c+d x)}{d}\right )\right )+2 f (c+d x) \left (\cosh \left (e-\frac {f (c+d x)}{d}\right )-\sinh \left (e-\frac {f (c+d x)}{d}\right )\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )\right )}{2 a d^2 (c+d x) (1+\coth (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 10.34, size = 91, normalized size = 0.57
method | result | size |
risch | \(-\frac {1}{2 d \left (d x +c \right ) a}+\frac {f \,{\mathrm e}^{-2 f x -2 e}}{2 a d \left (d x f +c f \right )}-\frac {f \,{\mathrm e}^{\frac {2 c f -2 d e}{d}} \expIntegral \left (1, 2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{a \,d^{2}}\) | \(91\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 57, normalized size = 0.36 \begin {gather*} -\frac {1}{2 \, {\left (a d^{2} x + a c d\right )}} + \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 d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 247, normalized size = 1.55 \begin {gather*} \frac {{\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 ({\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 ) - d\right )} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - {\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 ) + {\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 )\right )} \sinh \left (-\frac {2 \, {\left (c f - d \cosh \left (1\right ) - d \sinh \left (1\right )\right )}}{d}\right )}{{\left (a d^{3} x + a c d^{2}\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + {\left (a d^{3} x + a c d^{2}\right )} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\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 {\left (e + f x \right )} + c^{2} + 2 c d x \coth {\left (e + f x \right )} + 2 c d x + d^{2} x^{2} \coth {\left (e + f x \right )} + d^{2} x^{2}}\, dx}{a} \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. 321 vs.
\(2 (159) = 318\).
time = 0.44, size = 321, normalized size = 2.02 \begin {gather*} \frac {{\left (2 \, {\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 )} - 2 \, 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 )} + 2 \, 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 )} + 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}\right )} d^{2}}{2 \, {\left ({\left (d x + c\right )} a d^{4} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - a d^{5} e + a 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 = -0.01 \begin {gather*} \int \frac {1}{\left (a+a\,\mathrm {coth}\left (e+f\,x\right )\right )\,{\left (c+d\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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