Optimal. Leaf size=211 \[ \frac {f^2 \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{a d^3}-\frac {f^2 \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{a d^3}-\frac {f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{a d^3}+\frac {f^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{a d^3}+\frac {f}{d^2 (c+d x) (a \coth (e+f x)+a)}-\frac {f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a \coth (e+f x)+a)} \]
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Rubi [A] time = 0.30, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3725, 3724, 3303, 3298, 3301} \[ \frac {f^2 \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{a d^3}-\frac {f^2 \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{a d^3}-\frac {f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{a d^3}+\frac {f^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{a d^3}+\frac {f}{d^2 (c+d x) (a \coth (e+f x)+a)}-\frac {f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a \coth (e+f x)+a)} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 3724
Rule 3725
Rubi steps
\begin {align*} \int \frac {1}{(c+d x)^3 (a+a \coth (e+f x))} \, dx &=-\frac {f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+a \coth (e+f x))}-\frac {f \int \frac {1}{(c+d x)^2 (a+a \coth (e+f x))} \, dx}{d}\\ &=-\frac {f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+a \coth (e+f x))}+\frac {f}{d^2 (c+d x) (a+a \coth (e+f x))}+\frac {\left (i f^2\right ) \int \frac {\sin \left (2 \left (i e+\frac {\pi }{2}\right )+2 i f x\right )}{c+d x} \, dx}{a d^2}+\frac {f^2 \int \frac {\cos \left (2 \left (i e+\frac {\pi }{2}\right )+2 i f x\right )}{c+d x} \, dx}{a d^2}\\ &=-\frac {f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+a \coth (e+f x))}+\frac {f}{d^2 (c+d x) (a+a \coth (e+f x))}-\frac {\left (f^2 \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^2}+\frac {\left (f^2 \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^2}+\frac {\left (f^2 \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^2}-\frac {\left (f^2 \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^2}\\ &=-\frac {f}{2 a d^2 (c+d x)}-\frac {f^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}-\frac {1}{2 d (c+d x)^2 (a+a \coth (e+f x))}+\frac {f}{d^2 (c+d x) (a+a \coth (e+f x))}+\frac {f^2 \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{a d^3}+\frac {f^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}-\frac {f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}\\ \end {align*}
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Mathematica [A] time = 1.12, size = 265, normalized size = 1.26 \[ -\frac {\text {csch}(e+f x) \left (\sinh \left (\frac {c f}{d}\right )+\cosh \left (\frac {c f}{d}\right )\right ) \left (4 f^2 (c+d x)^2 \text {Chi}\left (\frac {2 f (c+d x)}{d}\right ) \left (\cosh \left (e-\frac {f (c+d x)}{d}\right )-\sinh \left (e-\frac {f (c+d x)}{d}\right )\right )+4 f^2 (c+d x)^2 \text {Shi}\left (\frac {2 f (c+d x)}{d}\right ) \left (\sinh \left (e-\frac {f (c+d x)}{d}\right )-\cosh \left (e-\frac {f (c+d x)}{d}\right )\right )+d \left (d \sinh \left (f \left (x-\frac {c}{d}\right )+e\right )+d \sinh \left (f \left (\frac {c}{d}+x\right )+e\right )-2 c f \sinh \left (f \left (\frac {c}{d}+x\right )+e\right )-2 d f x \sinh \left (f \left (\frac {c}{d}+x\right )+e\right )+d \cosh \left (f \left (x-\frac {c}{d}\right )+e\right )+(2 c f+2 d f x-d) \cosh \left (f \left (\frac {c}{d}+x\right )+e\right )\right )\right )}{4 a d^3 (c+d x)^2 (\coth (e+f x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 342, normalized size = 1.62 \[ -\frac {2 \, {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (f x + e\right ) \sinh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + {\left (d^{2} f x + c d f + 2 \, {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )\right )} \cosh \left (f x + e\right ) - {\left (d^{2} f x + c d f - 2 \, {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) - 2 \, {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) - d^{2}\right )} \sinh \left (f x + e\right )}{2 \, {\left ({\left (a d^{5} x^{2} + 2 \, a c d^{4} x + a c^{2} d^{3}\right )} \cosh \left (f x + e\right ) + {\left (a d^{5} x^{2} + 2 \, a c d^{4} x + a c^{2} d^{3}\right )} \sinh \left (f x + e\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 179, normalized size = 0.85 \[ -\frac {4 \, d^{2} f^{2} x^{2} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac {2 \, c f}{d}\right )} + 8 \, c d f^{2} x {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac {2 \, c f}{d}\right )} + 4 \, c^{2} f^{2} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac {2 \, c f}{d}\right )} + 2 \, d^{2} f x e^{\left (-2 \, f x\right )} + 2 \, c d f e^{\left (-2 \, f x\right )} - d^{2} e^{\left (-2 \, f x\right )} + d^{2} e^{\left (2 \, e\right )}}{4 \, {\left (a d^{5} x^{2} e^{\left (2 \, e\right )} + 2 \, a c d^{4} x e^{\left (2 \, e\right )} + a c^{2} d^{3} e^{\left (2 \, e\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.06, size = 210, normalized size = 1.00 \[ -\frac {1}{4 d a \left (d x +c \right )^{2}}-\frac {f^{3} {\mathrm e}^{-2 f x -2 e} x}{2 a d \left (d^{2} f^{2} x^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{3} {\mathrm e}^{-2 f x -2 e} c}{2 a \,d^{2} \left (d^{2} f^{2} x^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{2} {\mathrm e}^{-2 f x -2 e}}{4 a d \left (d^{2} f^{2} x^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{2} {\mathrm e}^{\frac {2 c f -2 d e}{d}} \Ei \left (1, 2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{a \,d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 68, normalized size = 0.32 \[ -\frac {1}{4 \, {\left (a d^{3} x^{2} + 2 \, a c d^{2} x + a c^{2} d\right )}} + \frac {e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{3}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{2 \, {\left (d x + c\right )}^{2} a d} \]
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 \[ \int \frac {1}{\left (a+a\,\mathrm {coth}\left (e+f\,x\right )\right )\,{\left (c+d\,x\right )}^3} \,d x \]
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 \[ \frac {\int \frac {1}{c^{3} \coth {\left (e + f x \right )} + c^{3} + 3 c^{2} d x \coth {\left (e + f x \right )} + 3 c^{2} d x + 3 c d^{2} x^{2} \coth {\left (e + f x \right )} + 3 c d^{2} x^{2} + d^{3} x^{3} \coth {\left (e + f x \right )} + d^{3} x^{3}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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