Optimal. Leaf size=34 \[ \text {Int}\left (\frac {\text {csch}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))},x\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\text {csch}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\text {csch}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx &=\int \frac {\text {csch}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx\\ \end {align*}
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Mathematica [A] time = 133.44, size = 0, normalized size = 0.00 \[ \int \frac {\text {csch}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.54, size = 0, normalized size = 0.00 \[ \frac {{\left (i \, a d f x + i \, a d e + {\left (a d f x + a d e\right )} e^{\left (3 \, d x + 3 \, c\right )} + {\left (-i \, a d f x - i \, a d e\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (a d f x + a d e\right )} e^{\left (d x + c\right )}\right )} {\rm integral}\left (\frac {{\left (-2 i \, d f x - 2 i \, d e - 2 i \, f\right )} e^{\left (2 \, d x + 2 \, c\right )} - 2 \, {\left (d f x + d e + f\right )} e^{\left (d x + c\right )} + 4 i \, f}{i \, a d f^{2} x^{2} + 2 i \, a d e f x + i \, a d e^{2} + {\left (a d f^{2} x^{2} + 2 \, a d e f x + a d e^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )} + {\left (-i \, a d f^{2} x^{2} - 2 i \, a d e f x - i \, a d e^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (a d f^{2} x^{2} + 2 \, a d e f x + a d e^{2}\right )} e^{\left (d x + c\right )}}, x\right ) - 2 i \, e^{\left (2 \, d x + 2 \, c\right )} - 2 \, e^{\left (d x + c\right )} + 4 i}{i \, a d f x + i \, a d e + {\left (a d f x + a d e\right )} e^{\left (3 \, d x + 3 \, c\right )} + {\left (-i \, a d f x - i \, a d e\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (a d f x + a d e\right )} e^{\left (d x + c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {csch}\left (d x +c \right )^{2}}{\left (f x +e \right ) \left (a +i a \sinh \left (d x +c \right )\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ -4 i \, f \int \frac {1}{-2 i \, a d f^{2} x^{2} - 4 i \, a d e f x - 2 i \, a d e^{2} + 2 \, {\left (a d f^{2} x^{2} e^{c} + 2 \, a d e f x e^{c} + a d e^{2} e^{c}\right )} e^{\left (d x\right )}}\,{d x} - \frac {4 \, {\left (i \, e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (d x + c\right )} - 2 i\right )}}{2 i \, a d f x + 2 i \, a d e + 2 \, {\left (a d f x e^{\left (3 \, c\right )} + a d e e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + {\left (-2 i \, a d f x e^{\left (2 \, c\right )} - 2 i \, a d e e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 2 \, {\left (a d f x e^{c} + a d e e^{c}\right )} e^{\left (d x\right )}} - 4 \, \int -\frac {i \, d f x + i \, d e + f}{4 \, {\left (a d f^{2} x^{2} + 2 \, a d e f x + a d e^{2} - {\left (a d f^{2} x^{2} e^{c} + 2 \, a d e f x e^{c} + a d e^{2} e^{c}\right )} e^{\left (d x\right )}\right )}}\,{d x} - 4 \, \int \frac {i \, d f x + i \, d e - f}{4 \, {\left (a d f^{2} x^{2} + 2 \, a d e f x + a d e^{2} + {\left (a d f^{2} x^{2} e^{c} + 2 \, a d e f x e^{c} + a d e^{2} e^{c}\right )} e^{\left (d x\right )}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (e+f\,x\right )\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {i \int \frac {\operatorname {csch}^{2}{\left (c + d x \right )}}{e \sinh {\left (c + d x \right )} - i e + f x \sinh {\left (c + d x \right )} - i f x}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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