Optimal. Leaf size=131 \[ -\frac {F^{c (a+b x)} \coth (d+e x) \text {csch}^2(d+e x)}{3 e}-\frac {b c F^{c (a+b x)} \text {csch}^2(d+e x) \log (F)}{6 e^2}-\frac {2 e^{2 (d+e x)} F^{c (a+b x)} \, _2F_1\left (2,1+\frac {b c \log (F)}{2 e};2+\frac {b c \log (F)}{2 e};e^{2 (d+e x)}\right ) (2 e-b c \log (F))}{3 e^2} \]
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Rubi [A]
time = 0.04, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5599, 5601}
\begin {gather*} -\frac {2 e^{2 (d+e x)} F^{c (a+b x)} (2 e-b c \log (F)) \, _2F_1\left (2,\frac {b c \log (F)}{2 e}+1;\frac {b c \log (F)}{2 e}+2;e^{2 (d+e x)}\right )}{3 e^2}-\frac {b c \log (F) \text {csch}^2(d+e x) F^{c (a+b x)}}{6 e^2}-\frac {\coth (d+e x) \text {csch}^2(d+e x) F^{c (a+b x)}}{3 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 5599
Rule 5601
Rubi steps
\begin {align*} \int F^{c (a+b x)} \text {csch}^4(d+e x) \, dx &=-\frac {F^{c (a+b x)} \coth (d+e x) \text {csch}^2(d+e x)}{3 e}-\frac {b c F^{c (a+b x)} \text {csch}^2(d+e x) \log (F)}{6 e^2}-\frac {1}{6} \left (4-\frac {b^2 c^2 \log ^2(F)}{e^2}\right ) \int F^{c (a+b x)} \text {csch}^2(d+e x) \, dx\\ &=-\frac {F^{c (a+b x)} \coth (d+e x) \text {csch}^2(d+e x)}{3 e}-\frac {b c F^{c (a+b x)} \text {csch}^2(d+e x) \log (F)}{6 e^2}-\frac {2 e^{2 (d+e x)} F^{c (a+b x)} \, _2F_1\left (2,1+\frac {b c \log (F)}{2 e};2+\frac {b c \log (F)}{2 e};e^{2 (d+e x)}\right ) (2 e-b c \log (F))}{3 e^2}\\ \end {align*}
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Mathematica [A]
time = 5.21, size = 202, normalized size = 1.54 \begin {gather*} \frac {F^{c (a+b x)} \left (-1+\coth (d)+2 \, _2F_1\left (1,\frac {b c \log (F)}{2 e};1+\frac {b c \log (F)}{2 e};\cosh (2 (d+e x))+\sinh (2 (d+e x))\right )\right ) \left (4 e^2-b^2 c^2 \log ^2(F)\right )}{6 e^3}-\frac {F^{a c+b c x} \text {csch}(d) \text {csch}^2(d+e x) (2 e \cosh (d)+b c \log (F) \sinh (d))}{6 e^2}+\frac {F^{a c+b c x} \text {csch}(d) \text {csch}^3(d+e x) \sinh (e x)}{3 e}-\frac {F^{a c+b c x} \text {csch}(d) \text {csch}(d+e x) \left (4 e^2-b^2 c^2 \log ^2(F)\right ) \sinh (e x)}{6 e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.32, size = 0, normalized size = 0.00 \[\int F^{c \left (b x +a \right )} \mathrm {csch}\left (e x +d \right )^{4}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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*} \int F^{c \left (a + b x\right )} \operatorname {csch}^{4}{\left (d + e x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {F^{c\,\left (a+b\,x\right )}}{{\mathrm {sinh}\left (d+e\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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