Optimal. Leaf size=132 \[ -\frac {b c \log (F) \cosh ^2(d+e x) F^{c (a+b x)}}{4 e^2-b^2 c^2 \log ^2(F)}+\frac {2 e \sinh (d+e x) \cosh (d+e x) F^{c (a+b x)}}{4 e^2-b^2 c^2 \log ^2(F)}+\frac {2 e^2 F^{c (a+b x)}}{b c \log (F) \left (4 e^2-b^2 c^2 \log ^2(F)\right )} \]
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Rubi [A] time = 0.05, antiderivative size = 132, 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 = {5477, 2194} \[ -\frac {b c \log (F) \cosh ^2(d+e x) F^{c (a+b x)}}{4 e^2-b^2 c^2 \log ^2(F)}+\frac {2 e \sinh (d+e x) \cosh (d+e x) F^{c (a+b x)}}{4 e^2-b^2 c^2 \log ^2(F)}+\frac {2 e^2 F^{c (a+b x)}}{b c \log (F) \left (4 e^2-b^2 c^2 \log ^2(F)\right )} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 5477
Rubi steps
\begin {align*} \int F^{c (a+b x)} \cosh ^2(d+e x) \, dx &=-\frac {b c F^{c (a+b x)} \cosh ^2(d+e x) \log (F)}{4 e^2-b^2 c^2 \log ^2(F)}+\frac {2 e F^{c (a+b x)} \cosh (d+e x) \sinh (d+e x)}{4 e^2-b^2 c^2 \log ^2(F)}+\frac {\left (2 e^2\right ) \int F^{c (a+b x)} \, dx}{4 e^2-b^2 c^2 \log ^2(F)}\\ &=\frac {2 e^2 F^{c (a+b x)}}{b c \log (F) \left (4 e^2-b^2 c^2 \log ^2(F)\right )}-\frac {b c F^{c (a+b x)} \cosh ^2(d+e x) \log (F)}{4 e^2-b^2 c^2 \log ^2(F)}+\frac {2 e F^{c (a+b x)} \cosh (d+e x) \sinh (d+e x)}{4 e^2-b^2 c^2 \log ^2(F)}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 85, normalized size = 0.64 \[ \frac {F^{c (a+b x)} \left (b^2 c^2 \log ^2(F) \cosh (2 (d+e x))+b^2 c^2 \log ^2(F)-2 b c e \log (F) \sinh (2 (d+e x))-4 e^2\right )}{2 b^3 c^3 \log ^3(F)-8 b c e^2 \log (F)} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.13, size = 699, normalized size = 5.30 \[ \frac {{\left ({\left (b^{2} c^{2} \log \relax (F)^{2} - 2 \, b c e \log \relax (F)\right )} \sinh \left (e x + d\right )^{4} - 8 \, e^{2} \cosh \left (e x + d\right )^{2} + 4 \, {\left (b^{2} c^{2} \cosh \left (e x + d\right ) \log \relax (F)^{2} - 2 \, b c e \cosh \left (e x + d\right ) \log \relax (F)\right )} \sinh \left (e x + d\right )^{3} + {\left (b^{2} c^{2} \cosh \left (e x + d\right )^{4} + 2 \, b^{2} c^{2} \cosh \left (e x + d\right )^{2} + b^{2} c^{2}\right )} \log \relax (F)^{2} - 2 \, {\left (6 \, b c e \cosh \left (e x + d\right )^{2} \log \relax (F) - {\left (3 \, b^{2} c^{2} \cosh \left (e x + d\right )^{2} + b^{2} c^{2}\right )} \log \relax (F)^{2} + 4 \, e^{2}\right )} \sinh \left (e x + d\right )^{2} - 2 \, {\left (b c e \cosh \left (e x + d\right )^{4} - b c e\right )} \log \relax (F) - 4 \, {\left (2 \, b c e \cosh \left (e x + d\right )^{3} \log \relax (F) + 4 \, e^{2} \cosh \left (e x + d\right ) - {\left (b^{2} c^{2} \cosh \left (e x + d\right )^{3} + b^{2} c^{2} \cosh \left (e x + d\right )\right )} \log \relax (F)^{2}\right )} \sinh \left (e x + d\right )\right )} \cosh \left ({\left (b c x + a c\right )} \log \relax (F)\right ) + {\left ({\left (b^{2} c^{2} \log \relax (F)^{2} - 2 \, b c e \log \relax (F)\right )} \sinh \left (e x + d\right )^{4} - 8 \, e^{2} \cosh \left (e x + d\right )^{2} + 4 \, {\left (b^{2} c^{2} \cosh \left (e x + d\right ) \log \relax (F)^{2} - 2 \, b c e \cosh \left (e x + d\right ) \log \relax (F)\right )} \sinh \left (e x + d\right )^{3} + {\left (b^{2} c^{2} \cosh \left (e x + d\right )^{4} + 2 \, b^{2} c^{2} \cosh \left (e x + d\right )^{2} + b^{2} c^{2}\right )} \log \relax (F)^{2} - 2 \, {\left (6 \, b c e \cosh \left (e x + d\right )^{2} \log \relax (F) - {\left (3 \, b^{2} c^{2} \cosh \left (e x + d\right )^{2} + b^{2} c^{2}\right )} \log \relax (F)^{2} + 4 \, e^{2}\right )} \sinh \left (e x + d\right )^{2} - 2 \, {\left (b c e \cosh \left (e x + d\right )^{4} - b c e\right )} \log \relax (F) - 4 \, {\left (2 \, b c e \cosh \left (e x + d\right )^{3} \log \relax (F) + 4 \, e^{2} \cosh \left (e x + d\right ) - {\left (b^{2} c^{2} \cosh \left (e x + d\right )^{3} + b^{2} c^{2} \cosh \left (e x + d\right )\right )} \log \relax (F)^{2}\right )} \sinh \left (e x + d\right )\right )} \sinh \left ({\left (b c x + a c\right )} \log \relax (F)\right )}{4 \, {\left (b^{3} c^{3} \cosh \left (e x + d\right )^{2} \log \relax (F)^{3} - 4 \, b c e^{2} \cosh \left (e x + d\right )^{2} \log \relax (F) + {\left (b^{3} c^{3} \log \relax (F)^{3} - 4 \, b c e^{2} \log \relax (F)\right )} \sinh \left (e x + d\right )^{2} + 2 \, {\left (b^{3} c^{3} \cosh \left (e x + d\right ) \log \relax (F)^{3} - 4 \, b c e^{2} \cosh \left (e x + d\right ) \log \relax (F)\right )} \sinh \left (e x + d\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.24, size = 903, normalized size = 6.84 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 143, normalized size = 1.08 \[ \frac {\left (\ln \relax (F )^{2} b^{2} c^{2} {\mathrm e}^{4 e x +4 d}+2 \ln \relax (F )^{2} b^{2} c^{2} {\mathrm e}^{2 e x +2 d}-2 \ln \relax (F ) b c e \,{\mathrm e}^{4 e x +4 d}+b^{2} c^{2} \ln \relax (F )^{2}+2 \ln \relax (F ) b c e -8 e^{2} {\mathrm e}^{2 e x +2 d}\right ) {\mathrm e}^{-2 e x -2 d} F^{c \left (b x +a \right )}}{4 \ln \relax (F ) b c \left (b c \ln \relax (F )-2 e \right ) \left (b c \ln \relax (F )+2 e \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 94, normalized size = 0.71 \[ \frac {F^{a c} e^{\left (b c x \log \relax (F) + 2 \, e x + 2 \, d\right )}}{4 \, {\left (b c \log \relax (F) + 2 \, e\right )}} + \frac {F^{a c} e^{\left (b c x \log \relax (F) - 2 \, e x\right )}}{4 \, {\left (b c e^{\left (2 \, d\right )} \log \relax (F) - 2 \, e e^{\left (2 \, d\right )}\right )}} + \frac {F^{b c x + a c}}{2 \, b c \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 100, normalized size = 0.76 \[ -\frac {2\,F^{a\,c+b\,c\,x}\,e^2-F^{a\,c+b\,c\,x}\,b^2\,c^2\,{\mathrm {cosh}\left (d+e\,x\right )}^2\,{\ln \relax (F)}^2+2\,F^{a\,c+b\,c\,x}\,b\,c\,e\,\mathrm {cosh}\left (d+e\,x\right )\,\mathrm {sinh}\left (d+e\,x\right )\,\ln \relax (F)}{b^3\,c^3\,{\ln \relax (F)}^3-4\,b\,c\,e^2\,\ln \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 30.57, size = 604, normalized size = 4.58 \[ \begin {cases} - \frac {x \sinh ^{2}{\left (d + e x \right )}}{2} + \frac {x \cosh ^{2}{\left (d + e x \right )}}{2} + \frac {\sinh {\left (d + e x \right )} \cosh {\left (d + e x \right )}}{2 e} & \text {for}\: F = 1 \\\tilde {\infty } e^{2} \left (e^{- \frac {2 e}{b c}}\right )^{a c} \left (e^{- \frac {2 e}{b c}}\right )^{b c x} \sinh ^{2}{\left (d + e x \right )} + \tilde {\infty } e^{2} \left (e^{- \frac {2 e}{b c}}\right )^{a c} \left (e^{- \frac {2 e}{b c}}\right )^{b c x} \sinh {\left (d + e x \right )} \cosh {\left (d + e x \right )} + \tilde {\infty } e^{2} \left (e^{- \frac {2 e}{b c}}\right )^{a c} \left (e^{- \frac {2 e}{b c}}\right )^{b c x} \cosh ^{2}{\left (d + e x \right )} & \text {for}\: F = e^{- \frac {2 e}{b c}} \\\tilde {\infty } e^{2} \left (e^{\frac {2 e}{b c}}\right )^{a c} \left (e^{\frac {2 e}{b c}}\right )^{b c x} \sinh ^{2}{\left (d + e x \right )} + \tilde {\infty } e^{2} \left (e^{\frac {2 e}{b c}}\right )^{a c} \left (e^{\frac {2 e}{b c}}\right )^{b c x} \sinh {\left (d + e x \right )} \cosh {\left (d + e x \right )} + \tilde {\infty } e^{2} \left (e^{\frac {2 e}{b c}}\right )^{a c} \left (e^{\frac {2 e}{b c}}\right )^{b c x} \cosh ^{2}{\left (d + e x \right )} & \text {for}\: F = e^{\frac {2 e}{b c}} \\F^{a c} \left (- \frac {x \sinh ^{2}{\left (d + e x \right )}}{2} + \frac {x \cosh ^{2}{\left (d + e x \right )}}{2} + \frac {\sinh {\left (d + e x \right )} \cosh {\left (d + e x \right )}}{2 e}\right ) & \text {for}\: b = 0 \\- \frac {x \sinh ^{2}{\left (d + e x \right )}}{2} + \frac {x \cosh ^{2}{\left (d + e x \right )}}{2} + \frac {\sinh {\left (d + e x \right )} \cosh {\left (d + e x \right )}}{2 e} & \text {for}\: c = 0 \\\frac {F^{a c} F^{b c x} b^{2} c^{2} \log {\relax (F )}^{2} \cosh ^{2}{\left (d + e x \right )}}{b^{3} c^{3} \log {\relax (F )}^{3} - 4 b c e^{2} \log {\relax (F )}} - \frac {2 F^{a c} F^{b c x} b c e \log {\relax (F )} \sinh {\left (d + e x \right )} \cosh {\left (d + e x \right )}}{b^{3} c^{3} \log {\relax (F )}^{3} - 4 b c e^{2} \log {\relax (F )}} + \frac {2 F^{a c} F^{b c x} e^{2} \sinh ^{2}{\left (d + e x \right )}}{b^{3} c^{3} \log {\relax (F )}^{3} - 4 b c e^{2} \log {\relax (F )}} - \frac {2 F^{a c} F^{b c x} e^{2} \cosh ^{2}{\left (d + e x \right )}}{b^{3} c^{3} \log {\relax (F )}^{3} - 4 b c e^{2} \log {\relax (F )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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