Optimal. Leaf size=250 \[ -\frac {e^{-4 c (a+b x)} \sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)}{128 b c}-\frac {5 e^{-2 c (a+b x)} \sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)}{64 b c}+\frac {5 e^{2 c (a+b x)} \sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)}{32 b c}+\frac {5 e^{4 c (a+b x)} \sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)}{128 b c}+\frac {e^{6 c (a+b x)} \sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)}{192 b c}+\frac {5}{16} x \sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x) \]
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Rubi [A] time = 0.23, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6720, 2282, 12, 266, 43} \[ -\frac {e^{-4 c (a+b x)} \sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)}{128 b c}-\frac {5 e^{-2 c (a+b x)} \sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)}{64 b c}+\frac {5 e^{2 c (a+b x)} \sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)}{32 b c}+\frac {5 e^{4 c (a+b x)} \sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)}{128 b c}+\frac {e^{6 c (a+b x)} \sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)}{192 b c}+\frac {5}{16} x \sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x) \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 266
Rule 2282
Rule 6720
Rubi steps
\begin {align*} \int e^{c (a+b x)} \cosh ^2(a c+b c x)^{5/2} \, dx &=\left (\sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)\right ) \int e^{c (a+b x)} \cosh ^5(a c+b c x) \, dx\\ &=\frac {\left (\sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^5}{32 x^5} \, dx,x,e^{c (a+b x)}\right )}{b c}\\ &=\frac {\left (\sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^5}{x^5} \, dx,x,e^{c (a+b x)}\right )}{32 b c}\\ &=\frac {\left (\sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)\right ) \operatorname {Subst}\left (\int \frac {(1+x)^5}{x^3} \, dx,x,e^{2 c (a+b x)}\right )}{64 b c}\\ &=\frac {\left (\sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)\right ) \operatorname {Subst}\left (\int \left (10+\frac {1}{x^3}+\frac {5}{x^2}+\frac {10}{x}+5 x+x^2\right ) \, dx,x,e^{2 c (a+b x)}\right )}{64 b c}\\ &=-\frac {e^{-4 c (a+b x)} \sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)}{128 b c}-\frac {5 e^{-2 c (a+b x)} \sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)}{64 b c}+\frac {5 e^{2 c (a+b x)} \sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)}{32 b c}+\frac {5 e^{4 c (a+b x)} \sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)}{128 b c}+\frac {e^{6 c (a+b x)} \sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)}{192 b c}+\frac {5}{16} x \sqrt {\cosh ^2(a c+b c x)} \text {sech}(a c+b c x)\\ \end {align*}
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Mathematica [A] time = 0.11, size = 106, normalized size = 0.42 \[ \frac {\left (-\frac {1}{2} e^{-4 c (a+b x)}-5 e^{-2 c (a+b x)}+10 e^{2 c (a+b x)}+\frac {5}{2} e^{4 c (a+b x)}+\frac {1}{3} e^{6 c (a+b x)}+20 b c x\right ) \cosh ^2(c (a+b x))^{5/2} \text {sech}^5(c (a+b x))}{64 b c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 218, normalized size = 0.87 \[ -\frac {\cosh \left (b c x + a c\right )^{5} + 5 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{4} - 5 \, \sinh \left (b c x + a c\right )^{5} - 5 \, {\left (10 \, \cosh \left (b c x + a c\right )^{2} + 9\right )} \sinh \left (b c x + a c\right )^{3} + 15 \, \cosh \left (b c x + a c\right )^{3} + 5 \, {\left (2 \, \cosh \left (b c x + a c\right )^{3} + 9 \, \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{2} - 60 \, {\left (2 \, b c x + 1\right )} \cosh \left (b c x + a c\right ) - 5 \, {\left (5 \, \cosh \left (b c x + a c\right )^{4} - 24 \, b c x + 27 \, \cosh \left (b c x + a c\right )^{2} + 12\right )} \sinh \left (b c x + a c\right )}{384 \, {\left (b c \cosh \left (b c x + a c\right ) - b c \sinh \left (b c x + a c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 101, normalized size = 0.40 \[ \frac {120 \, b c x - 3 \, {\left (30 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 10 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )} e^{\left (-4 \, b c x - 4 \, a c\right )} + {\left (2 \, e^{\left (6 \, b c x + 18 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 16 \, a c\right )} + 60 \, e^{\left (2 \, b c x + 14 \, a c\right )}\right )} e^{\left (-12 \, a c\right )}}{384 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{c \left (b x +a \right )} \left (\frac {\cosh \left (2 b c x +2 a c \right )}{2}+\frac {1}{2}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 112, normalized size = 0.45 \[ \frac {5 \, {\left (b c x + a c\right )}}{16 \, b c} + \frac {e^{\left (6 \, b c x + 6 \, a c\right )}}{192 \, b c} + \frac {5 \, e^{\left (4 \, b c x + 4 \, a c\right )}}{128 \, b c} + \frac {5 \, e^{\left (2 \, b c x + 2 \, a c\right )}}{32 \, b c} - \frac {5 \, e^{\left (-2 \, b c x - 2 \, a c\right )}}{64 \, b c} - \frac {e^{\left (-4 \, b c x - 4 \, a c\right )}}{128 \, b c} \]
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 \[ \int {\mathrm {e}}^{c\,\left (a+b\,x\right )}\,{\left ({\mathrm {cosh}\left (a\,c+b\,c\,x\right )}^2\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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