Optimal. Leaf size=65 \[ \frac {\cosh ^5\left (a+b \log \left (c x^n\right )\right )}{5 b n}-\frac {2 \cosh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\cosh \left (a+b \log \left (c x^n\right )\right )}{b n} \]
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Rubi [A] time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2633} \[ \frac {\cosh ^5\left (a+b \log \left (c x^n\right )\right )}{5 b n}-\frac {2 \cosh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\cosh \left (a+b \log \left (c x^n\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 2633
Rubi steps
\begin {align*} \int \frac {\sinh ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \sinh ^5(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ &=\frac {\cosh \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {2 \cosh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\cosh ^5\left (a+b \log \left (c x^n\right )\right )}{5 b n}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 68, normalized size = 1.05 \[ \frac {5 \cosh \left (a+b \log \left (c x^n\right )\right )}{8 b n}-\frac {5 \cosh \left (3 \left (a+b \log \left (c x^n\right )\right )\right )}{48 b n}+\frac {\cosh \left (5 \left (a+b \log \left (c x^n\right )\right )\right )}{80 b n} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.08, size = 130, normalized size = 2.00 \[ \frac {3 \, \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{5} + 15 \, \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{4} - 25 \, \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} + 15 \, {\left (2 \, \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} - 5 \, \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )\right )} \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + 150 \, \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{240 \, b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 115, normalized size = 1.77 \[ \frac {{\left (3 \, c^{10 \, b} x^{5 \, b n} e^{\left (10 \, a\right )} - 25 \, c^{8 \, b} x^{3 \, b n} e^{\left (8 \, a\right )} + 150 \, c^{6 \, b} x^{b n} e^{\left (6 \, a\right )} + \frac {150 \, c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} - 25 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 3}{x^{5 \, b n}}\right )} e^{\left (-5 \, a\right )}}{480 \, b c^{5 \, b} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 51, normalized size = 0.78 \[ \frac {\left (\frac {8}{15}+\frac {\left (\sinh ^{4}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{5}-\frac {4 \left (\sinh ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{15}\right ) \cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}{n b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 130, normalized size = 2.00 \[ \frac {e^{\left (5 \, b \log \left (c x^{n}\right ) + 5 \, a\right )}}{160 \, b n} - \frac {5 \, e^{\left (3 \, b \log \left (c x^{n}\right ) + 3 \, a\right )}}{96 \, b n} + \frac {5 \, e^{\left (b \log \left (c x^{n}\right ) + a\right )}}{16 \, b n} + \frac {5 \, e^{\left (-b \log \left (c x^{n}\right ) - a\right )}}{16 \, b n} - \frac {5 \, e^{\left (-3 \, b \log \left (c x^{n}\right ) - 3 \, a\right )}}{96 \, b n} + \frac {e^{\left (-5 \, b \log \left (c x^{n}\right ) - 5 \, a\right )}}{160 \, b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.84, size = 49, normalized size = 0.75 \[ \frac {\frac {{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^5}{5}-\frac {2\,{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{3}+\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}{b\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 94.65, size = 122, normalized size = 1.88 \[ \begin {cases} \log {\relax (x )} \sinh ^{5}{\relax (a )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\relax (x )} \sinh ^{5}{\left (a + b \log {\relax (c )} \right )} & \text {for}\: n = 0 \\\frac {\sinh ^{4}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )} \cosh {\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{b n} - \frac {4 \sinh ^{2}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )} \cosh ^{3}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{3 b n} + \frac {8 \cosh ^{5}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{15 b n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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