Optimal. Leaf size=42 \[ \frac {\sinh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\sinh \left (a+b \log \left (c x^n\right )\right )}{b n} \]
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Rubi [A] time = 0.03, antiderivative size = 42, 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} \[ \frac {\sinh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\sinh \left (a+b \log \left (c x^n\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 2633
Rubi steps
\begin {align*} \int \frac {\cosh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \cosh ^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {i \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ &=\frac {\sinh \left (a+b \log \left (c x^n\right )\right )}{b n}+\frac {\sinh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 1.00 \[ \frac {\sinh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\sinh \left (a+b \log \left (c x^n\right )\right )}{b n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 53, normalized size = 1.26 \[ \frac {\sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} + 3 \, {\left (\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + 3\right )} \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{12 \, b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 81, normalized size = 1.93 \[ \frac {{\left (c^{6 \, b} x^{3 \, b n} e^{\left (6 \, a\right )} + 9 \, c^{4 \, b} x^{b n} e^{\left (4 \, a\right )} - \frac {9 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1}{x^{3 \, b n}}\right )} e^{\left (-3 \, a\right )}}{24 \, b c^{3 \, b} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 36, normalized size = 0.86 \[ \frac {\left (\frac {2}{3}+\frac {\left (\cosh ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{3}\right ) \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}{n b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 86, normalized size = 2.05 \[ \frac {e^{\left (3 \, b \log \left (c x^{n}\right ) + 3 \, a\right )}}{24 \, b n} + \frac {3 \, e^{\left (b \log \left (c x^{n}\right ) + a\right )}}{8 \, b n} - \frac {3 \, e^{\left (-b \log \left (c x^{n}\right ) - a\right )}}{8 \, b n} - \frac {e^{\left (-3 \, b \log \left (c x^{n}\right ) - 3 \, a\right )}}{24 \, b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.05, size = 35, normalized size = 0.83 \[ \frac {{\mathrm {sinh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^3+3\,\mathrm {sinh}\left (a+b\,\ln \left (c\,x^n\right )\right )}{3\,b\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.57, size = 87, normalized size = 2.07 \[ \begin {cases} \log {\relax (x )} \cosh ^{3}{\relax (a )} & \text {for}\: b = 0 \wedge n = 0 \\\log {\relax (x )} \cosh ^{3}{\left (a + b \log {\relax (c )} \right )} & \text {for}\: n = 0 \\\log {\relax (x )} \cosh ^{3}{\relax (a )} & \text {for}\: b = 0 \\- \frac {2 \sinh ^{3}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{3 b n} + \frac {\sinh {\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )} \cosh ^{2}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{b n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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