Optimal. Leaf size=43 \[ -\frac {\cosh \left (a+b \log \left (c x^n\right )\right )}{b n}+\frac {\cosh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]
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Rubi [A]
time = 0.02, antiderivative size = 43, 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 = {2713}
\begin {gather*} \frac {\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2713
Rubi steps
\begin {align*} \int \frac {\sinh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\text {Subst}\left (\int \sinh ^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\text {Subst}\left (\int \left (1-x^2\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 {\cosh ^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 = 45, normalized size = 1.05 \begin {gather*} -\frac {3 \cosh \left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {\cosh \left (3 \left (a+b \log \left (c x^n\right )\right )\right )}{12 b n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sinh ^{3}\left (a +b \ln \left (c \,x^{n}\right )\right )}{x}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 86 vs.
\(2 (41) = 82\).
time = 0.26, size = 86, normalized size = 2.00 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 65, normalized size = 1.51 \begin {gather*} \frac {\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 9 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{12 \, b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 76 vs.
\(2 (32) = 64\).
time = 1.78, size = 76, normalized size = 1.77 \begin {gather*} \begin {cases} \log {\left (x \right )} \sinh ^{3}{\left (a \right )} & \text {for}\: b = 0 \wedge n = 0 \\\log {\left (x \right )} \sinh ^{3}{\left (a + b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\log {\left (x \right )} \sinh ^{3}{\left (a \right )} & \text {for}\: b = 0 \\\frac {\sinh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} \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} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 81, normalized size = 1.88 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.72, size = 37, normalized size = 0.86 \begin {gather*} -\frac {3\,\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )-{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{3\,b\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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