Optimal. Leaf size=89 \[ -\frac {x}{2}-\frac {2^{-2-\frac {1}{n}} e^{2 a} x \left (-b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-2 b x^n\right )}{n}-\frac {2^{-2-\frac {1}{n}} e^{-2 a} x \left (b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},2 b x^n\right )}{n} \]
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Rubi [A]
time = 0.05, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5416, 5415,
2239} \begin {gather*} -\frac {e^{2 a} 2^{-\frac {1}{n}-2} x \left (-b x^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},-2 b x^n\right )}{n}-\frac {e^{-2 a} 2^{-\frac {1}{n}-2} x \left (b x^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},2 b x^n\right )}{n}-\frac {x}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2239
Rule 5415
Rule 5416
Rubi steps
\begin {align*} \int \sinh ^2\left (a+b x^n\right ) \, dx &=\int \left (-\frac {1}{2}+\frac {1}{2} \cosh \left (2 a+2 b x^n\right )\right ) \, dx\\ &=-\frac {x}{2}+\frac {1}{2} \int \cosh \left (2 a+2 b x^n\right ) \, dx\\ &=-\frac {x}{2}+\frac {1}{4} \int e^{-2 a-2 b x^n} \, dx+\frac {1}{4} \int e^{2 a+2 b x^n} \, dx\\ &=-\frac {x}{2}-\frac {2^{-2-\frac {1}{n}} e^{2 a} x \left (-b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-2 b x^n\right )}{n}-\frac {2^{-2-\frac {1}{n}} e^{-2 a} x \left (b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},2 b x^n\right )}{n}\\ \end {align*}
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Mathematica [A]
time = 0.80, size = 81, normalized size = 0.91 \begin {gather*} -\frac {x \left (2 n+2^{-1/n} e^{2 a} \left (-b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-2 b x^n\right )+2^{-1/n} e^{-2 a} \left (b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},2 b x^n\right )\right )}{4 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.32, size = 0, normalized size = 0.00 \[\int \sinh ^{2}\left (a +b \,x^{n}\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.07, size = 68, normalized size = 0.76 \begin {gather*} -\frac {1}{2} \, x - \frac {x e^{\left (-2 \, a\right )} \Gamma \left (\frac {1}{n}, 2 \, b x^{n}\right )}{4 \, \left (2 \, b x^{n}\right )^{\left (\frac {1}{n}\right )} n} - \frac {x e^{\left (2 \, a\right )} \Gamma \left (\frac {1}{n}, -2 \, b x^{n}\right )}{4 \, \left (-2 \, b x^{n}\right )^{\left (\frac {1}{n}\right )} n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sinh ^{2}{\left (a + b x^{n} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {sinh}\left (a+b\,x^n\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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