Optimal. Leaf size=154 \[ -\frac {3^{\frac {1}{n}} e^{3 a} \left (-b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-3 b x^n\right )}{8 n x}+\frac {3 e^a \left (-b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-b x^n\right )}{8 n x}-\frac {3 e^{-a} \left (b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},b x^n\right )}{8 n x}+\frac {3^{\frac {1}{n}} e^{-3 a} \left (b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},3 b x^n\right )}{8 n x} \]
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Rubi [A]
time = 0.12, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5470, 5468,
2250} \begin {gather*} -\frac {e^{3 a} 3^{\frac {1}{n}} \left (-b x^n\right )^{\frac {1}{n}} \text {Gamma}\left (-\frac {1}{n},-3 b x^n\right )}{8 n x}+\frac {3 e^a \left (-b x^n\right )^{\frac {1}{n}} \text {Gamma}\left (-\frac {1}{n},-b x^n\right )}{8 n x}-\frac {3 e^{-a} \left (b x^n\right )^{\frac {1}{n}} \text {Gamma}\left (-\frac {1}{n},b x^n\right )}{8 n x}+\frac {e^{-3 a} 3^{\frac {1}{n}} \left (b x^n\right )^{\frac {1}{n}} \text {Gamma}\left (-\frac {1}{n},3 b x^n\right )}{8 n x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2250
Rule 5468
Rule 5470
Rubi steps
\begin {align*} \int \frac {\sinh ^3\left (a+b x^n\right )}{x^2} \, dx &=\int \left (-\frac {3 \sinh \left (a+b x^n\right )}{4 x^2}+\frac {\sinh \left (3 a+3 b x^n\right )}{4 x^2}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\sinh \left (3 a+3 b x^n\right )}{x^2} \, dx-\frac {3}{4} \int \frac {\sinh \left (a+b x^n\right )}{x^2} \, dx\\ &=-\left (\frac {1}{8} \int \frac {e^{-3 a-3 b x^n}}{x^2} \, dx\right )+\frac {1}{8} \int \frac {e^{3 a+3 b x^n}}{x^2} \, dx+\frac {3}{8} \int \frac {e^{-a-b x^n}}{x^2} \, dx-\frac {3}{8} \int \frac {e^{a+b x^n}}{x^2} \, dx\\ &=-\frac {3^{\frac {1}{n}} e^{3 a} \left (-b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-3 b x^n\right )}{8 n x}+\frac {3 e^a \left (-b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-b x^n\right )}{8 n x}-\frac {3 e^{-a} \left (b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},b x^n\right )}{8 n x}+\frac {3^{\frac {1}{n}} e^{-3 a} \left (b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},3 b x^n\right )}{8 n x}\\ \end {align*}
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Mathematica [A]
time = 1.03, size = 126, normalized size = 0.82 \begin {gather*} \frac {e^{-3 a} \left (-3^{\frac {1}{n}} e^{6 a} \left (-b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-3 b x^n\right )+3 e^{4 a} \left (-b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-b x^n\right )+\left (b x^n\right )^{\frac {1}{n}} \left (-3 e^{2 a} \Gamma \left (-\frac {1}{n},b x^n\right )+3^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},3 b x^n\right )\right )\right )}{8 n x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.81, size = 0, normalized size = 0.00 \[\int \frac {\sinh ^{3}\left (a +b \,x^{n}\right )}{x^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.10, size = 133, normalized size = 0.86 \begin {gather*} \frac {\left (3 \, b x^{n}\right )^{\left (\frac {1}{n}\right )} e^{\left (-3 \, a\right )} \Gamma \left (-\frac {1}{n}, 3 \, b x^{n}\right )}{8 \, n x} - \frac {3 \, \left (b x^{n}\right )^{\left (\frac {1}{n}\right )} e^{\left (-a\right )} \Gamma \left (-\frac {1}{n}, b x^{n}\right )}{8 \, n x} + \frac {3 \, \left (-b x^{n}\right )^{\left (\frac {1}{n}\right )} e^{a} \Gamma \left (-\frac {1}{n}, -b x^{n}\right )}{8 \, n x} - \frac {\left (-3 \, b x^{n}\right )^{\left (\frac {1}{n}\right )} e^{\left (3 \, a\right )} \Gamma \left (-\frac {1}{n}, -3 \, b x^{n}\right )}{8 \, n x} \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 \frac {\sinh ^{3}{\left (a + b x^{n} \right )}}{x^{2}}\, 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 \frac {{\mathrm {sinh}\left (a+b\,x^n\right )}^3}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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