Optimal. Leaf size=25 \[ \frac {\text {Chi}\left (b x^n\right ) \sinh (a)}{n}+\frac {\cosh (a) \text {Shi}\left (b x^n\right )}{n} \]
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Rubi [A]
time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5426, 5425,
5424} \begin {gather*} \frac {\sinh (a) \text {Chi}\left (b x^n\right )}{n}+\frac {\cosh (a) \text {Shi}\left (b x^n\right )}{n} \end {gather*}
Antiderivative was successfully verified.
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Rule 5424
Rule 5425
Rule 5426
Rubi steps
\begin {align*} \int \frac {\sinh \left (a+b x^n\right )}{x} \, dx &=\cosh (a) \int \frac {\sinh \left (b x^n\right )}{x} \, dx+\sinh (a) \int \frac {\cosh \left (b x^n\right )}{x} \, dx\\ &=\frac {\text {Chi}\left (b x^n\right ) \sinh (a)}{n}+\frac {\cosh (a) \text {Shi}\left (b x^n\right )}{n}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 23, normalized size = 0.92 \begin {gather*} \frac {\text {Chi}\left (b x^n\right ) \sinh (a)+\cosh (a) \text {Shi}\left (b x^n\right )}{n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.78, size = 33, normalized size = 1.32
method | result | size |
risch | \(\frac {{\mathrm e}^{-a} \expIntegral \left (1, b \,x^{n}\right )}{2 n}-\frac {{\mathrm e}^{a} \expIntegral \left (1, -b \,x^{n}\right )}{2 n}\) | \(33\) |
meijerg | \(\frac {\sqrt {\pi }\, \left (\frac {2 \hyperbolicCosineIntegral \left (b \,x^{n}\right )-2 \ln \left (b \,x^{n}\right )-2 \gamma }{\sqrt {\pi }}+\frac {2 \gamma +2 n \ln \left (x \right )+2 \ln \left (i b \right )}{\sqrt {\pi }}\right ) \sinh \left (a \right )}{2 n}+\frac {\cosh \left (a \right ) \hyperbolicSineIntegral \left (b \,x^{n}\right )}{n}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 30, normalized size = 1.20 \begin {gather*} -\frac {{\rm Ei}\left (-b x^{n}\right ) e^{\left (-a\right )}}{2 \, n} + \frac {{\rm Ei}\left (b x^{n}\right ) e^{a}}{2 \, n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 55 vs.
\(2 (25) = 50\).
time = 0.48, size = 55, normalized size = 2.20 \begin {gather*} \frac {{\left (\cosh \left (a\right ) + \sinh \left (a\right )\right )} {\rm Ei}\left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right )\right ) - {\left (\cosh \left (a\right ) - \sinh \left (a\right )\right )} {\rm Ei}\left (-b \cosh \left (n \log \left (x\right )\right ) - b \sinh \left (n \log \left (x\right )\right )\right )}{2 \, n} \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 {\left (a + b x^{n} \right )}}{x}\, 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 = -0.04 \begin {gather*} \int \frac {\mathrm {sinh}\left (a+b\,x^n\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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