Integrand size = 18, antiderivative size = 71 \[ \int x^{-1+\frac {n}{2}} \sinh \left (a+b x^n\right ) \, dx=-\frac {e^{-a} \sqrt {\pi } \text {erf}\left (\sqrt {b} x^{n/2}\right )}{2 \sqrt {b} n}+\frac {e^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2}\right )}{2 \sqrt {b} n} \]
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Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5464, 5406, 2235, 2236} \[ \int x^{-1+\frac {n}{2}} \sinh \left (a+b x^n\right ) \, dx=\frac {\sqrt {\pi } e^a \text {erfi}\left (\sqrt {b} x^{n/2}\right )}{2 \sqrt {b} n}-\frac {\sqrt {\pi } e^{-a} \text {erf}\left (\sqrt {b} x^{n/2}\right )}{2 \sqrt {b} n} \]
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Rule 2235
Rule 2236
Rule 5406
Rule 5464
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \sinh \left (a+b x^2\right ) \, dx,x,x^{n/2}\right )}{n} \\ & = -\frac {\text {Subst}\left (\int e^{-a-b x^2} \, dx,x,x^{n/2}\right )}{n}+\frac {\text {Subst}\left (\int e^{a+b x^2} \, dx,x,x^{n/2}\right )}{n} \\ & = -\frac {e^{-a} \sqrt {\pi } \text {erf}\left (\sqrt {b} x^{n/2}\right )}{2 \sqrt {b} n}+\frac {e^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2}\right )}{2 \sqrt {b} n} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.83 \[ \int x^{-1+\frac {n}{2}} \sinh \left (a+b x^n\right ) \, dx=\frac {e^{-a} \sqrt {\pi } \left (-\text {erf}\left (\sqrt {b} x^{n/2}\right )+e^{2 a} \text {erfi}\left (\sqrt {b} x^{n/2}\right )\right )}{2 \sqrt {b} n} \]
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Time = 0.56 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.76
method | result | size |
risch | \(-\frac {{\mathrm e}^{-a} \sqrt {\pi }\, \operatorname {erf}\left (x^{\frac {n}{2}} \sqrt {b}\right )}{2 n \sqrt {b}}+\frac {{\mathrm e}^{a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-b}\, x^{\frac {n}{2}}\right )}{2 n \sqrt {-b}}\) | \(54\) |
meijerg | \(\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\frac {\sqrt {i b}\, \sqrt {2}\, \operatorname {erf}\left (x^{\frac {n}{2}} \sqrt {b}\right )}{2 \sqrt {b}}+\frac {\sqrt {i b}\, \sqrt {2}\, \operatorname {erfi}\left (x^{\frac {n}{2}} \sqrt {b}\right )}{2 \sqrt {b}}\right ) \sinh \left (a \right )}{2 \sqrt {i b}\, n}-\frac {i \sqrt {2}\, \sqrt {\pi }\, \left (-\frac {\sqrt {2}\, \left (i b \right )^{\frac {3}{2}} \operatorname {erf}\left (x^{\frac {n}{2}} \sqrt {b}\right )}{2 b^{\frac {3}{2}}}+\frac {\sqrt {2}\, \left (i b \right )^{\frac {3}{2}} \operatorname {erfi}\left (x^{\frac {n}{2}} \sqrt {b}\right )}{2 b^{\frac {3}{2}}}\right ) \cosh \left (a \right )}{2 \sqrt {i b}\, n}\) | \(139\) |
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Time = 0.24 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.37 \[ \int x^{-1+\frac {n}{2}} \sinh \left (a+b x^n\right ) \, dx=-\frac {\sqrt {\pi } \sqrt {-b} {\left (\cosh \left (a\right ) + \sinh \left (a\right )\right )} \operatorname {erf}\left (\sqrt {-b} x \cosh \left (\frac {1}{2} \, {\left (n - 2\right )} \log \left (x\right )\right ) + \sqrt {-b} x \sinh \left (\frac {1}{2} \, {\left (n - 2\right )} \log \left (x\right )\right )\right ) + \sqrt {\pi } \sqrt {b} {\left (\cosh \left (a\right ) - \sinh \left (a\right )\right )} \operatorname {erf}\left (\sqrt {b} x \cosh \left (\frac {1}{2} \, {\left (n - 2\right )} \log \left (x\right )\right ) + \sqrt {b} x \sinh \left (\frac {1}{2} \, {\left (n - 2\right )} \log \left (x\right )\right )\right )}{2 \, b n} \]
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\[ \int x^{-1+\frac {n}{2}} \sinh \left (a+b x^n\right ) \, dx=\int x^{\frac {n}{2} - 1} \sinh {\left (a + b x^{n} \right )}\, dx \]
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Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.97 \[ \int x^{-1+\frac {n}{2}} \sinh \left (a+b x^n\right ) \, dx=-\frac {\sqrt {\pi } x^{\frac {1}{2} \, n} {\left (\operatorname {erf}\left (\sqrt {b x^{n}}\right ) - 1\right )} e^{\left (-a\right )}}{2 \, \sqrt {b x^{n}} n} + \frac {\sqrt {\pi } x^{\frac {1}{2} \, n} {\left (\operatorname {erf}\left (\sqrt {-b x^{n}}\right ) - 1\right )} e^{a}}{2 \, \sqrt {-b x^{n}} n} \]
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Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.75 \[ \int x^{-1+\frac {n}{2}} \sinh \left (a+b x^n\right ) \, dx=\frac {\frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} \sqrt {x^{n}}\right ) e^{\left (-a\right )}}{\sqrt {b}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-b} \sqrt {x^{n}}\right ) e^{a}}{\sqrt {-b}}}{2 \, n} \]
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Timed out. \[ \int x^{-1+\frac {n}{2}} \sinh \left (a+b x^n\right ) \, dx=\int x^{\frac {n}{2}-1}\,\mathrm {sinh}\left (a+b\,x^n\right ) \,d x \]
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