Integrand size = 10, antiderivative size = 78 \[ \int \sinh ^2\left (a+b x^2\right ) \, dx=-\frac {x}{2}+\frac {e^{-2 a} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {b} x\right )}{8 \sqrt {b}}+\frac {e^{2 a} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {b} x\right )}{8 \sqrt {b}} \]
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Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5408, 5407, 2235, 2236} \[ \int \sinh ^2\left (a+b x^2\right ) \, dx=\frac {\sqrt {\frac {\pi }{2}} e^{-2 a} \text {erf}\left (\sqrt {2} \sqrt {b} x\right )}{8 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{2}} e^{2 a} \text {erfi}\left (\sqrt {2} \sqrt {b} x\right )}{8 \sqrt {b}}-\frac {x}{2} \]
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Rule 2235
Rule 2236
Rule 5407
Rule 5408
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{2}+\frac {1}{2} \cosh \left (2 a+2 b x^2\right )\right ) \, dx \\ & = -\frac {x}{2}+\frac {1}{2} \int \cosh \left (2 a+2 b x^2\right ) \, dx \\ & = -\frac {x}{2}+\frac {1}{4} \int e^{-2 a-2 b x^2} \, dx+\frac {1}{4} \int e^{2 a+2 b x^2} \, dx \\ & = -\frac {x}{2}+\frac {e^{-2 a} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {b} x\right )}{8 \sqrt {b}}+\frac {e^{2 a} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {b} x\right )}{8 \sqrt {b}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.10 \[ \int \sinh ^2\left (a+b x^2\right ) \, dx=\frac {-4 \sqrt {2} \sqrt {b} x+\sqrt {\pi } \text {erf}\left (\sqrt {2} \sqrt {b} x\right ) (\cosh (2 a)-\sinh (2 a))+\sqrt {\pi } \text {erfi}\left (\sqrt {2} \sqrt {b} x\right ) (\cosh (2 a)+\sinh (2 a))}{8 \sqrt {2} \sqrt {b}} \]
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Time = 0.49 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.65
method | result | size |
risch | \(-\frac {x}{2}+\frac {{\mathrm e}^{-2 a} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (x \sqrt {2}\, \sqrt {b}\right )}{16 \sqrt {b}}+\frac {{\mathrm e}^{2 a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-2 b}\, x \right )}{8 \sqrt {-2 b}}\) | \(51\) |
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Time = 0.24 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.94 \[ \int \sinh ^2\left (a+b x^2\right ) \, dx=-\frac {\sqrt {2} \sqrt {\pi } \sqrt {-b} {\left (\cosh \left (2 \, a\right ) + \sinh \left (2 \, a\right )\right )} \operatorname {erf}\left (\sqrt {2} \sqrt {-b} x\right ) - \sqrt {2} \sqrt {\pi } \sqrt {b} {\left (\cosh \left (2 \, a\right ) - \sinh \left (2 \, a\right )\right )} \operatorname {erf}\left (\sqrt {2} \sqrt {b} x\right ) + 8 \, b x}{16 \, b} \]
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\[ \int \sinh ^2\left (a+b x^2\right ) \, dx=\int \sinh ^{2}{\left (a + b x^{2} \right )}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.72 \[ \int \sinh ^2\left (a+b x^2\right ) \, dx=\frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} \sqrt {-b} x\right ) e^{\left (2 \, a\right )}}{16 \, \sqrt {-b}} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} \sqrt {b} x\right ) e^{\left (-2 \, a\right )}}{16 \, \sqrt {b}} - \frac {1}{2} \, x \]
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Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.74 \[ \int \sinh ^2\left (a+b x^2\right ) \, dx=-\frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} \sqrt {-b} x\right ) e^{\left (2 \, a\right )}}{16 \, \sqrt {-b}} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} \sqrt {b} x\right ) e^{\left (-2 \, a\right )}}{16 \, \sqrt {b}} - \frac {1}{2} \, x \]
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Timed out. \[ \int \sinh ^2\left (a+b x^2\right ) \, dx=\int {\mathrm {sinh}\left (b\,x^2+a\right )}^2 \,d x \]
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