Integrand size = 20, antiderivative size = 128 \[ \int f^{a+c x^2} \sinh ^2\left (d+f x^2\right ) \, dx=-\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{-2 d} f^a \sqrt {\pi } \text {erf}\left (x \sqrt {2 f-c \log (f)}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {e^{2 d} f^a \sqrt {\pi } \text {erfi}\left (x \sqrt {2 f+c \log (f)}\right )}{8 \sqrt {2 f+c \log (f)}} \]
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Time = 0.16 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5623, 2235, 2325, 2236} \[ \int f^{a+c x^2} \sinh ^2\left (d+f x^2\right ) \, dx=\frac {\sqrt {\pi } e^{-2 d} f^a \text {erf}\left (x \sqrt {2 f-c \log (f)}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {\sqrt {\pi } e^{2 d} f^a \text {erfi}\left (x \sqrt {c \log (f)+2 f}\right )}{8 \sqrt {c \log (f)+2 f}}-\frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
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Rule 2235
Rule 2236
Rule 2325
Rule 5623
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{2} f^{a+c x^2}+\frac {1}{4} e^{-2 d-2 f x^2} f^{a+c x^2}+\frac {1}{4} e^{2 d+2 f x^2} f^{a+c x^2}\right ) \, dx \\ & = \frac {1}{4} \int e^{-2 d-2 f x^2} f^{a+c x^2} \, dx+\frac {1}{4} \int e^{2 d+2 f x^2} f^{a+c x^2} \, dx-\frac {1}{2} \int f^{a+c x^2} \, dx \\ & = -\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {1}{4} \int e^{-2 d+a \log (f)-x^2 (2 f-c \log (f))} \, dx+\frac {1}{4} \int e^{2 d+a \log (f)+x^2 (2 f+c \log (f))} \, dx \\ & = -\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{-2 d} f^a \sqrt {\pi } \text {erf}\left (x \sqrt {2 f-c \log (f)}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {e^{2 d} f^a \sqrt {\pi } \text {erfi}\left (x \sqrt {2 f+c \log (f)}\right )}{8 \sqrt {2 f+c \log (f)}} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.40 \[ \int f^{a+c x^2} \sinh ^2\left (d+f x^2\right ) \, dx=\frac {f^a \sqrt {\pi } \left (\text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right ) \left (8 f^2-2 c^2 \log ^2(f)\right )+\sqrt {c} \sqrt {\log (f)} \left (\text {erf}\left (x \sqrt {2 f-c \log (f)}\right ) \sqrt {2 f-c \log (f)} (2 f+c \log (f)) (-\cosh (2 d)+\sinh (2 d))-\text {erfi}\left (x \sqrt {2 f+c \log (f)}\right ) (2 f-c \log (f)) \sqrt {2 f+c \log (f)} (\cosh (2 d)+\sinh (2 d))\right )\right )}{8 \sqrt {c} \sqrt {\log (f)} \left (-4 f^2+c^2 \log ^2(f)\right )} \]
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Time = 0.31 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.79
method | result | size |
risch | \(\frac {f^{a} {\mathrm e}^{-2 d} \sqrt {\pi }\, \operatorname {erf}\left (x \sqrt {2 f -c \ln \left (f \right )}\right )}{8 \sqrt {2 f -c \ln \left (f \right )}}+\frac {f^{a} {\mathrm e}^{2 d} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-c \ln \left (f \right )-2 f}\, x \right )}{8 \sqrt {-c \ln \left (f \right )-2 f}}-\frac {f^{a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-c \ln \left (f \right )}\, x \right )}{4 \sqrt {-c \ln \left (f \right )}}\) | \(101\) |
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Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (98) = 196\).
Time = 0.29 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.98 \[ \int f^{a+c x^2} \sinh ^2\left (d+f x^2\right ) \, dx=-\frac {{\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} + 2 \, c f \log \left (f\right )\right )} \cosh \left (a \log \left (f\right ) - 2 \, d\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} + 2 \, c f \log \left (f\right )\right )} \sinh \left (a \log \left (f\right ) - 2 \, d\right )\right )} \sqrt {-c \log \left (f\right ) + 2 \, f} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + 2 \, f} x\right ) + {\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 2 \, c f \log \left (f\right )\right )} \cosh \left (a \log \left (f\right ) + 2 \, d\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 2 \, c f \log \left (f\right )\right )} \sinh \left (a \log \left (f\right ) + 2 \, d\right )\right )} \sqrt {-c \log \left (f\right ) - 2 \, f} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - 2 \, f} x\right ) - 2 \, {\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 4 \, f^{2}\right )} \cosh \left (a \log \left (f\right )\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 4 \, f^{2}\right )} \sinh \left (a \log \left (f\right )\right )\right )} \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x\right )}{8 \, {\left (c^{3} \log \left (f\right )^{3} - 4 \, c f^{2} \log \left (f\right )\right )}} \]
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\[ \int f^{a+c x^2} \sinh ^2\left (d+f x^2\right ) \, dx=\int f^{a + c x^{2}} \sinh ^{2}{\left (d + f x^{2} \right )}\, dx \]
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Time = 0.18 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.78 \[ \int f^{a+c x^2} \sinh ^2\left (d+f x^2\right ) \, dx=\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - 2 \, f} x\right ) e^{\left (2 \, d\right )}}{8 \, \sqrt {-c \log \left (f\right ) - 2 \, f}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + 2 \, f} x\right ) e^{\left (-2 \, d\right )}}{8 \, \sqrt {-c \log \left (f\right ) + 2 \, f}} - \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x\right )}{4 \, \sqrt {-c \log \left (f\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.84 \[ \int f^{a+c x^2} \sinh ^2\left (d+f x^2\right ) \, dx=\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-c \log \left (f\right )} x\right )}{4 \, \sqrt {-c \log \left (f\right )}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right ) - 2 \, f} x\right ) e^{\left (a \log \left (f\right ) + 2 \, d\right )}}{8 \, \sqrt {-c \log \left (f\right ) - 2 \, f}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right ) + 2 \, f} x\right ) e^{\left (a \log \left (f\right ) - 2 \, d\right )}}{8 \, \sqrt {-c \log \left (f\right ) + 2 \, f}} \]
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Timed out. \[ \int f^{a+c x^2} \sinh ^2\left (d+f x^2\right ) \, dx=\int f^{c\,x^2+a}\,{\mathrm {sinh}\left (f\,x^2+d\right )}^2 \,d x \]
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