Integrand size = 18, antiderivative size = 148 \[ \int f^{a+b x} \sinh ^2\left (d+f x^2\right ) \, dx=\frac {1}{8} e^{-2 d+\frac {b^2 \log ^2(f)}{8 f}} f^{-\frac {1}{2}+a} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {4 f x-b \log (f)}{2 \sqrt {2} \sqrt {f}}\right )+\frac {1}{8} e^{2 d-\frac {b^2 \log ^2(f)}{8 f}} f^{-\frac {1}{2}+a} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {4 f x+b \log (f)}{2 \sqrt {2} \sqrt {f}}\right )-\frac {f^{a+b x}}{2 b \log (f)} \]
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Time = 0.13 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5623, 2225, 2325, 2266, 2236, 2235} \[ \int f^{a+b x} \sinh ^2\left (d+f x^2\right ) \, dx=\frac {1}{8} \sqrt {\frac {\pi }{2}} f^{a-\frac {1}{2}} e^{\frac {b^2 \log ^2(f)}{8 f}-2 d} \text {erf}\left (\frac {4 f x-b \log (f)}{2 \sqrt {2} \sqrt {f}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} f^{a-\frac {1}{2}} e^{2 d-\frac {b^2 \log ^2(f)}{8 f}} \text {erfi}\left (\frac {b \log (f)+4 f x}{2 \sqrt {2} \sqrt {f}}\right )-\frac {f^{a+b x}}{2 b \log (f)} \]
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Rule 2225
Rule 2235
Rule 2236
Rule 2266
Rule 2325
Rule 5623
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{2} f^{a+b x}+\frac {1}{4} e^{-2 d-2 f x^2} f^{a+b x}+\frac {1}{4} e^{2 d+2 f x^2} f^{a+b x}\right ) \, dx \\ & = \frac {1}{4} \int e^{-2 d-2 f x^2} f^{a+b x} \, dx+\frac {1}{4} \int e^{2 d+2 f x^2} f^{a+b x} \, dx-\frac {1}{2} \int f^{a+b x} \, dx \\ & = -\frac {f^{a+b x}}{2 b \log (f)}+\frac {1}{4} \int e^{-2 d-2 f x^2+a \log (f)+b x \log (f)} \, dx+\frac {1}{4} \int e^{2 d+2 f x^2+a \log (f)+b x \log (f)} \, dx \\ & = -\frac {f^{a+b x}}{2 b \log (f)}+\frac {1}{4} \left (e^{2 d-\frac {b^2 \log ^2(f)}{8 f}} f^a\right ) \int e^{\frac {(4 f x+b \log (f))^2}{8 f}} \, dx+\frac {1}{4} \left (e^{-2 d+\frac {b^2 \log ^2(f)}{8 f}} f^a\right ) \int e^{-\frac {(-4 f x+b \log (f))^2}{8 f}} \, dx \\ & = \frac {1}{8} e^{-2 d+\frac {b^2 \log ^2(f)}{8 f}} f^{-\frac {1}{2}+a} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {4 f x-b \log (f)}{2 \sqrt {2} \sqrt {f}}\right )+\frac {1}{8} e^{2 d-\frac {b^2 \log ^2(f)}{8 f}} f^{-\frac {1}{2}+a} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {4 f x+b \log (f)}{2 \sqrt {2} \sqrt {f}}\right )-\frac {f^{a+b x}}{2 b \log (f)} \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.01 \[ \int f^{a+b x} \sinh ^2\left (d+f x^2\right ) \, dx=\frac {1}{16} f^a \left (-\frac {8 f^{b x}}{b \log (f)}+\frac {e^{\frac {b^2 \log ^2(f)}{8 f}} \sqrt {2 \pi } \text {erf}\left (\frac {4 f x-b \log (f)}{2 \sqrt {2} \sqrt {f}}\right ) (\cosh (2 d)-\sinh (2 d))}{\sqrt {f}}+\frac {e^{-\frac {b^2 \log ^2(f)}{8 f}} \sqrt {2 \pi } \text {erfi}\left (\frac {4 f x+b \log (f)}{2 \sqrt {2} \sqrt {f}}\right ) (\cosh (2 d)+\sinh (2 d))}{\sqrt {f}}\right ) \]
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Time = 0.77 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.85
method | result | size |
risch | \(-\frac {\operatorname {erf}\left (-\sqrt {2}\, \sqrt {f}\, x +\frac {\ln \left (f \right ) b \sqrt {2}}{4 \sqrt {f}}\right ) \sqrt {2}\, \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {b^{2} \ln \left (f \right )^{2}-16 d f}{8 f}}}{16 \sqrt {f}}-\frac {\operatorname {erf}\left (-\sqrt {-2 f}\, x +\frac {\ln \left (f \right ) b}{2 \sqrt {-2 f}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}-16 d f}{8 f}}}{8 \sqrt {-2 f}}-\frac {f^{a} f^{b x}}{2 b \ln \left (f \right )}\) | \(126\) |
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Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (114) = 228\).
Time = 0.31 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.88 \[ \int f^{a+b x} \sinh ^2\left (d+f x^2\right ) \, dx=-\frac {\sqrt {2} \sqrt {\pi } b \sqrt {-f} \cosh \left (\frac {b^{2} \log \left (f\right )^{2} - 8 \, a f \log \left (f\right ) - 16 \, d f}{8 \, f}\right ) \operatorname {erf}\left (\frac {\sqrt {2} {\left (4 \, f x + b \log \left (f\right )\right )} \sqrt {-f}}{4 \, f}\right ) \log \left (f\right ) + \sqrt {2} \sqrt {\pi } b \sqrt {f} \cosh \left (\frac {b^{2} \log \left (f\right )^{2} + 8 \, a f \log \left (f\right ) - 16 \, d f}{8 \, f}\right ) \operatorname {erf}\left (-\frac {\sqrt {2} {\left (4 \, f x - b \log \left (f\right )\right )}}{4 \, \sqrt {f}}\right ) \log \left (f\right ) + \sqrt {2} \sqrt {\pi } b \sqrt {f} \operatorname {erf}\left (-\frac {\sqrt {2} {\left (4 \, f x - b \log \left (f\right )\right )}}{4 \, \sqrt {f}}\right ) \log \left (f\right ) \sinh \left (\frac {b^{2} \log \left (f\right )^{2} + 8 \, a f \log \left (f\right ) - 16 \, d f}{8 \, f}\right ) - \sqrt {2} \sqrt {\pi } b \sqrt {-f} \operatorname {erf}\left (\frac {\sqrt {2} {\left (4 \, f x + b \log \left (f\right )\right )} \sqrt {-f}}{4 \, f}\right ) \log \left (f\right ) \sinh \left (\frac {b^{2} \log \left (f\right )^{2} - 8 \, a f \log \left (f\right ) - 16 \, d f}{8 \, f}\right ) + 8 \, f \cosh \left ({\left (b x + a\right )} \log \left (f\right )\right ) + 8 \, f \sinh \left ({\left (b x + a\right )} \log \left (f\right )\right )}{16 \, b f \log \left (f\right )} \]
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\[ \int f^{a+b x} \sinh ^2\left (d+f x^2\right ) \, dx=\int f^{a + b x} \sinh ^{2}{\left (d + f x^{2} \right )}\, dx \]
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none
Time = 0.26 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.86 \[ \int f^{a+b x} \sinh ^2\left (d+f x^2\right ) \, dx=\frac {\sqrt {2} \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {2} \sqrt {f} x - \frac {\sqrt {2} b \log \left (f\right )}{4 \, \sqrt {f}}\right ) e^{\left (\frac {b^{2} \log \left (f\right )^{2}}{8 \, f} - 2 \, d\right )}}{16 \, \sqrt {f}} + \frac {\sqrt {2} \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {2} \sqrt {-f} x - \frac {\sqrt {2} b \log \left (f\right )}{4 \, \sqrt {-f}}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2}}{8 \, f} + 2 \, d\right )}}{16 \, \sqrt {-f}} - \frac {f^{b x + a}}{2 \, b \log \left (f\right )} \]
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.41 \[ \int f^{a+b x} \sinh ^2\left (d+f x^2\right ) \, dx=-\frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{4} \, \sqrt {2} \sqrt {f} {\left (4 \, x - \frac {b \log \left (f\right )}{f}\right )}\right ) e^{\left (\frac {b^{2} \log \left (f\right )^{2} + 8 \, a f \log \left (f\right ) - 16 \, d f}{8 \, f}\right )}}{16 \, \sqrt {f}} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{4} \, \sqrt {2} \sqrt {-f} {\left (4 \, x + \frac {b \log \left (f\right )}{f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 8 \, a f \log \left (f\right ) - 16 \, d f}{8 \, f}\right )}}{16 \, \sqrt {-f}} - {\left (\frac {2 \, b \cos \left (-\frac {1}{2} \, \pi b x \mathrm {sgn}\left (f\right ) + \frac {1}{2} \, \pi b x - \frac {1}{2} \, \pi a \mathrm {sgn}\left (f\right ) + \frac {1}{2} \, \pi a\right ) \log \left ({\left | f \right |}\right )}{4 \, b^{2} \log \left ({\left | f \right |}\right )^{2} + {\left (\pi b \mathrm {sgn}\left (f\right ) - \pi b\right )}^{2}} - \frac {{\left (\pi b \mathrm {sgn}\left (f\right ) - \pi b\right )} \sin \left (-\frac {1}{2} \, \pi b x \mathrm {sgn}\left (f\right ) + \frac {1}{2} \, \pi b x - \frac {1}{2} \, \pi a \mathrm {sgn}\left (f\right ) + \frac {1}{2} \, \pi a\right )}{4 \, b^{2} \log \left ({\left | f \right |}\right )^{2} + {\left (\pi b \mathrm {sgn}\left (f\right ) - \pi b\right )}^{2}}\right )} e^{\left (b x \log \left ({\left | f \right |}\right ) + a \log \left ({\left | f \right |}\right )\right )} + i \, {\left (-\frac {i \, e^{\left (\frac {1}{2} i \, \pi b x \mathrm {sgn}\left (f\right ) - \frac {1}{2} i \, \pi b x + \frac {1}{2} i \, \pi a \mathrm {sgn}\left (f\right ) - \frac {1}{2} i \, \pi a\right )}}{2 i \, \pi b \mathrm {sgn}\left (f\right ) - 2 i \, \pi b + 4 \, b \log \left ({\left | f \right |}\right )} + \frac {i \, e^{\left (-\frac {1}{2} i \, \pi b x \mathrm {sgn}\left (f\right ) + \frac {1}{2} i \, \pi b x - \frac {1}{2} i \, \pi a \mathrm {sgn}\left (f\right ) + \frac {1}{2} i \, \pi a\right )}}{-2 i \, \pi b \mathrm {sgn}\left (f\right ) + 2 i \, \pi b + 4 \, b \log \left ({\left | f \right |}\right )}\right )} e^{\left (b x \log \left ({\left | f \right |}\right ) + a \log \left ({\left | f \right |}\right )\right )} \]
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Timed out. \[ \int f^{a+b x} \sinh ^2\left (d+f x^2\right ) \, dx=\int f^{a+b\,x}\,{\mathrm {sinh}\left (f\,x^2+d\right )}^2 \,d x \]
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