Optimal. Leaf size=257 \[ \frac {3}{16} \sqrt {\pi } f^{a-\frac {1}{2}} e^{\frac {(e-b \log (f))^2}{4 f}-d} \text {erf}\left (\frac {-b \log (f)+e+2 f x}{2 \sqrt {f}}\right )-\frac {1}{16} \sqrt {\frac {\pi }{3}} f^{a-\frac {1}{2}} e^{\frac {(3 e-b \log (f))^2}{12 f}-3 d} \text {erf}\left (\frac {-b \log (f)+3 e+6 f x}{2 \sqrt {3} \sqrt {f}}\right )-\frac {3}{16} \sqrt {\pi } f^{a-\frac {1}{2}} e^{d-\frac {(b \log (f)+e)^2}{4 f}} \text {erfi}\left (\frac {b \log (f)+e+2 f x}{2 \sqrt {f}}\right )+\frac {1}{16} \sqrt {\frac {\pi }{3}} f^{a-\frac {1}{2}} e^{3 d-\frac {(b \log (f)+3 e)^2}{12 f}} \text {erfi}\left (\frac {b \log (f)+3 e+6 f x}{2 \sqrt {3} \sqrt {f}}\right ) \]
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Rubi [A] time = 0.48, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5512, 2287, 2234, 2205, 2204} \[ \frac {3}{16} \sqrt {\pi } f^{a-\frac {1}{2}} e^{\frac {(e-b \log (f))^2}{4 f}-d} \text {Erf}\left (\frac {-b \log (f)+e+2 f x}{2 \sqrt {f}}\right )-\frac {1}{16} \sqrt {\frac {\pi }{3}} f^{a-\frac {1}{2}} e^{\frac {(3 e-b \log (f))^2}{12 f}-3 d} \text {Erf}\left (\frac {-b \log (f)+3 e+6 f x}{2 \sqrt {3} \sqrt {f}}\right )-\frac {3}{16} \sqrt {\pi } f^{a-\frac {1}{2}} e^{d-\frac {(b \log (f)+e)^2}{4 f}} \text {Erfi}\left (\frac {b \log (f)+e+2 f x}{2 \sqrt {f}}\right )+\frac {1}{16} \sqrt {\frac {\pi }{3}} f^{a-\frac {1}{2}} e^{3 d-\frac {(b \log (f)+3 e)^2}{12 f}} \text {Erfi}\left (\frac {b \log (f)+3 e+6 f x}{2 \sqrt {3} \sqrt {f}}\right ) \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 2234
Rule 2287
Rule 5512
Rubi steps
\begin {align*} \int f^{a+b x} \sinh ^3\left (d+e x+f x^2\right ) \, dx &=\int \left (-\frac {1}{8} e^{-3 \left (d+e x+f x^2\right )} f^{a+b x}+\frac {3}{8} \exp \left (2 d+2 e x+2 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+b x}-\frac {3}{8} \exp \left (4 d+4 e x+4 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+b x}+\frac {1}{8} \exp \left (6 d+6 e x+6 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+b x}\right ) \, dx\\ &=-\left (\frac {1}{8} \int e^{-3 \left (d+e x+f x^2\right )} f^{a+b x} \, dx\right )+\frac {1}{8} \int \exp \left (6 d+6 e x+6 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+b x} \, dx+\frac {3}{8} \int \exp \left (2 d+2 e x+2 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+b x} \, dx-\frac {3}{8} \int \exp \left (4 d+4 e x+4 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+b x} \, dx\\ &=-\left (\frac {1}{8} \int \exp \left (-3 d-3 f x^2+a \log (f)-x (3 e-b \log (f))\right ) \, dx\right )+\frac {1}{8} \int \exp \left (3 d+3 f x^2+a \log (f)+x (3 e+b \log (f))\right ) \, dx+\frac {3}{8} \int e^{-d-f x^2+a \log (f)-x (e-b \log (f))} \, dx-\frac {3}{8} \int e^{d+f x^2+a \log (f)+x (e+b \log (f))} \, dx\\ &=\frac {1}{8} \left (3 e^{-d+\frac {(e-b \log (f))^2}{4 f}} f^a\right ) \int e^{-\frac {(-e-2 f x+b \log (f))^2}{4 f}} \, dx-\frac {1}{8} \left (e^{-3 d+\frac {(3 e-b \log (f))^2}{12 f}} f^a\right ) \int e^{-\frac {(-3 e-6 f x+b \log (f))^2}{12 f}} \, dx-\frac {1}{8} \left (3 e^{d-\frac {(e+b \log (f))^2}{4 f}} f^a\right ) \int e^{\frac {(e+2 f x+b \log (f))^2}{4 f}} \, dx+\frac {1}{8} \left (e^{3 d-\frac {(3 e+b \log (f))^2}{12 f}} f^a\right ) \int e^{\frac {(3 e+6 f x+b \log (f))^2}{12 f}} \, dx\\ &=\frac {3}{16} e^{-d+\frac {(e-b \log (f))^2}{4 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erf}\left (\frac {e+2 f x-b \log (f)}{2 \sqrt {f}}\right )-\frac {1}{16} e^{-3 d+\frac {(3 e-b \log (f))^2}{12 f}} f^{-\frac {1}{2}+a} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {3 e+6 f x-b \log (f)}{2 \sqrt {3} \sqrt {f}}\right )-\frac {3}{16} e^{d-\frac {(e+b \log (f))^2}{4 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erfi}\left (\frac {e+2 f x+b \log (f)}{2 \sqrt {f}}\right )+\frac {1}{16} e^{3 d-\frac {(3 e+b \log (f))^2}{12 f}} f^{-\frac {1}{2}+a} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {3 e+6 f x+b \log (f)}{2 \sqrt {3} \sqrt {f}}\right )\\ \end {align*}
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Mathematica [A] time = 0.78, size = 354, normalized size = 1.38 \[ \frac {1}{16} \sqrt {\frac {\pi }{3}} f^{a-\frac {b e+f}{2 f}} e^{-\frac {b^2 \log ^2(f)+3 e^2}{4 f}} \left (3 \sqrt {3} (\cosh (d)-\sinh (d)) e^{\frac {b^2 \log ^2(f)+2 e^2}{2 f}} \text {erf}\left (\frac {-b \log (f)+e+2 f x}{2 \sqrt {f}}\right )-(\cosh (3 d)-\sinh (3 d)) e^{\frac {2 b^2 \log ^2(f)+9 e^2}{6 f}} \text {erf}\left (\frac {-b \log (f)+3 e+6 f x}{2 \sqrt {3} \sqrt {f}}\right )+\sinh (3 d) e^{\frac {b^2 \log ^2(f)}{6 f}} \text {erfi}\left (\frac {b \log (f)+3 e+6 f x}{2 \sqrt {3} \sqrt {f}}\right )+\cosh (3 d) e^{\frac {b^2 \log ^2(f)}{6 f}} \text {erfi}\left (\frac {b \log (f)+3 e+6 f x}{2 \sqrt {3} \sqrt {f}}\right )-3 \sqrt {3} \sinh (d) e^{\frac {e^2}{2 f}} \text {erfi}\left (\frac {b \log (f)+e+2 f x}{2 \sqrt {f}}\right )-3 \sqrt {3} \cosh (d) e^{\frac {e^2}{2 f}} \text {erfi}\left (\frac {b \log (f)+e+2 f x}{2 \sqrt {f}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 541, normalized size = 2.11 \[ -\frac {\sqrt {3} \sqrt {\pi } \sqrt {-f} \cosh \left (\frac {b^{2} \log \relax (f)^{2} + 9 \, e^{2} - 36 \, d f + 6 \, {\left (b e - 2 \, a f\right )} \log \relax (f)}{12 \, f}\right ) \operatorname {erf}\left (\frac {\sqrt {3} {\left (6 \, f x + b \log \relax (f) + 3 \, e\right )} \sqrt {-f}}{6 \, f}\right ) - \sqrt {3} \sqrt {\pi } \sqrt {f} \cosh \left (\frac {b^{2} \log \relax (f)^{2} + 9 \, e^{2} - 36 \, d f - 6 \, {\left (b e - 2 \, a f\right )} \log \relax (f)}{12 \, f}\right ) \operatorname {erf}\left (-\frac {\sqrt {3} {\left (6 \, f x - b \log \relax (f) + 3 \, e\right )}}{6 \, \sqrt {f}}\right ) - \sqrt {3} \sqrt {\pi } \sqrt {-f} \operatorname {erf}\left (\frac {\sqrt {3} {\left (6 \, f x + b \log \relax (f) + 3 \, e\right )} \sqrt {-f}}{6 \, f}\right ) \sinh \left (\frac {b^{2} \log \relax (f)^{2} + 9 \, e^{2} - 36 \, d f + 6 \, {\left (b e - 2 \, a f\right )} \log \relax (f)}{12 \, f}\right ) - \sqrt {3} \sqrt {\pi } \sqrt {f} \operatorname {erf}\left (-\frac {\sqrt {3} {\left (6 \, f x - b \log \relax (f) + 3 \, e\right )}}{6 \, \sqrt {f}}\right ) \sinh \left (\frac {b^{2} \log \relax (f)^{2} + 9 \, e^{2} - 36 \, d f - 6 \, {\left (b e - 2 \, a f\right )} \log \relax (f)}{12 \, f}\right ) - 9 \, \sqrt {\pi } \sqrt {-f} \cosh \left (\frac {b^{2} \log \relax (f)^{2} + e^{2} - 4 \, d f + 2 \, {\left (b e - 2 \, a f\right )} \log \relax (f)}{4 \, f}\right ) \operatorname {erf}\left (\frac {{\left (2 \, f x + b \log \relax (f) + e\right )} \sqrt {-f}}{2 \, f}\right ) + 9 \, \sqrt {\pi } \sqrt {f} \cosh \left (\frac {b^{2} \log \relax (f)^{2} + e^{2} - 4 \, d f - 2 \, {\left (b e - 2 \, a f\right )} \log \relax (f)}{4 \, f}\right ) \operatorname {erf}\left (-\frac {2 \, f x - b \log \relax (f) + e}{2 \, \sqrt {f}}\right ) + 9 \, \sqrt {\pi } \sqrt {-f} \operatorname {erf}\left (\frac {{\left (2 \, f x + b \log \relax (f) + e\right )} \sqrt {-f}}{2 \, f}\right ) \sinh \left (\frac {b^{2} \log \relax (f)^{2} + e^{2} - 4 \, d f + 2 \, {\left (b e - 2 \, a f\right )} \log \relax (f)}{4 \, f}\right ) + 9 \, \sqrt {\pi } \sqrt {f} \operatorname {erf}\left (-\frac {2 \, f x - b \log \relax (f) + e}{2 \, \sqrt {f}}\right ) \sinh \left (\frac {b^{2} \log \relax (f)^{2} + e^{2} - 4 \, d f - 2 \, {\left (b e - 2 \, a f\right )} \log \relax (f)}{4 \, f}\right )}{48 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 285, normalized size = 1.11 \[ \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{6} \, \sqrt {3} \sqrt {f} {\left (6 \, x - \frac {b \log \relax (f) - 3 \, e}{f}\right )}\right ) e^{\left (\frac {b^{2} \log \relax (f)^{2} + 12 \, a f \log \relax (f) - 6 \, b e \log \relax (f) - 36 \, d f + 9 \, e^{2}}{12 \, f}\right )}}{48 \, \sqrt {f}} - \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{6} \, \sqrt {3} \sqrt {-f} {\left (6 \, x + \frac {b \log \relax (f) + 3 \, e}{f}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f)^{2} - 12 \, a f \log \relax (f) + 6 \, b e \log \relax (f) - 36 \, d f + 9 \, e^{2}}{12 \, f}\right )}}{48 \, \sqrt {-f}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {f} {\left (2 \, x - \frac {b \log \relax (f) - e}{f}\right )}\right ) e^{\left (\frac {b^{2} \log \relax (f)^{2} + 4 \, a f \log \relax (f) - 2 \, b e \log \relax (f) - 4 \, d f + e^{2}}{4 \, f}\right )}}{16 \, \sqrt {f}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-f} {\left (2 \, x + \frac {b \log \relax (f) + e}{f}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f)^{2} - 4 \, a f \log \relax (f) + 2 \, b e \log \relax (f) - 4 \, d f + e^{2}}{4 \, f}\right )}}{16 \, \sqrt {-f}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 265, normalized size = 1.03 \[ -\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}+6 \ln \relax (f ) b e -36 d f +9 e^{2}}{12 f}} \erf \left (-\sqrt {-3 f}\, x +\frac {3 e +b \ln \relax (f )}{2 \sqrt {-3 f}}\right )}{16 \sqrt {-3 f}}+\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {\ln \relax (f )^{2} b^{2}-6 \ln \relax (f ) b e -36 d f +9 e^{2}}{12 f}} \sqrt {3}\, \erf \left (-\sqrt {3}\, \sqrt {f}\, x +\frac {\left (b \ln \relax (f )-3 e \right ) \sqrt {3}}{6 \sqrt {f}}\right )}{48 \sqrt {f}}-\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {\ln \relax (f )^{2} b^{2}-2 \ln \relax (f ) b e -4 d f +e^{2}}{4 f}} \erf \left (-\sqrt {f}\, x +\frac {b \ln \relax (f )-e}{2 \sqrt {f}}\right )}{16 \sqrt {f}}+\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}+2 \ln \relax (f ) b e -4 d f +e^{2}}{4 f}} \erf \left (-\sqrt {-f}\, x +\frac {e +b \ln \relax (f )}{2 \sqrt {-f}}\right )}{16 \sqrt {-f}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 228, normalized size = 0.89 \[ \frac {\sqrt {3} \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {3} \sqrt {-f} x - \frac {\sqrt {3} {\left (b \log \relax (f) + 3 \, e\right )}}{6 \, \sqrt {-f}}\right ) e^{\left (3 \, d - \frac {{\left (b \log \relax (f) + 3 \, e\right )}^{2}}{12 \, f}\right )}}{48 \, \sqrt {-f}} + \frac {3}{16} \, \sqrt {\pi } f^{a - \frac {1}{2}} \operatorname {erf}\left (\sqrt {f} x - \frac {b \log \relax (f) - e}{2 \, \sqrt {f}}\right ) e^{\left (-d + \frac {{\left (b \log \relax (f) - e\right )}^{2}}{4 \, f}\right )} - \frac {\sqrt {3} \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {3} \sqrt {f} x - \frac {\sqrt {3} {\left (b \log \relax (f) - 3 \, e\right )}}{6 \, \sqrt {f}}\right ) e^{\left (-3 \, d + \frac {{\left (b \log \relax (f) - 3 \, e\right )}^{2}}{12 \, f}\right )}}{48 \, \sqrt {f}} - \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-f} x - \frac {b \log \relax (f) + e}{2 \, \sqrt {-f}}\right ) e^{\left (d - \frac {{\left (b \log \relax (f) + e\right )}^{2}}{4 \, f}\right )}}{16 \, \sqrt {-f}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int f^{a+b\,x}\,{\mathrm {sinh}\left (f\,x^2+e\,x+d\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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