Optimal. Leaf size=300 \[ \frac {3 e^{-d+\frac {e^2}{4 f-4 c \log (f)}} f^a \sqrt {\pi } \text {Erf}\left (\frac {e+2 x (f-c \log (f))}{2 \sqrt {f-c \log (f)}}\right )}{16 \sqrt {f-c \log (f)}}+\frac {e^{-3 d+\frac {9 e^2}{12 f-4 c \log (f)}} f^a \sqrt {\pi } \text {Erf}\left (\frac {3 e+2 x (3 f-c \log (f))}{2 \sqrt {3 f-c \log (f)}}\right )}{16 \sqrt {3 f-c \log (f)}}+\frac {3 e^{d-\frac {e^2}{4 (f+c \log (f))}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {e+2 x (f+c \log (f))}{2 \sqrt {f+c \log (f)}}\right )}{16 \sqrt {f+c \log (f)}}+\frac {e^{3 d-\frac {9 e^2}{4 (3 f+c \log (f))}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {3 e+2 x (3 f+c \log (f))}{2 \sqrt {3 f+c \log (f)}}\right )}{16 \sqrt {3 f+c \log (f)}} \]
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Rubi [A]
time = 0.48, antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {5624, 2325,
2266, 2236, 2235} \begin {gather*} \frac {3 \sqrt {\pi } f^a e^{\frac {e^2}{4 f-4 c \log (f)}-d} \text {Erf}\left (\frac {2 x (f-c \log (f))+e}{2 \sqrt {f-c \log (f)}}\right )}{16 \sqrt {f-c \log (f)}}+\frac {\sqrt {\pi } f^a e^{\frac {9 e^2}{12 f-4 c \log (f)}-3 d} \text {Erf}\left (\frac {2 x (3 f-c \log (f))+3 e}{2 \sqrt {3 f-c \log (f)}}\right )}{16 \sqrt {3 f-c \log (f)}}+\frac {3 \sqrt {\pi } f^a e^{d-\frac {e^2}{4 (c \log (f)+f)}} \text {Erfi}\left (\frac {2 x (c \log (f)+f)+e}{2 \sqrt {c \log (f)+f}}\right )}{16 \sqrt {c \log (f)+f}}+\frac {\sqrt {\pi } f^a e^{3 d-\frac {9 e^2}{4 (c \log (f)+3 f)}} \text {Erfi}\left (\frac {2 x (c \log (f)+3 f)+3 e}{2 \sqrt {c \log (f)+3 f}}\right )}{16 \sqrt {c \log (f)+3 f}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2236
Rule 2266
Rule 2325
Rule 5624
Rubi steps
\begin {align*} \int f^{a+c x^2} \cosh ^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+c x^2}+\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+c x^2}+\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+c x^2}+\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+c x^2}\right ) \, dx\\ &=\frac {1}{8} \int e^{-3 \left (d+e x+f x^2\right )} f^{a+c x^2} \, dx+\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+c x^2} \, 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+c x^2} \, 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+c x^2} \, dx\\ &=\frac {1}{8} \int \exp \left (-3 d-3 e x+a \log (f)-x^2 (3 f-c \log (f))\right ) \, dx+\frac {1}{8} \int \exp \left (3 d+3 e x+a \log (f)+x^2 (3 f+c \log (f))\right ) \, dx+\frac {3}{8} \int e^{-d-e x+a \log (f)-x^2 (f-c \log (f))} \, dx+\frac {3}{8} \int e^{d+e x+a \log (f)+x^2 (f+c \log (f))} \, dx\\ &=\frac {1}{8} \left (3 e^{-d+\frac {e^2}{4 f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(-e+2 x (-f+c \log (f)))^2}{4 (-f+c \log (f))}\right ) \, dx+\frac {1}{8} \left (e^{-3 d+\frac {9 e^2}{12 f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(-3 e+2 x (-3 f+c \log (f)))^2}{4 (-3 f+c \log (f))}\right ) \, dx+\frac {1}{8} \left (3 e^{d-\frac {e^2}{4 (f+c \log (f))}} f^a\right ) \int \exp \left (\frac {(e+2 x (f+c \log (f)))^2}{4 (f+c \log (f))}\right ) \, dx+\frac {1}{8} \left (e^{3 d-\frac {9 e^2}{4 (3 f+c \log (f))}} f^a\right ) \int \exp \left (\frac {(3 e+2 x (3 f+c \log (f)))^2}{4 (3 f+c \log (f))}\right ) \, dx\\ &=\frac {3 e^{-d+\frac {e^2}{4 f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {e+2 x (f-c \log (f))}{2 \sqrt {f-c \log (f)}}\right )}{16 \sqrt {f-c \log (f)}}+\frac {e^{-3 d+\frac {9 e^2}{12 f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {3 e+2 x (3 f-c \log (f))}{2 \sqrt {3 f-c \log (f)}}\right )}{16 \sqrt {3 f-c \log (f)}}+\frac {3 e^{d-\frac {e^2}{4 (f+c \log (f))}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e+2 x (f+c \log (f))}{2 \sqrt {f+c \log (f)}}\right )}{16 \sqrt {f+c \log (f)}}+\frac {e^{3 d-\frac {9 e^2}{4 (3 f+c \log (f))}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 e+2 x (3 f+c \log (f))}{2 \sqrt {3 f+c \log (f)}}\right )}{16 \sqrt {3 f+c \log (f)}}\\ \end {align*}
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Mathematica [A]
time = 4.06, size = 478, normalized size = 1.59 \begin {gather*} \frac {e^{-\frac {1}{4} e^2 \left (\frac {1}{f+c \log (f)}+\frac {9}{3 f+c \log (f)}\right )} f^a \sqrt {\pi } \left (3 e^{\frac {1}{4} e^2 \left (\frac {1}{f-c \log (f)}+\frac {1}{f+c \log (f)}+\frac {9}{3 f+c \log (f)}\right )} \text {Erf}\left (\frac {e+2 f x-2 c x \log (f)}{2 \sqrt {f-c \log (f)}}\right ) \sqrt {f-c \log (f)} \left (9 f^3+9 c f^2 \log (f)-c^2 f \log ^2(f)-c^3 \log ^3(f)\right ) (\cosh (d)-\sinh (d))+(f-c \log (f)) \left (e^{\frac {1}{4} e^2 \left (\frac {9}{3 f-c \log (f)}+\frac {1}{f+c \log (f)}+\frac {9}{3 f+c \log (f)}\right )} \text {Erf}\left (\frac {3 e+6 f x-2 c x \log (f)}{2 \sqrt {3 f-c \log (f)}}\right ) \sqrt {3 f-c \log (f)} \left (3 f^2+4 c f \log (f)+c^2 \log ^2(f)\right ) (\cosh (3 d)-\sinh (3 d))+(3 f-c \log (f)) \left (3 e^{\frac {9 e^2}{4 (3 f+c \log (f))}} \text {Erfi}\left (\frac {e+2 f x+2 c x \log (f)}{2 \sqrt {f+c \log (f)}}\right ) \sqrt {f+c \log (f)} (3 f+c \log (f)) (\cosh (d)+\sinh (d))+e^{\frac {e^2}{4 (f+c \log (f))}} \text {Erfi}\left (\frac {3 e+6 f x+2 c x \log (f)}{2 \sqrt {3 f+c \log (f)}}\right ) (f+c \log (f)) \sqrt {3 f+c \log (f)} (\cosh (3 d)+\sinh (3 d))\right )\right )\right )}{16 \left (9 f^4-10 c^2 f^2 \log ^2(f)+c^4 \log ^4(f)\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 2.70, size = 302, normalized size = 1.01
method | result | size |
risch | \(\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {3 \left (4 d \ln \left (f \right ) c -12 d f +3 e^{2}\right )}{4 \left (-3 f +c \ln \left (f \right )\right )}} \erf \left (x \sqrt {3 f -c \ln \left (f \right )}+\frac {3 e}{2 \sqrt {3 f -c \ln \left (f \right )}}\right )}{16 \sqrt {3 f -c \ln \left (f \right )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {3 d \ln \left (f \right ) c +9 d f -\frac {9 e^{2}}{4}}{3 f +c \ln \left (f \right )}} \erf \left (-\sqrt {-c \ln \left (f \right )-3 f}\, x +\frac {3 e}{2 \sqrt {-c \ln \left (f \right )-3 f}}\right )}{16 \sqrt {-c \ln \left (f \right )-3 f}}+\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {4 d \ln \left (f \right ) c -4 d f +e^{2}}{4 \left (-f +c \ln \left (f \right )\right )}} \erf \left (x \sqrt {f -c \ln \left (f \right )}+\frac {e}{2 \sqrt {f -c \ln \left (f \right )}}\right )}{16 \sqrt {f -c \ln \left (f \right )}}-\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {4 d \ln \left (f \right ) c +4 d f -e^{2}}{4 c \ln \left (f \right )+4 f}} \erf \left (-\sqrt {-c \ln \left (f \right )-f}\, x +\frac {e}{2 \sqrt {-c \ln \left (f \right )-f}}\right )}{16 \sqrt {-c \ln \left (f \right )-f}}\) | \(302\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 263, normalized size = 0.88 \begin {gather*} \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - 3 \, f} x - \frac {3 \, e}{2 \, \sqrt {-c \log \left (f\right ) - 3 \, f}}\right ) e^{\left (3 \, d - \frac {9 \, e^{2}}{4 \, {\left (c \log \left (f\right ) + 3 \, f\right )}}\right )}}{16 \, \sqrt {-c \log \left (f\right ) - 3 \, f}} + \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - f} x - \frac {e}{2 \, \sqrt {-c \log \left (f\right ) - f}}\right ) e^{\left (d - \frac {e^{2}}{4 \, {\left (c \log \left (f\right ) + f\right )}}\right )}}{16 \, \sqrt {-c \log \left (f\right ) - f}} + \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + f} x + \frac {e}{2 \, \sqrt {-c \log \left (f\right ) + f}}\right ) e^{\left (-d - \frac {e^{2}}{4 \, {\left (c \log \left (f\right ) - f\right )}}\right )}}{16 \, \sqrt {-c \log \left (f\right ) + f}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + 3 \, f} x + \frac {3 \, e}{2 \, \sqrt {-c \log \left (f\right ) + 3 \, f}}\right ) e^{\left (-3 \, d - \frac {9 \, e^{2}}{4 \, {\left (c \log \left (f\right ) - 3 \, f\right )}}\right )}}{16 \, \sqrt {-c \log \left (f\right ) + 3 \, f}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 969 vs.
\(2 (253) = 506\).
time = 0.50, size = 969, normalized size = 3.23 \begin {gather*} -\frac {{\left (\sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} + 3 \, c^{2} f \log \left (f\right )^{2} - c f^{2} \log \left (f\right ) - 3 \, f^{3}\right )} \cosh \left (\frac {4 \, a c \log \left (f\right )^{2} + 36 \, d f - 9 \, \cosh \left (1\right )^{2} - 12 \, {\left (c d + a f\right )} \log \left (f\right ) - 18 \, \cosh \left (1\right ) \sinh \left (1\right ) - 9 \, \sinh \left (1\right )^{2}}{4 \, {\left (c \log \left (f\right ) - 3 \, f\right )}}\right ) + \sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} + 3 \, c^{2} f \log \left (f\right )^{2} - c f^{2} \log \left (f\right ) - 3 \, f^{3}\right )} \sinh \left (\frac {4 \, a c \log \left (f\right )^{2} + 36 \, d f - 9 \, \cosh \left (1\right )^{2} - 12 \, {\left (c d + a f\right )} \log \left (f\right ) - 18 \, \cosh \left (1\right ) \sinh \left (1\right ) - 9 \, \sinh \left (1\right )^{2}}{4 \, {\left (c \log \left (f\right ) - 3 \, f\right )}}\right )\right )} \sqrt {-c \log \left (f\right ) + 3 \, f} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) - 6 \, f x - 3 \, \cosh \left (1\right ) - 3 \, \sinh \left (1\right )\right )} \sqrt {-c \log \left (f\right ) + 3 \, f}}{2 \, {\left (c \log \left (f\right ) - 3 \, f\right )}}\right ) + 3 \, {\left (\sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} + c^{2} f \log \left (f\right )^{2} - 9 \, c f^{2} \log \left (f\right ) - 9 \, f^{3}\right )} \cosh \left (\frac {4 \, a c \log \left (f\right )^{2} + 4 \, d f - \cosh \left (1\right )^{2} - 4 \, {\left (c d + a f\right )} \log \left (f\right ) - 2 \, \cosh \left (1\right ) \sinh \left (1\right ) - \sinh \left (1\right )^{2}}{4 \, {\left (c \log \left (f\right ) - f\right )}}\right ) + \sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} + c^{2} f \log \left (f\right )^{2} - 9 \, c f^{2} \log \left (f\right ) - 9 \, f^{3}\right )} \sinh \left (\frac {4 \, a c \log \left (f\right )^{2} + 4 \, d f - \cosh \left (1\right )^{2} - 4 \, {\left (c d + a f\right )} \log \left (f\right ) - 2 \, \cosh \left (1\right ) \sinh \left (1\right ) - \sinh \left (1\right )^{2}}{4 \, {\left (c \log \left (f\right ) - f\right )}}\right )\right )} \sqrt {-c \log \left (f\right ) + f} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) - 2 \, f x - \cosh \left (1\right ) - \sinh \left (1\right )\right )} \sqrt {-c \log \left (f\right ) + f}}{2 \, {\left (c \log \left (f\right ) - f\right )}}\right ) + 3 \, {\left (\sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} - c^{2} f \log \left (f\right )^{2} - 9 \, c f^{2} \log \left (f\right ) + 9 \, f^{3}\right )} \cosh \left (\frac {4 \, a c \log \left (f\right )^{2} + 4 \, d f - \cosh \left (1\right )^{2} + 4 \, {\left (c d + a f\right )} \log \left (f\right ) - 2 \, \cosh \left (1\right ) \sinh \left (1\right ) - \sinh \left (1\right )^{2}}{4 \, {\left (c \log \left (f\right ) + f\right )}}\right ) + \sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} - c^{2} f \log \left (f\right )^{2} - 9 \, c f^{2} \log \left (f\right ) + 9 \, f^{3}\right )} \sinh \left (\frac {4 \, a c \log \left (f\right )^{2} + 4 \, d f - \cosh \left (1\right )^{2} + 4 \, {\left (c d + a f\right )} \log \left (f\right ) - 2 \, \cosh \left (1\right ) \sinh \left (1\right ) - \sinh \left (1\right )^{2}}{4 \, {\left (c \log \left (f\right ) + f\right )}}\right )\right )} \sqrt {-c \log \left (f\right ) - f} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) + 2 \, f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )} \sqrt {-c \log \left (f\right ) - f}}{2 \, {\left (c \log \left (f\right ) + f\right )}}\right ) + {\left (\sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} - 3 \, c^{2} f \log \left (f\right )^{2} - c f^{2} \log \left (f\right ) + 3 \, f^{3}\right )} \cosh \left (\frac {4 \, a c \log \left (f\right )^{2} + 36 \, d f - 9 \, \cosh \left (1\right )^{2} + 12 \, {\left (c d + a f\right )} \log \left (f\right ) - 18 \, \cosh \left (1\right ) \sinh \left (1\right ) - 9 \, \sinh \left (1\right )^{2}}{4 \, {\left (c \log \left (f\right ) + 3 \, f\right )}}\right ) + \sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} - 3 \, c^{2} f \log \left (f\right )^{2} - c f^{2} \log \left (f\right ) + 3 \, f^{3}\right )} \sinh \left (\frac {4 \, a c \log \left (f\right )^{2} + 36 \, d f - 9 \, \cosh \left (1\right )^{2} + 12 \, {\left (c d + a f\right )} \log \left (f\right ) - 18 \, \cosh \left (1\right ) \sinh \left (1\right ) - 9 \, \sinh \left (1\right )^{2}}{4 \, {\left (c \log \left (f\right ) + 3 \, f\right )}}\right )\right )} \sqrt {-c \log \left (f\right ) - 3 \, f} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) + 6 \, f x + 3 \, \cosh \left (1\right ) + 3 \, \sinh \left (1\right )\right )} \sqrt {-c \log \left (f\right ) - 3 \, f}}{2 \, {\left (c \log \left (f\right ) + 3 \, f\right )}}\right )}{16 \, {\left (c^{4} \log \left (f\right )^{4} - 10 \, c^{2} f^{2} \log \left (f\right )^{2} + 9 \, f^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 352, normalized size = 1.17 \begin {gather*} -\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) - 3 \, f} {\left (2 \, x + \frac {3 \, e}{c \log \left (f\right ) + 3 \, f}\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} + 12 \, c d \log \left (f\right ) + 12 \, a f \log \left (f\right ) - 9 \, e^{2} + 36 \, d f}{4 \, {\left (c \log \left (f\right ) + 3 \, f\right )}}\right )}}{16 \, \sqrt {-c \log \left (f\right ) - 3 \, f}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) - f} {\left (2 \, x + \frac {e}{c \log \left (f\right ) + f}\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} + 4 \, c d \log \left (f\right ) + 4 \, a f \log \left (f\right ) - e^{2} + 4 \, d f}{4 \, {\left (c \log \left (f\right ) + f\right )}}\right )}}{16 \, \sqrt {-c \log \left (f\right ) - f}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) + f} {\left (2 \, x - \frac {e}{c \log \left (f\right ) - f}\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) - 4 \, a f \log \left (f\right ) - e^{2} + 4 \, d f}{4 \, {\left (c \log \left (f\right ) - f\right )}}\right )}}{16 \, \sqrt {-c \log \left (f\right ) + f}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) + 3 \, f} {\left (2 \, x - \frac {3 \, e}{c \log \left (f\right ) - 3 \, f}\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} - 12 \, c d \log \left (f\right ) - 12 \, a f \log \left (f\right ) - 9 \, e^{2} + 36 \, d f}{4 \, {\left (c \log \left (f\right ) - 3 \, f\right )}}\right )}}{16 \, \sqrt {-c \log \left (f\right ) + 3 \, f}} \end {gather*}
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 \begin {gather*} \int f^{c\,x^2+a}\,{\mathrm {cosh}\left (f\,x^2+e\,x+d\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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