Optimal. Leaf size=271 \[ -\frac {3 e^{-d-\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {e^{-3 d-\frac {9 e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {3 e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {3 e^{d-\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{3 d-\frac {9 e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {3 e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}} \]
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Rubi [A]
time = 0.27, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5624, 2325,
2266, 2235} \begin {gather*} -\frac {3 \sqrt {\pi } f^a e^{-\frac {e^2}{4 c \log (f)}-d} \text {Erfi}\left (\frac {e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {\sqrt {\pi } f^a e^{-\frac {9 e^2}{4 c \log (f)}-3 d} \text {Erfi}\left (\frac {3 e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {3 \sqrt {\pi } f^a e^{d-\frac {e^2}{4 c \log (f)}} \text {Erfi}\left (\frac {2 c x \log (f)+e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{3 d-\frac {9 e^2}{4 c \log (f)}} \text {Erfi}\left (\frac {2 c x \log (f)+3 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2266
Rule 2325
Rule 5624
Rubi steps
\begin {align*} \int f^{a+c x^2} \cosh ^3(d+e x) \, dx &=\int \left (\frac {1}{8} e^{-3 d-3 e x} f^{a+c x^2}+\frac {3}{8} e^{-d-e x} f^{a+c x^2}+\frac {3}{8} e^{d+e x} f^{a+c x^2}+\frac {1}{8} e^{3 d+3 e x} f^{a+c x^2}\right ) \, dx\\ &=\frac {1}{8} \int e^{-3 d-3 e x} f^{a+c x^2} \, dx+\frac {1}{8} \int e^{3 d+3 e x} f^{a+c x^2} \, dx+\frac {3}{8} \int e^{-d-e x} f^{a+c x^2} \, dx+\frac {3}{8} \int e^{d+e x} f^{a+c x^2} \, dx\\ &=\frac {1}{8} \int e^{-3 d-3 e x+a \log (f)+c x^2 \log (f)} \, dx+\frac {1}{8} \int e^{3 d+3 e x+a \log (f)+c x^2 \log (f)} \, dx+\frac {3}{8} \int e^{-d-e x+a \log (f)+c x^2 \log (f)} \, dx+\frac {3}{8} \int e^{d+e x+a \log (f)+c x^2 \log (f)} \, dx\\ &=\frac {1}{8} \left (e^{-3 d-\frac {9 e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(-3 e+2 c x \log (f))^2}{4 c \log (f)}} \, dx+\frac {1}{8} \left (e^{3 d-\frac {9 e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(3 e+2 c x \log (f))^2}{4 c \log (f)}} \, dx+\frac {1}{8} \left (3 e^{-d-\frac {e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(-e+2 c x \log (f))^2}{4 c \log (f)}} \, dx+\frac {1}{8} \left (3 e^{d-\frac {e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(e+2 c x \log (f))^2}{4 c \log (f)}} \, dx\\ &=-\frac {3 e^{-d-\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {e^{-3 d-\frac {9 e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {3 e^{d-\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{3 d-\frac {9 e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 214, normalized size = 0.79 \begin {gather*} \frac {e^{-\frac {9 e^2}{4 c \log (f)}} f^a \sqrt {\pi } \left ((\cosh (d)+\sinh (d)) \left (3 e^{\frac {2 e^2}{c \log (f)}} \text {Erfi}\left (\frac {e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )+3 e^{\frac {2 e^2}{c \log (f)}} \text {Erfi}\left (\frac {-e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cosh (2 d)-\sinh (2 d))+\text {Erfi}\left (\frac {3 e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cosh (2 d)+\sinh (2 d))\right )+\text {Erfi}\left (\frac {-3 e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cosh (3 d)-\sinh (3 d))\right )}{16 \sqrt {c} \sqrt {\log (f)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.27, size = 234, normalized size = 0.86
method | result | size |
risch | \(\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {3 \left (4 d \ln \left (f \right ) c +3 e^{2}\right )}{4 \ln \left (f \right ) c}} \erf \left (\sqrt {-c \ln \left (f \right )}\, x +\frac {3 e}{2 \sqrt {-c \ln \left (f \right )}}\right )}{16 \sqrt {-c \ln \left (f \right )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {3 d \ln \left (f \right ) c -\frac {9 e^{2}}{4}}{c \ln \left (f \right )}} \erf \left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {3 e}{2 \sqrt {-c \ln \left (f \right )}}\right )}{16 \sqrt {-c \ln \left (f \right )}}+\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {4 d \ln \left (f \right ) c +e^{2}}{4 \ln \left (f \right ) c}} \erf \left (\sqrt {-c \ln \left (f \right )}\, x +\frac {e}{2 \sqrt {-c \ln \left (f \right )}}\right )}{16 \sqrt {-c \ln \left (f \right )}}-\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {4 d \ln \left (f \right ) c -e^{2}}{4 \ln \left (f \right ) c}} \erf \left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {e}{2 \sqrt {-c \ln \left (f \right )}}\right )}{16 \sqrt {-c \ln \left (f \right )}}\) | \(234\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 211, normalized size = 0.78 \begin {gather*} \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x - \frac {3 \, e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (3 \, d - \frac {9 \, e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt {-c \log \left (f\right )}} + \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x - \frac {e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (d - \frac {e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt {-c \log \left (f\right )}} + \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x + \frac {e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (-d - \frac {e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt {-c \log \left (f\right )}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x + \frac {3 \, e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (-3 \, d - \frac {9 \, e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt {-c \log \left (f\right )}} \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. 548 vs.
\(2 (205) = 410\).
time = 0.41, size = 548, normalized size = 2.02 \begin {gather*} -\frac {\sqrt {-c \log \left (f\right )} {\left (\sqrt {\pi } \cosh \left (\frac {4 \, a c \log \left (f\right )^{2} + 12 \, c d \log \left (f\right ) - 9 \, \cosh \left (1\right )^{2} - 18 \, \cosh \left (1\right ) \sinh \left (1\right ) - 9 \, \sinh \left (1\right )^{2}}{4 \, c \log \left (f\right )}\right ) + \sqrt {\pi } \sinh \left (\frac {4 \, a c \log \left (f\right )^{2} + 12 \, c d \log \left (f\right ) - 9 \, \cosh \left (1\right )^{2} - 18 \, \cosh \left (1\right ) \sinh \left (1\right ) - 9 \, \sinh \left (1\right )^{2}}{4 \, c \log \left (f\right )}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) + 3 \, \cosh \left (1\right ) + 3 \, \sinh \left (1\right )\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) + 3 \, \sqrt {-c \log \left (f\right )} {\left (\sqrt {\pi } \cosh \left (\frac {4 \, a c \log \left (f\right )^{2} + 4 \, c d \log \left (f\right ) - \cosh \left (1\right )^{2} - 2 \, \cosh \left (1\right ) \sinh \left (1\right ) - \sinh \left (1\right )^{2}}{4 \, c \log \left (f\right )}\right ) + \sqrt {\pi } \sinh \left (\frac {4 \, a c \log \left (f\right )^{2} + 4 \, c d \log \left (f\right ) - \cosh \left (1\right )^{2} - 2 \, \cosh \left (1\right ) \sinh \left (1\right ) - \sinh \left (1\right )^{2}}{4 \, c \log \left (f\right )}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) + \cosh \left (1\right ) + \sinh \left (1\right )\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) + 3 \, \sqrt {-c \log \left (f\right )} {\left (\sqrt {\pi } \cosh \left (\frac {4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) - \cosh \left (1\right )^{2} - 2 \, \cosh \left (1\right ) \sinh \left (1\right ) - \sinh \left (1\right )^{2}}{4 \, c \log \left (f\right )}\right ) + \sqrt {\pi } \sinh \left (\frac {4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) - \cosh \left (1\right )^{2} - 2 \, \cosh \left (1\right ) \sinh \left (1\right ) - \sinh \left (1\right )^{2}}{4 \, c \log \left (f\right )}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) - \cosh \left (1\right ) - \sinh \left (1\right )\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) + \sqrt {-c \log \left (f\right )} {\left (\sqrt {\pi } \cosh \left (\frac {4 \, a c \log \left (f\right )^{2} - 12 \, c d \log \left (f\right ) - 9 \, \cosh \left (1\right )^{2} - 18 \, \cosh \left (1\right ) \sinh \left (1\right ) - 9 \, \sinh \left (1\right )^{2}}{4 \, c \log \left (f\right )}\right ) + \sqrt {\pi } \sinh \left (\frac {4 \, a c \log \left (f\right )^{2} - 12 \, c d \log \left (f\right ) - 9 \, \cosh \left (1\right )^{2} - 18 \, \cosh \left (1\right ) \sinh \left (1\right ) - 9 \, \sinh \left (1\right )^{2}}{4 \, c \log \left (f\right )}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \left (f\right ) - 3 \, \cosh \left (1\right ) - 3 \, \sinh \left (1\right )\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right )}{16 \, c \log \left (f\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int f^{a + c x^{2}} \cosh ^{3}{\left (d + e x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 264, normalized size = 0.97 \begin {gather*} -\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x + \frac {3 \, e}{c \log \left (f\right )}\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} + 12 \, c d \log \left (f\right ) - 9 \, e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt {-c \log \left (f\right )}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x + \frac {e}{c \log \left (f\right )}\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} + 4 \, c d \log \left (f\right ) - e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt {-c \log \left (f\right )}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x - \frac {e}{c \log \left (f\right )}\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) - e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt {-c \log \left (f\right )}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x - \frac {3 \, e}{c \log \left (f\right )}\right )}\right ) e^{\left (\frac {4 \, a c \log \left (f\right )^{2} - 12 \, c d \log \left (f\right ) - 9 \, e^{2}}{4 \, c \log \left (f\right )}\right )}}{16 \, \sqrt {-c \log \left (f\right )}} \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 (d+e\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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