Optimal. Leaf size=133 \[ -\frac {e^{-d-\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{d-\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.16, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5624, 2325,
2266, 2235} \begin {gather*} \frac {\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 )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {\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 )}{4 \sqrt {c} \sqrt {\log (f)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2235
Rule 2266
Rule 2325
Rule 5624
Rubi steps
\begin {align*} \int f^{a+c x^2} \cosh (d+e x) \, dx &=\int \left (\frac {1}{2} e^{-d-e x} f^{a+c x^2}+\frac {1}{2} e^{d+e x} f^{a+c x^2}\right ) \, dx\\ &=\frac {1}{2} \int e^{-d-e x} f^{a+c x^2} \, dx+\frac {1}{2} \int e^{d+e x} f^{a+c x^2} \, dx\\ &=\frac {1}{2} \int e^{-d-e x+a \log (f)+c x^2 \log (f)} \, dx+\frac {1}{2} \int e^{d+e x+a \log (f)+c x^2 \log (f)} \, dx\\ &=\frac {1}{2} \left (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}{2} \left (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 {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 )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {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 )}{4 \sqrt {c} \sqrt {\log (f)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.11, size = 104, normalized size = 0.78 \begin {gather*} \frac {e^{-\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \left (\text {Erfi}\left (\frac {-e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cosh (d)-\sinh (d))+\text {Erfi}\left (\frac {e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cosh (d)+\sinh (d))\right )}{4 \sqrt {c} \sqrt {\log (f)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.77, size = 117, normalized size = 0.88
method | result | size |
risch | \(\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {4 d \ln \left (f \right ) c +e^{2}}{4 \ln \left (f \right ) c}} \erf \left (\sqrt {-c \ln \left (f \right )}\, x +\frac {e}{2 \sqrt {-c \ln \left (f \right )}}\right )}{4 \sqrt {-c \ln \left (f \right )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {4 d \ln \left (f \right ) c -e^{2}}{4 \ln \left (f \right ) c}} \erf \left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {e}{2 \sqrt {-c \ln \left (f \right )}}\right )}{4 \sqrt {-c \ln \left (f \right )}}\) | \(117\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 105, normalized size = 0.79 \begin {gather*} \frac {\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 )}}{4 \, \sqrt {-c \log \left (f\right )}} + \frac {\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 )}}{4 \, \sqrt {-c \log \left (f\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 276 vs.
\(2 (101) = 202\).
time = 0.37, size = 276, normalized size = 2.08 \begin {gather*} -\frac {\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} - 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 )}{4 \, c \log \left (f\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int f^{a + c x^{2}} \cosh {\left (d + e x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.41, size = 132, normalized size = 0.99 \begin {gather*} -\frac {\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 )}}{4 \, \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 {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 )}}{4 \, \sqrt {-c \log \left (f\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int f^{c\,x^2+a}\,\mathrm {cosh}\left (d+e\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________