Optimal. Leaf size=57 \[ \frac {e^c \sqrt {\pi } \text {Erfi}(b x)^2}{8 b}+\frac {b e^{-c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{2 \sqrt {\pi }} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {6550, 6510, 30,
6513} \begin {gather*} \frac {b e^{-c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{2 \sqrt {\pi }}+\frac {\sqrt {\pi } e^c \text {Erfi}(b x)^2}{8 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 6510
Rule 6513
Rule 6550
Rubi steps
\begin {align*} \int \cosh \left (c+b^2 x^2\right ) \text {erfi}(b x) \, dx &=\frac {1}{2} \int e^{-c-b^2 x^2} \text {erfi}(b x) \, dx+\frac {1}{2} \int e^{c+b^2 x^2} \text {erfi}(b x) \, dx\\ &=\frac {b e^{-c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{2 \sqrt {\pi }}+\frac {\left (e^c \sqrt {\pi }\right ) \text {Subst}(\int x \, dx,x,\text {erfi}(b x))}{4 b}\\ &=\frac {e^c \sqrt {\pi } \text {erfi}(b x)^2}{8 b}+\frac {b e^{-c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{2 \sqrt {\pi }}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 7.53, size = 74, normalized size = 1.30 \begin {gather*} \frac {4 b^2 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right ) (-\cosh (c)+\sinh (c))+\pi \text {Erfi}(b x) (2 \text {Erf}(b x) (\cosh (c)-\sinh (c))+\text {Erfi}(b x) (\cosh (c)+\sinh (c)))}{8 b \sqrt {\pi }} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.17, size = 0, normalized size = 0.00 \[\int \cosh \left (b^{2} x^{2}+c \right ) \erfi \left (b x \right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 \cosh {\left (b^{2} x^{2} + c \right )} \operatorname {erfi}{\left (b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \mathrm {cosh}\left (b^2\,x^2+c\right )\,\mathrm {erfi}\left (b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________