Optimal. Leaf size=65 \[ \frac {\sqrt {\pi } e^a \text {erfi}\left (\sqrt {c+1} x\right )}{4 \sqrt {c+1}}-\frac {\sqrt {\pi } e^{-a} \text {erfi}\left (\sqrt {1-c} x\right )}{4 \sqrt {1-c}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5512, 2204} \[ \frac {\sqrt {\pi } e^a \text {Erfi}\left (\sqrt {c+1} x\right )}{4 \sqrt {c+1}}-\frac {\sqrt {\pi } e^{-a} \text {Erfi}\left (\sqrt {1-c} x\right )}{4 \sqrt {1-c}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2204
Rule 5512
Rubi steps
\begin {align*} \int e^{x^2} \sinh \left (a+c x^2\right ) \, dx &=\int \left (-\frac {1}{2} e^{-a+(1-c) x^2}+\frac {1}{2} e^{a+(1+c) x^2}\right ) \, dx\\ &=-\left (\frac {1}{2} \int e^{-a+(1-c) x^2} \, dx\right )+\frac {1}{2} \int e^{a+(1+c) x^2} \, dx\\ &=-\frac {e^{-a} \sqrt {\pi } \text {erfi}\left (\sqrt {1-c} x\right )}{4 \sqrt {1-c}}+\frac {e^a \sqrt {\pi } \text {erfi}\left (\sqrt {1+c} x\right )}{4 \sqrt {1+c}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 72, normalized size = 1.11 \[ \frac {\sqrt {\pi } \left ((c-1) \sqrt {c+1} (\sinh (a)+\cosh (a)) \text {erfi}\left (\sqrt {c+1} x\right )-\sqrt {c-1} (c+1) (\cosh (a)-\sinh (a)) \text {erf}\left (\sqrt {c-1} x\right )\right )}{4 \left (c^2-1\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.96, size = 75, normalized size = 1.15 \[ -\frac {\sqrt {\pi } {\left ({\left (c + 1\right )} \cosh \relax (a) - {\left (c + 1\right )} \sinh \relax (a)\right )} \sqrt {c - 1} \operatorname {erf}\left (\sqrt {c - 1} x\right ) + \sqrt {\pi } {\left ({\left (c - 1\right )} \cosh \relax (a) + {\left (c - 1\right )} \sinh \relax (a)\right )} \sqrt {-c - 1} \operatorname {erf}\left (\sqrt {-c - 1} x\right )}{4 \, {\left (c^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.14, size = 49, normalized size = 0.75 \[ \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {c - 1} x\right ) e^{\left (-a\right )}}{4 \, \sqrt {c - 1}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c - 1} x\right ) e^{a}}{4 \, \sqrt {-c - 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.35, size = 48, normalized size = 0.74 \[ -\frac {\sqrt {\pi }\, {\mathrm e}^{-a} \erf \left (\sqrt {c -1}\, x \right )}{4 \sqrt {c -1}}+\frac {\sqrt {\pi }\, {\mathrm e}^{a} \erf \left (\sqrt {-1-c}\, x \right )}{4 \sqrt {-1-c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 47, normalized size = 0.72 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {c - 1} x\right ) e^{\left (-a\right )}}{4 \, \sqrt {c - 1}} + \frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {-c - 1} x\right ) e^{a}}{4 \, \sqrt {-c - 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {e}}^{x^2}\,\mathrm {sinh}\left (c\,x^2+a\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{x^{2}} \sinh {\left (a + c x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________