Optimal. Leaf size=37 \[ \frac{\sqrt{\pi } \text{Erfi}(a+b x)}{4 b}-\frac{\sqrt{\pi } \text{Erf}(a+b x)}{4 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0170969, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5310, 5298, 2204, 2205} \[ \frac{\sqrt{\pi } \text{Erfi}(a+b x)}{4 b}-\frac{\sqrt{\pi } \text{Erf}(a+b x)}{4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5310
Rule 5298
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \sinh \left ((a+b x)^2\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \sinh \left (x^2\right ) \, dx,x,a+b x\right )}{b}\\ &=-\frac{\operatorname{Subst}\left (\int e^{-x^2} \, dx,x,a+b x\right )}{2 b}+\frac{\operatorname{Subst}\left (\int e^{x^2} \, dx,x,a+b x\right )}{2 b}\\ &=-\frac{\sqrt{\pi } \text{erf}(a+b x)}{4 b}+\frac{\sqrt{\pi } \text{erfi}(a+b x)}{4 b}\\ \end{align*}
Mathematica [A] time = 0.0045263, size = 27, normalized size = 0.73 \[ \frac{\sqrt{\pi } (\text{Erfi}(a+b x)-\text{Erf}(a+b x))}{4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.028, size = 36, normalized size = 1. \begin{align*} -{\frac{{\it Erf} \left ( bx+a \right ) \sqrt{\pi }}{4\,b}}-{\frac{{\frac{i}{4}}\sqrt{\pi }{\it Erf} \left ( ibx+ia \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.61128, size = 695, normalized size = 18.78 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.80087, size = 144, normalized size = 3.89 \begin{align*} -\frac{\sqrt{\pi } \sqrt{b^{2}} \operatorname{erf}\left (\frac{\sqrt{b^{2}}{\left (b x + a\right )}}{b}\right ) - \sqrt{\pi } \sqrt{b^{2}} \operatorname{erfi}\left (\frac{\sqrt{b^{2}}{\left (b x + a\right )}}{b}\right )}{4 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh{\left (\left (a + b x\right )^{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [C] time = 1.25509, size = 53, normalized size = 1.43 \begin{align*} -\frac{i \, \sqrt{\pi } \operatorname{erf}\left (i \, b{\left (x + \frac{a}{b}\right )}\right )}{4 \, b} + \frac{\sqrt{\pi } \operatorname{erf}\left (-b{\left (x + \frac{a}{b}\right )}\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]