Optimal. Leaf size=115 \[ \frac{\sqrt{\pi } \sqrt{b} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d}-\frac{\sqrt{\pi } \sqrt{b} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d}+\frac{(c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.246222, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5863, 5653, 5779, 3308, 2180, 2204, 2205} \[ \frac{\sqrt{\pi } \sqrt{b} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d}-\frac{\sqrt{\pi } \sqrt{b} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d}+\frac{(c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5863
Rule 5653
Rule 5779
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \sqrt{a+b \sinh ^{-1}(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{a+b \sinh ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac{(c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{d}-\frac{b \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+x^2} \sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{2 d}\\ &=\frac{(c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{d}-\frac{b \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{2 d}\\ &=\frac{(c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{d}+\frac{b \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{4 d}-\frac{b \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{4 d}\\ &=\frac{(c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{d}+\frac{\operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{2 d}-\frac{\operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{2 d}\\ &=\frac{(c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{d}+\frac{\sqrt{b} e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d}-\frac{\sqrt{b} e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{4 d}\\ \end{align*}
Mathematica [A] time = 0.0816329, size = 111, normalized size = 0.97 \[ \frac{e^{-\frac{a}{b}} \sqrt{a+b \sinh ^{-1}(c+d x)} \left (\frac{\text{Gamma}\left (\frac{3}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{\sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}}}-\frac{e^{\frac{2 a}{b}} \text{Gamma}\left (\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right )}{\sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)}}\right )}{2 d} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+b{\it Arcsinh} \left ( dx+c \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \operatorname{arsinh}\left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 \sqrt{a + b \operatorname{asinh}{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \operatorname{arsinh}\left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]