Optimal. Leaf size=155 \[ \frac{\sqrt{\frac{\pi }{2}} e e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{b^{3/2} d}+\frac{\sqrt{\frac{\pi }{2}} e e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{b^{3/2} d}-\frac{2 e \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)}{b d \sqrt{a+b \cosh ^{-1}(c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.225043, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {5866, 12, 5666, 3307, 2180, 2204, 2205} \[ \frac{\sqrt{\frac{\pi }{2}} e e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{b^{3/2} d}+\frac{\sqrt{\frac{\pi }{2}} e e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{b^{3/2} d}-\frac{2 e \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)}{b d \sqrt{a+b \cosh ^{-1}(c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5866
Rule 12
Rule 5666
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{c e+d e x}{\left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e x}{\left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac{e \operatorname{Subst}\left (\int \frac{x}{\left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{2 e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{b d \sqrt{a+b \cosh ^{-1}(c+d x)}}+\frac{(2 e) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b d}\\ &=-\frac{2 e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{b d \sqrt{a+b \cosh ^{-1}(c+d x)}}+\frac{e \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b d}+\frac{e \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b d}\\ &=-\frac{2 e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{b d \sqrt{a+b \cosh ^{-1}(c+d x)}}+\frac{(2 e) \operatorname{Subst}\left (\int e^{\frac{2 a}{b}-\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{b^2 d}+\frac{(2 e) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b}+\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{b^2 d}\\ &=-\frac{2 e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{b d \sqrt{a+b \cosh ^{-1}(c+d x)}}+\frac{e e^{\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{b^{3/2} d}+\frac{e e^{-\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{b^{3/2} d}\\ \end{align*}
Mathematica [B] time = 6.38246, size = 314, normalized size = 2.03 \[ \frac{e \left (-\frac{2 \sqrt{b} e^{-\frac{a}{b}} \left (c e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c+d x)} \text{Gamma}\left (\frac{1}{2},\frac{a}{b}+\cosh ^{-1}(c+d x)\right )-c \sqrt{-\frac{a+b \cosh ^{-1}(c+d x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )+e^{a/b} \sinh \left (2 \cosh ^{-1}(c+d x)\right )\right )}{\sqrt{a+b \cosh ^{-1}(c+d x)}}-2 \sqrt{\pi } c \left (\sinh \left (\frac{a}{b}\right )+\cosh \left (\frac{a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )+\sqrt{2 \pi } \left (\sinh \left (\frac{2 a}{b}\right )+\cosh \left (\frac{2 a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )-2 \sqrt{\pi } c e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )+\sqrt{2 \pi } e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )\right )}{2 b^{3/2} d} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.129, size = 0, normalized size = 0. \begin{align*} \int{(dex+ce) \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) ^{-{\frac{3}{2}}}}\, 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 \frac{d e x + c e}{{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{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*} e \left (\int \frac{c}{a \sqrt{a + b \operatorname{acosh}{\left (c + d x \right )}} + b \sqrt{a + b \operatorname{acosh}{\left (c + d x \right )}} \operatorname{acosh}{\left (c + d x \right )}}\, dx + \int \frac{d x}{a \sqrt{a + b \operatorname{acosh}{\left (c + d x \right )}} + b \sqrt{a + b \operatorname{acosh}{\left (c + d x \right )}} \operatorname{acosh}{\left (c + d x \right )}}\, dx\right ) \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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]