Optimal. Leaf size=180 \[ -\frac{2 \sqrt{a^2 x^2-1} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a^2 x^2-1}}{a \sqrt{c+d x^2}}\right )}{3 c^2 \sqrt{d} \sqrt{a x-1} \sqrt{a x+1}}+\frac{a \left (1-a^2 x^2\right )}{3 c \sqrt{a x-1} \sqrt{a x+1} \left (a^2 c+d\right ) \sqrt{c+d x^2}}+\frac{2 x \cosh ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}+\frac{x \cosh ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.175823, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 10, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {192, 191, 5705, 12, 519, 571, 78, 63, 217, 206} \[ -\frac{2 \sqrt{a^2 x^2-1} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a^2 x^2-1}}{a \sqrt{c+d x^2}}\right )}{3 c^2 \sqrt{d} \sqrt{a x-1} \sqrt{a x+1}}+\frac{a \left (1-a^2 x^2\right )}{3 c \sqrt{a x-1} \sqrt{a x+1} \left (a^2 c+d\right ) \sqrt{c+d x^2}}+\frac{2 x \cosh ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}+\frac{x \cosh ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 192
Rule 191
Rule 5705
Rule 12
Rule 519
Rule 571
Rule 78
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{\cosh ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx &=\frac{x \cosh ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac{2 x \cosh ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}-a \int \frac{x \left (3 c+2 d x^2\right )}{3 c^2 \sqrt{-1+a x} \sqrt{1+a x} \left (c+d x^2\right )^{3/2}} \, dx\\ &=\frac{x \cosh ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac{2 x \cosh ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}-\frac{a \int \frac{x \left (3 c+2 d x^2\right )}{\sqrt{-1+a x} \sqrt{1+a x} \left (c+d x^2\right )^{3/2}} \, dx}{3 c^2}\\ &=\frac{x \cosh ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac{2 x \cosh ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}-\frac{\left (a \sqrt{-1+a^2 x^2}\right ) \int \frac{x \left (3 c+2 d x^2\right )}{\sqrt{-1+a^2 x^2} \left (c+d x^2\right )^{3/2}} \, dx}{3 c^2 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{x \cosh ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac{2 x \cosh ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}-\frac{\left (a \sqrt{-1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{3 c+2 d x}{\sqrt{-1+a^2 x} (c+d x)^{3/2}} \, dx,x,x^2\right )}{6 c^2 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{a \left (1-a^2 x^2\right )}{3 c \left (a^2 c+d\right ) \sqrt{-1+a x} \sqrt{1+a x} \sqrt{c+d x^2}}+\frac{x \cosh ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac{2 x \cosh ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}-\frac{\left (a \sqrt{-1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+a^2 x} \sqrt{c+d x}} \, dx,x,x^2\right )}{3 c^2 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{a \left (1-a^2 x^2\right )}{3 c \left (a^2 c+d\right ) \sqrt{-1+a x} \sqrt{1+a x} \sqrt{c+d x^2}}+\frac{x \cosh ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac{2 x \cosh ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}-\frac{\left (2 \sqrt{-1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+\frac{d}{a^2}+\frac{d x^2}{a^2}}} \, dx,x,\sqrt{-1+a^2 x^2}\right )}{3 a c^2 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{a \left (1-a^2 x^2\right )}{3 c \left (a^2 c+d\right ) \sqrt{-1+a x} \sqrt{1+a x} \sqrt{c+d x^2}}+\frac{x \cosh ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac{2 x \cosh ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}-\frac{\left (2 \sqrt{-1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{a^2}} \, dx,x,\frac{\sqrt{-1+a^2 x^2}}{\sqrt{c+d x^2}}\right )}{3 a c^2 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{a \left (1-a^2 x^2\right )}{3 c \left (a^2 c+d\right ) \sqrt{-1+a x} \sqrt{1+a x} \sqrt{c+d x^2}}+\frac{x \cosh ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac{2 x \cosh ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}-\frac{2 \sqrt{-1+a^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{-1+a^2 x^2}}{a \sqrt{c+d x^2}}\right )}{3 c^2 \sqrt{d} \sqrt{-1+a x} \sqrt{1+a x}}\\ \end{align*}
Mathematica [C] time = 1.97651, size = 609, normalized size = 3.38 \[ \frac{\frac{4 (a x-1)^{3/2} \left (c+d x^2\right ) \sqrt{\frac{(a x+1) \left (a \sqrt{c}-i \sqrt{d}\right )}{(a x-1) \left (a \sqrt{c}+i \sqrt{d}\right )}} \left (\frac{a \left (\sqrt{d}-i a \sqrt{c}\right ) \left (\sqrt{d} x+i \sqrt{c}\right ) \sqrt{\frac{\frac{i a \sqrt{c}}{\sqrt{d}}+a (-x)+\frac{i \sqrt{d} x}{\sqrt{c}}+1}{1-a x}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{-\frac{a \left (x+\frac{i \sqrt{c}}{\sqrt{d}}\right )+\frac{i \sqrt{d} x}{\sqrt{c}}-1}{2-2 a x}}\right ),\frac{4 i a \sqrt{c} \sqrt{d}}{\left (a \sqrt{c}+i \sqrt{d}\right )^2}\right )}{a x-1}+a \sqrt{c} \left (-a \sqrt{c}+i \sqrt{d}\right ) \sqrt{\frac{\left (a^2 c+d\right ) \left (c+d x^2\right )}{c d (a x-1)^2}} \sqrt{-\frac{a \left (x+\frac{i \sqrt{c}}{\sqrt{d}}\right )+\frac{i \sqrt{d} x}{\sqrt{c}}-1}{1-a x}} \Pi \left (\frac{2 a \sqrt{c}}{\sqrt{c} a+i \sqrt{d}};\sin ^{-1}\left (\sqrt{-\frac{\frac{i \sqrt{d} x}{\sqrt{c}}+a \left (x+\frac{i \sqrt{c}}{\sqrt{d}}\right )-1}{2-2 a x}}\right )|\frac{4 i a \sqrt{c} \sqrt{d}}{\left (\sqrt{c} a+i \sqrt{d}\right )^2}\right )\right )}{a \sqrt{a x+1} \left (a^2 c+d\right ) \sqrt{-\frac{a \left (x+\frac{i \sqrt{c}}{\sqrt{d}}\right )+\frac{i \sqrt{d} x}{\sqrt{c}}-1}{1-a x}}}-\frac{a c \sqrt{a x+1} \sqrt{a x-1} \left (c+d x^2\right )}{a^2 c+d}+x \cosh ^{-1}(a x) \left (3 c+2 d x^2\right )}{3 c^2 \left (c+d x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.151, size = 0, normalized size = 0. \begin{align*} \int{{\rm arccosh} \left (ax\right ) \left ( d{x}^{2}+c \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.73017, size = 1270, normalized size = 7.06 \begin{align*} \left [\frac{{\left (a^{2} c^{3} +{\left (a^{2} c d^{2} + d^{3}\right )} x^{4} + c^{2} d + 2 \,{\left (a^{2} c^{2} d + c d^{2}\right )} x^{2}\right )} \sqrt{d} \log \left (8 \, a^{4} d^{2} x^{4} + a^{4} c^{2} - 6 \, a^{2} c d + 8 \,{\left (a^{4} c d - a^{2} d^{2}\right )} x^{2} - 4 \,{\left (2 \, a^{3} d x^{2} + a^{3} c - a d\right )} \sqrt{a^{2} x^{2} - 1} \sqrt{d x^{2} + c} \sqrt{d} + d^{2}\right ) + 2 \,{\left (2 \,{\left (a^{2} c d^{2} + d^{3}\right )} x^{3} + 3 \,{\left (a^{2} c^{2} d + c d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - 2 \,{\left (a c d^{2} x^{2} + a c^{2} d\right )} \sqrt{a^{2} x^{2} - 1} \sqrt{d x^{2} + c}}{6 \,{\left (a^{2} c^{5} d + c^{4} d^{2} +{\left (a^{2} c^{3} d^{3} + c^{2} d^{4}\right )} x^{4} + 2 \,{\left (a^{2} c^{4} d^{2} + c^{3} d^{3}\right )} x^{2}\right )}}, \frac{{\left (a^{2} c^{3} +{\left (a^{2} c d^{2} + d^{3}\right )} x^{4} + c^{2} d + 2 \,{\left (a^{2} c^{2} d + c d^{2}\right )} x^{2}\right )} \sqrt{-d} \arctan \left (\frac{{\left (2 \, a^{2} d x^{2} + a^{2} c - d\right )} \sqrt{a^{2} x^{2} - 1} \sqrt{d x^{2} + c} \sqrt{-d}}{2 \,{\left (a^{3} d^{2} x^{4} - a c d +{\left (a^{3} c d - a d^{2}\right )} x^{2}\right )}}\right ) +{\left (2 \,{\left (a^{2} c d^{2} + d^{3}\right )} x^{3} + 3 \,{\left (a^{2} c^{2} d + c d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) -{\left (a c d^{2} x^{2} + a c^{2} d\right )} \sqrt{a^{2} x^{2} - 1} \sqrt{d x^{2} + c}}{3 \,{\left (a^{2} c^{5} d + c^{4} d^{2} +{\left (a^{2} c^{3} d^{3} + c^{2} d^{4}\right )} x^{4} + 2 \,{\left (a^{2} c^{4} d^{2} + c^{3} d^{3}\right )} x^{2}\right )}}\right ] \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 \frac{\operatorname{acosh}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]