Optimal. Leaf size=149 \[ -\frac{\sinh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{b^2 c}-\frac{\sinh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{2 b^2 c}+\frac{\cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{b^2 c}+\frac{\cosh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{2 b^2 c}-\frac{\left (c^2 x^2+1\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.317802, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5696, 5779, 5448, 3303, 3298, 3301} \[ -\frac{\sinh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right )}{b^2 c}-\frac{\sinh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 a}{b}+4 \sinh ^{-1}(c x)\right )}{2 b^2 c}+\frac{\cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right )}{b^2 c}+\frac{\cosh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 a}{b}+4 \sinh ^{-1}(c x)\right )}{2 b^2 c}-\frac{\left (c^2 x^2+1\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5696
Rule 5779
Rule 5448
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\left (1+c^2 x^2\right )^{3/2}}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=-\frac{\left (1+c^2 x^2\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{(4 c) \int \frac{x \left (1+c^2 x^2\right )}{a+b \sinh ^{-1}(c x)} \, dx}{b}\\ &=-\frac{\left (1+c^2 x^2\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{4 \operatorname{Subst}\left (\int \frac{\cosh ^3(x) \sinh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c}\\ &=-\frac{\left (1+c^2 x^2\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{4 \operatorname{Subst}\left (\int \left (\frac{\sinh (2 x)}{4 (a+b x)}+\frac{\sinh (4 x)}{8 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c}\\ &=-\frac{\left (1+c^2 x^2\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{\sinh (4 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c}+\frac{\operatorname{Subst}\left (\int \frac{\sinh (2 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c}\\ &=-\frac{\left (1+c^2 x^2\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{\cosh \left (\frac{2 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c}+\frac{\cosh \left (\frac{4 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c}-\frac{\sinh \left (\frac{2 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c}-\frac{\sinh \left (\frac{4 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c}\\ &=-\frac{\left (1+c^2 x^2\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{\text{Chi}\left (\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right ) \sinh \left (\frac{2 a}{b}\right )}{b^2 c}-\frac{\text{Chi}\left (\frac{4 a}{b}+4 \sinh ^{-1}(c x)\right ) \sinh \left (\frac{4 a}{b}\right )}{2 b^2 c}+\frac{\cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right )}{b^2 c}+\frac{\cosh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 a}{b}+4 \sinh ^{-1}(c x)\right )}{2 b^2 c}\\ \end{align*}
Mathematica [A] time = 0.479051, size = 122, normalized size = 0.82 \[ \frac{-\frac{2 b \left (c^2 x^2+1\right )^2}{a+b \sinh ^{-1}(c x)}-2 \sinh \left (\frac{2 a}{b}\right ) \text{Chi}\left (2 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-\sinh \left (\frac{4 a}{b}\right ) \text{Chi}\left (4 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+2 \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (2 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+\cosh \left (\frac{4 a}{b}\right ) \text{Shi}\left (4 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )}{2 b^2 c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.191, size = 420, normalized size = 2.8 \begin{align*} -{\frac{3}{8\,bc \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) }}-{\frac{1}{16\,bc \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) } \left ( 8\,{c}^{4}{x}^{4}-8\,{c}^{3}{x}^{3}\sqrt{{c}^{2}{x}^{2}+1}+8\,{c}^{2}{x}^{2}-4\,cx\sqrt{{c}^{2}{x}^{2}+1}+1 \right ) }+{\frac{1}{4\,c{b}^{2}}{{\rm e}^{4\,{\frac{a}{b}}}}{\it Ei} \left ( 1,4\,{\it Arcsinh} \left ( cx \right ) +4\,{\frac{a}{b}} \right ) }-{\frac{1}{4\,bc \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) } \left ( 2\,{c}^{2}{x}^{2}-2\,cx\sqrt{{c}^{2}{x}^{2}+1}+1 \right ) }+{\frac{1}{2\,c{b}^{2}}{{\rm e}^{2\,{\frac{a}{b}}}}{\it Ei} \left ( 1,2\,{\it Arcsinh} \left ( cx \right ) +2\,{\frac{a}{b}} \right ) }-{\frac{1}{4\,c{b}^{2} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) } \left ( 2\,{x}^{2}b{c}^{2}+2\,bc\sqrt{{c}^{2}{x}^{2}+1}x+2\,{\it Arcsinh} \left ( cx \right ){{\rm e}^{-2\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-2\,{\it Arcsinh} \left ( cx \right ) -2\,{\frac{a}{b}} \right ) b+2\,{{\rm e}^{-2\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-2\,{\it Arcsinh} \left ( cx \right ) -2\,{\frac{a}{b}} \right ) a+b \right ) }-{\frac{1}{16\,c{b}^{2} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) } \left ( 8\,{x}^{4}b{c}^{4}+8\,\sqrt{{c}^{2}{x}^{2}+1}{x}^{3}b{c}^{3}+8\,{x}^{2}b{c}^{2}+4\,bc\sqrt{{c}^{2}{x}^{2}+1}x+4\,{\it Arcsinh} \left ( cx \right ){\it Ei} \left ( 1,-4\,{\it Arcsinh} \left ( cx \right ) -4\,{\frac{a}{b}} \right ){{\rm e}^{-4\,{\frac{a}{b}}}}b+4\,{\it Ei} \left ( 1,-4\,{\it Arcsinh} \left ( cx \right ) -4\,{\frac{a}{b}} \right ){{\rm e}^{-4\,{\frac{a}{b}}}}a+b \right ) } \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*} -\frac{{\left (c^{4} x^{4} + 2 \, c^{2} x^{2} + 1\right )}{\left (c^{2} x^{2} + 1\right )} +{\left (c^{5} x^{5} + 2 \, c^{3} x^{3} + c x\right )} \sqrt{c^{2} x^{2} + 1}}{a b c^{3} x^{2} + \sqrt{c^{2} x^{2} + 1} a b c^{2} x + a b c +{\left (b^{2} c^{3} x^{2} + \sqrt{c^{2} x^{2} + 1} b^{2} c^{2} x + b^{2} c\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )} + \int \frac{{\left (4 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1\right )}{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 4 \,{\left (2 \, c^{5} x^{5} + 3 \, c^{3} x^{3} + c x\right )}{\left (c^{2} x^{2} + 1\right )} +{\left (4 \, c^{6} x^{6} + 9 \, c^{4} x^{4} + 6 \, c^{2} x^{2} + 1\right )} \sqrt{c^{2} x^{2} + 1}}{a b c^{4} x^{4} +{\left (c^{2} x^{2} + 1\right )} a b c^{2} x^{2} + 2 \, a b c^{2} x^{2} + a b +{\left (b^{2} c^{4} x^{4} +{\left (c^{2} x^{2} + 1\right )} b^{2} c^{2} x^{2} + 2 \, b^{2} c^{2} x^{2} + b^{2} + 2 \,{\left (b^{2} c^{3} x^{3} + b^{2} c x\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + 2 \,{\left (a b c^{3} x^{3} + a b c x\right )} \sqrt{c^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{b^{2} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname{arsinh}\left (c x\right ) + a^{2}}, x\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{\left (c^{2} x^{2} + 1\right )^{\frac{3}{2}}}{\left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{2}}\, 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 \frac{{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]