Optimal. Leaf size=143 \[ -\frac{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt{d-c^2 d x^2}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{c x-1} \sqrt{c x+1} \log \left (1-c^2 x^2\right )}{2 c^3 d \sqrt{d-c^2 d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.453895, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {5798, 5752, 5676, 260} \[ -\frac{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt{d-c^2 d x^2}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{c x-1} \sqrt{c x+1} \log \left (1-c^2 x^2\right )}{2 c^3 d \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5798
Rule 5752
Rule 5676
Rule 260
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{\left (b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x}{-1+c^2 x^2} \, dx}{c d \sqrt{d-c^2 d x^2}}\\ &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{-1+c x} \sqrt{1+c x} \log \left (1-c^2 x^2\right )}{2 c^3 d \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.65547, size = 159, normalized size = 1.11 \[ \frac{2 a \sqrt{d} \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )+2 a c d x+b d \left (2 c x \cosh ^{-1}(c x)-\sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (2 \log \left (\sqrt{\frac{c x-1}{c x+1}} (c x+1)\right )+\cosh ^{-1}(c x)^2\right )\right )}{2 c^3 d^2 \sqrt{d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.18, size = 279, normalized size = 2. \begin{align*}{\frac{ax}{{c}^{2}d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}}-{\frac{a}{{c}^{2}d}\arctan \left ({x\sqrt{{c}^{2}d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}} \right ){\frac{1}{\sqrt{{c}^{2}d}}}}+{\frac{b \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{2\,{d}^{2}{c}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{\rm arccosh} \left (cx\right )}{{d}^{2}{c}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{\rm arccosh} \left (cx\right )x}{{c}^{2}{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{b}{{d}^{2}{c}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}\ln \left ( \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2}-1 \right ) } \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-c^{2} d x^{2} + d}{\left (b x^{2} \operatorname{arcosh}\left (c x\right ) + a x^{2}\right )}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{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{x^{2} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{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 (b \operatorname{arcosh}\left (c x\right ) + a\right )} x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]