Optimal. Leaf size=89 \[ -\frac{b x \left (a+b \sin ^{-1}(c x)\right )}{c d^2 \sqrt{1-c^2 x^2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{b^2 \log \left (1-c^2 x^2\right )}{2 c^2 d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0985197, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {4677, 4651, 260} \[ -\frac{b x \left (a+b \sin ^{-1}(c x)\right )}{c d^2 \sqrt{1-c^2 x^2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{b^2 \log \left (1-c^2 x^2\right )}{2 c^2 d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4677
Rule 4651
Rule 260
Rubi steps
\begin{align*} \int \frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^2} \, dx &=\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{b \int \frac{a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{c d^2}\\ &=-\frac{b x \left (a+b \sin ^{-1}(c x)\right )}{c d^2 \sqrt{1-c^2 x^2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac{b^2 \int \frac{x}{1-c^2 x^2} \, dx}{d^2}\\ &=-\frac{b x \left (a+b \sin ^{-1}(c x)\right )}{c d^2 \sqrt{1-c^2 x^2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{b^2 \log \left (1-c^2 x^2\right )}{2 c^2 d^2}\\ \end{align*}
Mathematica [A] time = 0.189782, size = 75, normalized size = 0.84 \[ -\frac{\frac{2 b c x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{c^2 x^2-1}+b^2 \log \left (1-c^2 x^2\right )}{2 c^2 d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.03, size = 205, normalized size = 2.3 \begin{align*} -{\frac{{a}^{2}}{2\,{c}^{2}{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{2\,{c}^{2}{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{{b}^{2}\arcsin \left ( cx \right ) x}{c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{{b}^{2}\ln \left ( -{c}^{2}{x}^{2}+1 \right ) }{2\,{c}^{2}{d}^{2}}}-{\frac{ab\arcsin \left ( cx \right ) }{{c}^{2}{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{ab}{2\,{c}^{2}{d}^{2} \left ( cx-1 \right ) }\sqrt{- \left ( cx-1 \right ) ^{2}-2\,cx+2}}+{\frac{ab}{2\,{c}^{2}{d}^{2} \left ( cx+1 \right ) }\sqrt{- \left ( cx+1 \right ) ^{2}+2\,cx+2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.67815, size = 495, normalized size = 5.56 \begin{align*} \frac{1}{2} \,{\left (\frac{{\left (\frac{\sqrt{-c^{2} x^{2} + 1} c^{2} d^{2}}{c^{6} d^{4} + \sqrt{c^{6} d^{4}} c^{4} d^{2} x} - \frac{\sqrt{-c^{2} x^{2} + 1} c^{2} d^{2}}{c^{6} d^{4} - \sqrt{c^{6} d^{4}} c^{4} d^{2} x}\right )} c^{5} d^{2}}{\sqrt{c^{6} d^{4}}} - \frac{2 \, \arcsin \left (c x\right )}{c^{4} d^{2} x^{2} - c^{2} d^{2}}\right )} a b - \frac{1}{2} \,{\left (\frac{c^{6} d^{2}{\left (\frac{\log \left (c x + 1\right )}{c^{5} d^{2}} + \frac{\log \left (c x - 1\right )}{c^{5} d^{2}}\right )}}{\sqrt{c^{6} d^{4}}} - \frac{{\left (\frac{\sqrt{-c^{2} x^{2} + 1} c^{2} d^{2}}{c^{6} d^{4} + \sqrt{c^{6} d^{4}} c^{4} d^{2} x} - \frac{\sqrt{-c^{2} x^{2} + 1} c^{2} d^{2}}{c^{6} d^{4} - \sqrt{c^{6} d^{4}} c^{4} d^{2} x}\right )} c^{5} d^{2} \arcsin \left (c x\right )}{\sqrt{c^{6} d^{4}}}\right )} b^{2} - \frac{b^{2} \arcsin \left (c x\right )^{2}}{2 \,{\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\right )}} - \frac{a^{2}}{2 \,{\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.65029, size = 230, normalized size = 2.58 \begin{align*} -\frac{b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2} +{\left (b^{2} c^{2} x^{2} - b^{2}\right )} \log \left (c^{2} x^{2} - 1\right ) - 2 \,{\left (b^{2} c x \arcsin \left (c x\right ) + a b c x\right )} \sqrt{-c^{2} x^{2} + 1}}{2 \,{\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\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*} \frac{\int \frac{a^{2} x}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac{b^{2} x \operatorname{asin}^{2}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac{2 a b x \operatorname{asin}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.55962, size = 275, normalized size = 3.09 \begin{align*} -\frac{b^{2} x^{2} \arcsin \left (c x\right )^{2}}{2 \,{\left (c^{2} x^{2} - 1\right )} d^{2}} - \frac{a b x^{2} \arcsin \left (c x\right )}{{\left (c^{2} x^{2} - 1\right )} d^{2}} - \frac{a^{2} x^{2}}{2 \,{\left (c^{2} x^{2} - 1\right )} d^{2}} - \frac{b^{2} x \arcsin \left (c x\right )}{\sqrt{-c^{2} x^{2} + 1} c d^{2}} + \frac{b^{2} \arcsin \left (c x\right )^{2}}{2 \, c^{2} d^{2}} - \frac{a b x}{\sqrt{-c^{2} x^{2} + 1} c d^{2}} + \frac{a b \arcsin \left (c x\right )}{c^{2} d^{2}} - \frac{b^{2} \log \left (2\right )}{c^{2} d^{2}} - \frac{b^{2} \log \left ({\left | -c^{2} x^{2} + 1 \right |}\right )}{2 \, c^{2} d^{2}} + \frac{a^{2}}{2 \, c^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]