Optimal. Leaf size=56 \[ -\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{c}+\frac{1}{2} x^2 \left (a+b \sec ^{-1}(c x)\right )^2+\frac{b^2 \log (x)}{c^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0682639, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5222, 4409, 4184, 3475} \[ -\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{c}+\frac{1}{2} x^2 \left (a+b \sec ^{-1}(c x)\right )^2+\frac{b^2 \log (x)}{c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5222
Rule 4409
Rule 4184
Rule 3475
Rubi steps
\begin{align*} \int x \left (a+b \sec ^{-1}(c x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int (a+b x)^2 \sec ^2(x) \tan (x) \, dx,x,\sec ^{-1}(c x)\right )}{c^2}\\ &=\frac{1}{2} x^2 \left (a+b \sec ^{-1}(c x)\right )^2-\frac{b \operatorname{Subst}\left (\int (a+b x) \sec ^2(x) \, dx,x,\sec ^{-1}(c x)\right )}{c^2}\\ &=-\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x \left (a+b \sec ^{-1}(c x)\right )}{c}+\frac{1}{2} x^2 \left (a+b \sec ^{-1}(c x)\right )^2+\frac{b^2 \operatorname{Subst}\left (\int \tan (x) \, dx,x,\sec ^{-1}(c x)\right )}{c^2}\\ &=-\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x \left (a+b \sec ^{-1}(c x)\right )}{c}+\frac{1}{2} x^2 \left (a+b \sec ^{-1}(c x)\right )^2+\frac{b^2 \log (x)}{c^2}\\ \end{align*}
Mathematica [A] time = 0.149093, size = 90, normalized size = 1.61 \[ \frac{a c x \left (a c x-2 b \sqrt{1-\frac{1}{c^2 x^2}}\right )+2 b c x \sec ^{-1}(c x) \left (a c x-b \sqrt{1-\frac{1}{c^2 x^2}}\right )+b^2 c^2 x^2 \sec ^{-1}(c x)^2+2 b^2 \log (c x)}{2 c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.249, size = 134, normalized size = 2.4 \begin{align*}{\frac{{a}^{2}{x}^{2}}{2}}+{\frac{{x}^{2}{b}^{2} \left ({\rm arcsec} \left (cx\right ) \right ) ^{2}}{2}}-{\frac{{b}^{2}{\rm arcsec} \left (cx\right )x}{c}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}-{\frac{{b}^{2}}{{c}^{2}}\ln \left ({\frac{1}{cx}} \right ) }+ab{x}^{2}{\rm arcsec} \left (cx\right )-{\frac{xab}{c}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{ab}{{c}^{3}x}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.0201, size = 117, normalized size = 2.09 \begin{align*} \frac{1}{2} \, b^{2} x^{2} \operatorname{arcsec}\left (c x\right )^{2} + \frac{1}{2} \, a^{2} x^{2} +{\left (x^{2} \operatorname{arcsec}\left (c x\right ) - \frac{x \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c}\right )} a b -{\left (\frac{x \sqrt{-\frac{1}{c^{2} x^{2}} + 1} \operatorname{arcsec}\left (c x\right )}{c} - \frac{\log \left (x\right )}{c^{2}}\right )} b^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.75695, size = 266, normalized size = 4.75 \begin{align*} \frac{b^{2} c^{2} x^{2} \operatorname{arcsec}\left (c x\right )^{2} + a^{2} c^{2} x^{2} + 4 \, a b c^{2} \arctan \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) + 2 \, b^{2} \log \left (x\right ) + 2 \,{\left (a b c^{2} x^{2} - a b c^{2}\right )} \operatorname{arcsec}\left (c x\right ) - 2 \, \sqrt{c^{2} x^{2} - 1}{\left (b^{2} \operatorname{arcsec}\left (c x\right ) + a b\right )}}{2 \, c^{2}} \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 x \left (a + b \operatorname{asec}{\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{\left (b \operatorname{arcsec}\left (c x\right ) + a\right )}^{2} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]