Optimal. Leaf size=70 \[ \frac{2 b \tan ^{-1}\left (\frac{\sqrt{a+1} \tan \left (\frac{1}{2} \sec ^{-1}(a+b x)\right )}{\sqrt{1-a}}\right )}{a \sqrt{1-a^2}}-\frac{b \sec ^{-1}(a+b x)}{a}-\frac{\sec ^{-1}(a+b x)}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0974052, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5258, 4426, 3783, 2659, 205} \[ \frac{2 b \tan ^{-1}\left (\frac{\sqrt{a+1} \tan \left (\frac{1}{2} \sec ^{-1}(a+b x)\right )}{\sqrt{1-a}}\right )}{a \sqrt{1-a^2}}-\frac{b \sec ^{-1}(a+b x)}{a}-\frac{\sec ^{-1}(a+b x)}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5258
Rule 4426
Rule 3783
Rule 2659
Rule 205
Rubi steps
\begin{align*} \int \frac{\sec ^{-1}(a+b x)}{x^2} \, dx &=b \operatorname{Subst}\left (\int \frac{x \sec (x) \tan (x)}{(-a+\sec (x))^2} \, dx,x,\sec ^{-1}(a+b x)\right )\\ &=-\frac{\sec ^{-1}(a+b x)}{x}+b \operatorname{Subst}\left (\int \frac{1}{-a+\sec (x)} \, dx,x,\sec ^{-1}(a+b x)\right )\\ &=-\frac{b \sec ^{-1}(a+b x)}{a}-\frac{\sec ^{-1}(a+b x)}{x}+\frac{b \operatorname{Subst}\left (\int \frac{1}{1-a \cos (x)} \, dx,x,\sec ^{-1}(a+b x)\right )}{a}\\ &=-\frac{b \sec ^{-1}(a+b x)}{a}-\frac{\sec ^{-1}(a+b x)}{x}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{1-a+(1+a) x^2} \, dx,x,\tan \left (\frac{1}{2} \sec ^{-1}(a+b x)\right )\right )}{a}\\ &=-\frac{b \sec ^{-1}(a+b x)}{a}-\frac{\sec ^{-1}(a+b x)}{x}+\frac{2 b \tan ^{-1}\left (\frac{\sqrt{1+a} \tan \left (\frac{1}{2} \sec ^{-1}(a+b x)\right )}{\sqrt{1-a}}\right )}{a \sqrt{1-a^2}}\\ \end{align*}
Mathematica [C] time = 0.301405, size = 112, normalized size = 1.6 \[ -\frac{\sec ^{-1}(a+b x)}{x}+\frac{b \left (\sin ^{-1}\left (\frac{1}{a+b x}\right )-\frac{i \log \left (\frac{2 \left (a \sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} (a+b x)+\frac{i a \left (a^2+a b x-1\right )}{\sqrt{1-a^2}}\right )}{b x}\right )}{\sqrt{1-a^2}}\right )}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.219, size = 154, normalized size = 2.2 \begin{align*} -{\frac{{\rm arcsec} \left (bx+a\right )}{x}}+{\frac{b}{a \left ( bx+a \right ) }\sqrt{-1+ \left ( bx+a \right ) ^{2}}\arctan \left ({\frac{1}{\sqrt{-1+ \left ( bx+a \right ) ^{2}}}} \right ){\frac{1}{\sqrt{{\frac{-1+ \left ( bx+a \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}}}}}}-{\frac{b}{a \left ( bx+a \right ) }\sqrt{-1+ \left ( bx+a \right ) ^{2}}\ln \left ( 2\,{\frac{\sqrt{{a}^{2}-1}\sqrt{-1+ \left ( bx+a \right ) ^{2}}+a \left ( bx+a \right ) -1}{bx}} \right ){\frac{1}{\sqrt{{\frac{-1+ \left ( bx+a \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}}}}}{\frac{1}{\sqrt{{a}^{2}-1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [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]
________________________________________________________________________________________
Fricas [B] time = 2.58253, size = 666, normalized size = 9.51 \begin{align*} \left [-\frac{2 \,{\left (a^{2} - 1\right )} b x \arctan \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - \sqrt{a^{2} - 1} b x \log \left (\frac{a^{2} b x + a^{3} + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{\left (a^{2} - \sqrt{a^{2} - 1} a - 1\right )} -{\left (a b x + a^{2} - 1\right )} \sqrt{a^{2} - 1} - a}{x}\right ) +{\left (a^{3} - a\right )} \operatorname{arcsec}\left (b x + a\right )}{{\left (a^{3} - a\right )} x}, -\frac{2 \,{\left (a^{2} - 1\right )} b x \arctan \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - 2 \, \sqrt{-a^{2} + 1} b x \arctan \left (-\frac{\sqrt{-a^{2} + 1} b x - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1} \sqrt{-a^{2} + 1}}{a^{2} - 1}\right ) +{\left (a^{3} - a\right )} \operatorname{arcsec}\left (b x + a\right )}{{\left (a^{3} - a\right )} 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{\operatorname{asec}{\left (a + b x \right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.63546, size = 174, normalized size = 2.49 \begin{align*} -2 \, b{\left (\frac{\arctan \left (-\frac{{\left (x{\left | b \right |} - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )} b + a{\left | b \right |}}{b}\right )}{a \mathrm{sgn}\left (b x + a\right )} - \frac{\arctan \left (-\frac{x{\left | b \right |} - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{\sqrt{-a^{2} + 1}}\right )}{\sqrt{-a^{2} + 1} a \mathrm{sgn}\left (b x + a\right )}\right )} - \frac{\arccos \left (\frac{1}{b x + a}\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]