Optimal. Leaf size=67 \[ \frac{2 b^2 \log (x)}{(1-a)^3}-\frac{2 b^2 \log (-a-b x+1)}{(1-a)^3}-\frac{2 b}{(1-a)^2 x}-\frac{a+1}{2 (1-a) x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.050526, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6163, 77} \[ \frac{2 b^2 \log (x)}{(1-a)^3}-\frac{2 b^2 \log (-a-b x+1)}{(1-a)^3}-\frac{2 b}{(1-a)^2 x}-\frac{a+1}{2 (1-a) x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6163
Rule 77
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a+b x)}}{x^3} \, dx &=\int \frac{1+a+b x}{x^3 (1-a-b x)} \, dx\\ &=\int \left (\frac{-1-a}{(-1+a) x^3}+\frac{2 b}{(-1+a)^2 x^2}-\frac{2 b^2}{(-1+a)^3 x}+\frac{2 b^3}{(-1+a)^3 (-1+a+b x)}\right ) \, dx\\ &=-\frac{1+a}{2 (1-a) x^2}-\frac{2 b}{(1-a)^2 x}+\frac{2 b^2 \log (x)}{(1-a)^3}-\frac{2 b^2 \log (1-a-b x)}{(1-a)^3}\\ \end{align*}
Mathematica [A] time = 0.0333783, size = 54, normalized size = 0.81 \[ \frac{(a-1) \left (a^2-4 b x-1\right )+4 b^2 x^2 \log (-a-b x+1)-4 b^2 x^2 \log (x)}{2 (a-1)^3 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.037, size = 63, normalized size = 0.9 \begin{align*}{\frac{1}{ \left ( 2\,a-2 \right ){x}^{2}}}+{\frac{a}{ \left ( 2\,a-2 \right ){x}^{2}}}-2\,{\frac{b}{ \left ( a-1 \right ) ^{2}x}}-2\,{\frac{{b}^{2}\ln \left ( x \right ) }{ \left ( a-1 \right ) ^{3}}}+2\,{\frac{{b}^{2}\ln \left ( bx+a-1 \right ) }{ \left ( a-1 \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.958169, size = 100, normalized size = 1.49 \begin{align*} \frac{2 \, b^{2} \log \left (b x + a - 1\right )}{a^{3} - 3 \, a^{2} + 3 \, a - 1} - \frac{2 \, b^{2} \log \left (x\right )}{a^{3} - 3 \, a^{2} + 3 \, a - 1} + \frac{a^{2} - 4 \, b x - 1}{2 \,{\left (a^{2} - 2 \, a + 1\right )} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.88922, size = 161, normalized size = 2.4 \begin{align*} \frac{4 \, b^{2} x^{2} \log \left (b x + a - 1\right ) - 4 \, b^{2} x^{2} \log \left (x\right ) + a^{3} - 4 \,{\left (a - 1\right )} b x - a^{2} - a + 1}{2 \,{\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.669218, size = 209, normalized size = 3.12 \begin{align*} - \frac{2 b^{2} \log{\left (x + \frac{- \frac{2 a^{4} b^{2}}{\left (a - 1\right )^{3}} + \frac{8 a^{3} b^{2}}{\left (a - 1\right )^{3}} - \frac{12 a^{2} b^{2}}{\left (a - 1\right )^{3}} + 2 a b^{2} + \frac{8 a b^{2}}{\left (a - 1\right )^{3}} - 2 b^{2} - \frac{2 b^{2}}{\left (a - 1\right )^{3}}}{4 b^{3}} \right )}}{\left (a - 1\right )^{3}} + \frac{2 b^{2} \log{\left (x + \frac{\frac{2 a^{4} b^{2}}{\left (a - 1\right )^{3}} - \frac{8 a^{3} b^{2}}{\left (a - 1\right )^{3}} + \frac{12 a^{2} b^{2}}{\left (a - 1\right )^{3}} + 2 a b^{2} - \frac{8 a b^{2}}{\left (a - 1\right )^{3}} - 2 b^{2} + \frac{2 b^{2}}{\left (a - 1\right )^{3}}}{4 b^{3}} \right )}}{\left (a - 1\right )^{3}} - \frac{- a^{2} + 4 b x + 1}{x^{2} \left (2 a^{2} - 4 a + 2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1441, size = 123, normalized size = 1.84 \begin{align*} \frac{2 \, b^{3} \log \left ({\left | b x + a - 1 \right |}\right )}{a^{3} b - 3 \, a^{2} b + 3 \, a b - b} - \frac{2 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{3} - 3 \, a^{2} + 3 \, a - 1} + \frac{a^{3} - a^{2} - 4 \,{\left (a b - b\right )} x - a + 1}{2 \,{\left (a - 1\right )}^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]