Optimal. Leaf size=103 \[ -\frac{2 (1-a)^2 \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{-a-b x+1}}\right )}{(a+1) \sqrt{1-a^2}}+\frac{4 \sqrt{-a-b x+1}}{(a+1) \sqrt{a+b x+1}}+\sin ^{-1}(a+b x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0853212, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {6163, 98, 157, 53, 619, 216, 93, 208} \[ -\frac{2 (1-a)^2 \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{-a-b x+1}}\right )}{(a+1) \sqrt{1-a^2}}+\frac{4 \sqrt{-a-b x+1}}{(a+1) \sqrt{a+b x+1}}+\sin ^{-1}(a+b x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6163
Rule 98
Rule 157
Rule 53
Rule 619
Rule 216
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-3 \tanh ^{-1}(a+b x)}}{x} \, dx &=\int \frac{(1-a-b x)^{3/2}}{x (1+a+b x)^{3/2}} \, dx\\ &=\frac{4 \sqrt{1-a-b x}}{(1+a) \sqrt{1+a+b x}}+\frac{2 \int \frac{\frac{1}{2} (1-a)^2 b+\frac{1}{2} (1+a) b^2 x}{x \sqrt{1-a-b x} \sqrt{1+a+b x}} \, dx}{(1+a) b}\\ &=\frac{4 \sqrt{1-a-b x}}{(1+a) \sqrt{1+a+b x}}+\frac{(1-a)^2 \int \frac{1}{x \sqrt{1-a-b x} \sqrt{1+a+b x}} \, dx}{1+a}+b \int \frac{1}{\sqrt{1-a-b x} \sqrt{1+a+b x}} \, dx\\ &=\frac{4 \sqrt{1-a-b x}}{(1+a) \sqrt{1+a+b x}}+\frac{\left (2 (1-a)^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-a-(-1+a) x^2} \, dx,x,\frac{\sqrt{1+a+b x}}{\sqrt{1-a-b x}}\right )}{1+a}+b \int \frac{1}{\sqrt{(1-a) (1+a)-2 a b x-b^2 x^2}} \, dx\\ &=\frac{4 \sqrt{1-a-b x}}{(1+a) \sqrt{1+a+b x}}-\frac{2 (1-a)^2 \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{1+a+b x}}{\sqrt{1+a} \sqrt{1-a-b x}}\right )}{(1+a) \sqrt{1-a^2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{4 b^2}}} \, dx,x,-2 a b-2 b^2 x\right )}{2 b}\\ &=\frac{4 \sqrt{1-a-b x}}{(1+a) \sqrt{1+a+b x}}+\sin ^{-1}(a+b x)-\frac{2 (1-a)^2 \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{1+a+b x}}{\sqrt{1+a} \sqrt{1-a-b x}}\right )}{(1+a) \sqrt{1-a^2}}\\ \end{align*}
Mathematica [A] time = 0.442148, size = 131, normalized size = 1.27 \[ -\frac{4 (a+b x-1)}{(a+1) \sqrt{-(a+b x-1) (a+b x+1)}}+2 \left (\frac{a-1}{a+1}\right )^{3/2} \tan ^{-1}\left (\frac{\sqrt{-a-b x+1}}{\sqrt{\frac{a-1}{a+1}} \sqrt{a+b x+1}}\right )+\frac{2 \sqrt{-b} \sinh ^{-1}\left (\frac{\sqrt{-b} \sqrt{-a-b x+1}}{\sqrt{2} \sqrt{b}}\right )}{\sqrt{b}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.066, size = 1062, normalized size = 10.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac{3}{2}}}{{\left (b x + a + 1\right )}^{3} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.83458, size = 1057, normalized size = 10.26 \begin{align*} \left [\frac{{\left ({\left (a - 1\right )} b x + a^{2} - 1\right )} \sqrt{-\frac{a - 1}{a + 1}} \log \left (\frac{{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \,{\left (a^{3} - a\right )} b x - 4 \, a^{2} - 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (a^{3} +{\left (a^{2} + a\right )} b x + a^{2} - a - 1\right )} \sqrt{-\frac{a - 1}{a + 1}} + 2}{x^{2}}\right ) - 2 \,{\left ({\left (a + 1\right )} b x + a^{2} + 2 \, a + 1\right )} \arctan \left (\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + 8 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \,{\left ({\left (a + 1\right )} b x + a^{2} + 2 \, a + 1\right )}}, \frac{{\left ({\left (a - 1\right )} b x + a^{2} - 1\right )} \sqrt{\frac{a - 1}{a + 1}} \arctan \left (\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (a b x + a^{2} - 1\right )} \sqrt{\frac{a - 1}{a + 1}}}{{\left (a - 1\right )} b^{2} x^{2} + a^{3} + 2 \,{\left (a^{2} - a\right )} b x - a^{2} - a + 1}\right ) -{\left ({\left (a + 1\right )} b x + a^{2} + 2 \, a + 1\right )} \arctan \left (\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + 4 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{{\left (a + 1\right )} b x + a^{2} + 2 \, a + 1}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [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]
________________________________________________________________________________________
Giac [A] time = 1.21332, size = 208, normalized size = 2.02 \begin{align*} -\frac{b \arcsin \left (-b x - a\right ) \mathrm{sgn}\left (b\right )}{{\left | b \right |}} + \frac{2 \,{\left (a^{2} b - 2 \, a b + b\right )} \arctan \left (\frac{\frac{{\left (\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left | b \right |} + b\right )} a}{b^{2} x + a b} - 1}{\sqrt{a^{2} - 1}}\right )}{\sqrt{a^{2} - 1}{\left (a{\left | b \right |} +{\left | b \right |}\right )}} - \frac{8 \, b}{{\left (a{\left | b \right |} +{\left | b \right |}\right )}{\left (\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left | b \right |} + b}{b^{2} x + a b} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]