Optimal. Leaf size=50 \[ \frac{c \sqrt{1-a^2 x^2}}{a}+\frac{c \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{a}-\frac{2 c \sin ^{-1}(a x)}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.163831, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {6131, 6128, 852, 1809, 844, 216, 266, 63, 208} \[ \frac{c \sqrt{1-a^2 x^2}}{a}+\frac{c \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{a}-\frac{2 c \sin ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6131
Rule 6128
Rule 852
Rule 1809
Rule 844
Rule 216
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{3 \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right ) \, dx &=-\frac{c \int \frac{e^{3 \tanh ^{-1}(a x)} (1-a x)}{x} \, dx}{a}\\ &=-\frac{c \int \frac{\left (1-a^2 x^2\right )^{3/2}}{x (1-a x)^2} \, dx}{a}\\ &=-\frac{c \int \frac{(1+a x)^2}{x \sqrt{1-a^2 x^2}} \, dx}{a}\\ &=\frac{c \sqrt{1-a^2 x^2}}{a}+\frac{c \int \frac{-a^2-2 a^3 x}{x \sqrt{1-a^2 x^2}} \, dx}{a^3}\\ &=\frac{c \sqrt{1-a^2 x^2}}{a}-(2 c) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx-\frac{c \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{a}\\ &=\frac{c \sqrt{1-a^2 x^2}}{a}-\frac{2 c \sin ^{-1}(a x)}{a}-\frac{c \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{2 a}\\ &=\frac{c \sqrt{1-a^2 x^2}}{a}-\frac{2 c \sin ^{-1}(a x)}{a}+\frac{c \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )}{a^3}\\ &=\frac{c \sqrt{1-a^2 x^2}}{a}-\frac{2 c \sin ^{-1}(a x)}{a}+\frac{c \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.053473, size = 47, normalized size = 0.94 \[ \frac{c \left (\sqrt{1-a^2 x^2}+\log \left (\sqrt{1-a^2 x^2}+1\right )-2 \sin ^{-1}(a x)-\log (x)\right )}{a} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.041, size = 84, normalized size = 1.7 \begin{align*} -{ac{x}^{2}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{c}{a}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-2\,{\frac{c}{\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{c}{a}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.48107, size = 205, normalized size = 4.1 \begin{align*} -a^{3} c{\left (\frac{x^{2}}{\sqrt{-a^{2} x^{2} + 1} a^{2}} - \frac{2}{\sqrt{-a^{2} x^{2} + 1} a^{4}}\right )} + 2 \, a^{2} c{\left (\frac{x}{\sqrt{-a^{2} x^{2} + 1} a^{2}} - \frac{\arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}} a^{2}}\right )} - \frac{2 \, c x}{\sqrt{-a^{2} x^{2} + 1}} - \frac{c{\left (\frac{1}{\sqrt{-a^{2} x^{2} + 1}} - \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right )\right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.11897, size = 144, normalized size = 2.88 \begin{align*} \frac{4 \, c \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) - c \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) + \sqrt{-a^{2} x^{2} + 1} c}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 7.76892, size = 104, normalized size = 2.08 \begin{align*} - a c \left (\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right ) - 2 c \left (\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left (x \sqrt{a^{2}} \right )} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left (x \sqrt{- a^{2}} \right )} & \text{for}\: a^{2} < 0 \end{cases}\right ) - \frac{c \left (\begin{cases} - \operatorname{acosh}{\left (\frac{1}{a x} \right )} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname{asin}{\left (\frac{1}{a x} \right )} & \text{otherwise} \end{cases}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.19219, size = 92, normalized size = 1.84 \begin{align*} -\frac{2 \, c \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} + \frac{c \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{{\left | a \right |}} + \frac{\sqrt{-a^{2} x^{2} + 1} c}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]