Optimal. Leaf size=92 \[ -\frac{1}{2} a c x^2 \sqrt{1-\frac{1}{a^2 x^2}}+4 c x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{8 c \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{15 c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.243798, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6175, 6178, 1805, 1807, 807, 266, 63, 208} \[ -\frac{1}{2} a c x^2 \sqrt{1-\frac{1}{a^2 x^2}}+4 c x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{8 c \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{15 c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6175
Rule 6178
Rule 1805
Rule 1807
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{-3 \coth ^{-1}(a x)} (c-a c x) \, dx &=-\left ((a c) \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right ) x \, dx\right )\\ &=(a c) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^4}{x^3 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{8 c \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-(a c) \operatorname{Subst}\left (\int \frac{-1+\frac{4 x}{a}-\frac{7 x^2}{a^2}}{x^3 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{8 c \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{1}{2} a c \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{1}{2} (a c) \operatorname{Subst}\left (\int \frac{-\frac{8}{a}+\frac{15 x}{a^2}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{8 c \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+4 c \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{1}{2} a c \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{(15 c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=\frac{8 c \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+4 c \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{1}{2} a c \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{(15 c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{4 a}\\ &=\frac{8 c \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+4 c \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{1}{2} a c \sqrt{1-\frac{1}{a^2 x^2}} x^2-\frac{1}{2} (15 a c) \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )\\ &=\frac{8 c \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+4 c \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{1}{2} a c \sqrt{1-\frac{1}{a^2 x^2}} x^2-\frac{15 c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a}\\ \end{align*}
Mathematica [A] time = 0.133689, size = 68, normalized size = 0.74 \[ \frac{1}{2} c \left (\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (-a^2 x^2+7 a x+24\right )}{a x+1}-\frac{15 \log \left (a x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{a}\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.16, size = 422, normalized size = 4.6 \begin{align*} -{\frac{c}{2\, \left ( ax-1 \right ) a} \left ( \sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{3}{a}^{3}+2\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}-\ln \left ({ \left ({a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \right ){x}^{2}{a}^{3}-16\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}+16\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}+\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa-2\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) x{a}^{2}+8\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-32\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa+32\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) x{a}^{2}-\ln \left ({ \left ({a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \right ) a-16\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }+16\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.06275, size = 211, normalized size = 2.29 \begin{align*} \frac{1}{2} \, a{\left (\frac{2 \,{\left (9 \, c \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 7 \, c \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{\frac{2 \,{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac{{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} - \frac{15 \, c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac{15 \, c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} + \frac{16 \, c \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.66708, size = 201, normalized size = 2.18 \begin{align*} -\frac{15 \, c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 15 \, c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (a^{2} c x^{2} - 7 \, a c x - 24 \, c\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{2 \, a} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]