Optimal. Leaf size=145 \[ -\frac{1}{3} a^2 c x^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}-\frac{5}{6} a c x^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}-\frac{5}{2} c x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}-\frac{5 c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{2 a} \]
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Rubi [A] time = 0.119083, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6191, 6195, 94, 92, 208} \[ -\frac{1}{3} a^2 c x^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}-\frac{5}{6} a c x^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}-\frac{5}{2} c x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}-\frac{5 c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{2 a} \]
Antiderivative was successfully verified.
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Rule 6191
Rule 6195
Rule 94
Rule 92
Rule 208
Rubi steps
\begin{align*} \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx &=-\left (\left (a^2 c\right ) \int e^{3 \coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right ) x^2 \, dx\right )\\ &=\left (a^2 c\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{5/2}}{x^4 \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{3} a^2 c \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3+\frac{1}{3} (5 a c) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{3/2}}{x^3 \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{5}{6} a c \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2-\frac{1}{3} a^2 c \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3+\frac{1}{2} (5 c) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{x^2 \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{5}{2} c \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x-\frac{5}{6} a c \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2-\frac{1}{3} a^2 c \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3+\frac{(5 c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=-\frac{5}{2} c \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x-\frac{5}{6} a c \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2-\frac{1}{3} a^2 c \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3-\frac{(5 c) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a}-\frac{x^2}{a}} \, dx,x,\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{2 a^2}\\ &=-\frac{5}{2} c \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x-\frac{5}{6} a c \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2-\frac{1}{3} a^2 c \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3-\frac{5 c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0755309, size = 61, normalized size = 0.42 \[ -\frac{c \left (a x \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a^2 x^2+9 a x+22\right )+15 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )\right )}{6 a} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.175, size = 183, normalized size = 1.3 \begin{align*} -{\frac{ \left ( ax-1 \right ) ^{2}c}{ \left ( 6\,ax+6 \right ) a} \left ( 2\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}+9\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa+24\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }-9\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) a+24\,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{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04643, size = 231, normalized size = 1.59 \begin{align*} \frac{1}{6} \, a{\left (\frac{2 \,{\left (15 \, c \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - 40 \, c \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 33 \, c \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{\frac{3 \,{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac{3 \,{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac{{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - 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}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51622, size = 225, normalized size = 1.55 \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 (2 \, a^{3} c x^{3} + 11 \, a^{2} c x^{2} + 31 \, a c x + 22 \, c\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{6 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - c \left (\int \frac{a^{2} x^{2}}{\frac{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx + \int - \frac{1}{\frac{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18765, size = 211, normalized size = 1.46 \begin{align*} -\frac{1}{6} \, a{\left (\frac{15 \, c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{15 \, c \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} + \frac{2 \,{\left (\frac{40 \,{\left (a x - 1\right )} c \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} - \frac{15 \,{\left (a x - 1\right )}^{2} c \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} - 33 \, c \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{2}{\left (\frac{a x - 1}{a x + 1} - 1\right )}^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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