Optimal. Leaf size=94 \[ -\frac{\left (a+\frac{1}{x}\right )^7}{9 a^8 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}+\frac{16 \left (a+\frac{1}{x}\right )^6}{63 a^7 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{47 \left (a+\frac{1}{x}\right )^5}{315 a^6 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}} \]
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Rubi [A] time = 0.275223, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6175, 6178, 852, 1635, 789, 651} \[ -\frac{\left (a+\frac{1}{x}\right )^7}{9 a^8 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}+\frac{16 \left (a+\frac{1}{x}\right )^6}{63 a^7 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{47 \left (a+\frac{1}{x}\right )^5}{315 a^6 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6178
Rule 852
Rule 1635
Rule 789
Rule 651
Rubi steps
\begin{align*} \int \frac{e^{3 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx &=\frac{\int \frac{e^{3 \coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right )^4 x^4} \, dx}{a^4 c^4}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}}{\left (1-\frac{x}{a}\right )^7} \, dx,x,\frac{1}{x}\right )}{a^4 c^4}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (1+\frac{x}{a}\right )^7}{\left (1-\frac{x^2}{a^2}\right )^{11/2}} \, dx,x,\frac{1}{x}\right )}{a^4 c^4}\\ &=-\frac{\left (a+\frac{1}{x}\right )^7}{9 a^8 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}+\frac{\operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^6 \left (7 a^2+9 a x\right )}{\left (1-\frac{x^2}{a^2}\right )^{9/2}} \, dx,x,\frac{1}{x}\right )}{9 a^4 c^4}\\ &=\frac{16 \left (a+\frac{1}{x}\right )^6}{63 a^7 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{\left (a+\frac{1}{x}\right )^7}{9 a^8 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}-\frac{47 \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^5}{\left (1-\frac{x^2}{a^2}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{63 a^2 c^4}\\ &=-\frac{47 \left (a+\frac{1}{x}\right )^5}{315 a^6 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}+\frac{16 \left (a+\frac{1}{x}\right )^6}{63 a^7 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{\left (a+\frac{1}{x}\right )^7}{9 a^8 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.0615454, size = 50, normalized size = 0.53 \[ -\frac{x \sqrt{1-\frac{1}{a^2 x^2}} (a x+1)^2 \left (2 a^2 x^2-14 a x+47\right )}{315 c^4 (a x-1)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.123, size = 50, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,{a}^{2}{x}^{2}-14\,ax+47 \right ) \left ( ax+1 \right ) }{315\,{c}^{4} \left ( ax-1 \right ) ^{3}a} \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02957, size = 74, normalized size = 0.79 \begin{align*} \frac{\frac{90 \,{\left (a x - 1\right )}}{a x + 1} - \frac{63 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 35}{1260 \, a c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57475, size = 251, normalized size = 2.67 \begin{align*} -\frac{{\left (2 \, a^{5} x^{5} - 8 \, a^{4} x^{4} + 11 \, a^{3} x^{3} + 101 \, a^{2} x^{2} + 127 \, a x + 47\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{315 \,{\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A] time = 1.38352, size = 93, normalized size = 0.99 \begin{align*} \frac{{\left (a x + 1\right )}^{4}{\left (\frac{90 \,{\left (a x - 1\right )}}{a x + 1} - \frac{63 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 35\right )}}{1260 \,{\left (a x - 1\right )}^{4} a c^{4} \sqrt{\frac{a x - 1}{a x + 1}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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