Optimal. Leaf size=127 \[ -\frac{10 (3-4 a x) e^{3 \coth ^{-1}(a x)}}{63 a c^4 \left (1-a^2 x^2\right )^2}+\frac{8 (3-2 a x) e^{3 \coth ^{-1}(a x)}}{21 a c^4 \left (1-a^2 x^2\right )}-\frac{(1-2 a x) e^{3 \coth ^{-1}(a x)}}{9 a c^4 \left (1-a^2 x^2\right )^3}-\frac{16 e^{3 \coth ^{-1}(a x)}}{63 a c^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.138726, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {6185, 6183} \[ -\frac{10 (3-4 a x) e^{3 \coth ^{-1}(a x)}}{63 a c^4 \left (1-a^2 x^2\right )^2}+\frac{8 (3-2 a x) e^{3 \coth ^{-1}(a x)}}{21 a c^4 \left (1-a^2 x^2\right )}-\frac{(1-2 a x) e^{3 \coth ^{-1}(a x)}}{9 a c^4 \left (1-a^2 x^2\right )^3}-\frac{16 e^{3 \coth ^{-1}(a x)}}{63 a c^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6185
Rule 6183
Rubi steps
\begin{align*} \int \frac{e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx &=-\frac{e^{3 \coth ^{-1}(a x)} (1-2 a x)}{9 a c^4 \left (1-a^2 x^2\right )^3}+\frac{10 \int \frac{e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx}{9 c}\\ &=-\frac{e^{3 \coth ^{-1}(a x)} (1-2 a x)}{9 a c^4 \left (1-a^2 x^2\right )^3}-\frac{10 e^{3 \coth ^{-1}(a x)} (3-4 a x)}{63 a c^4 \left (1-a^2 x^2\right )^2}+\frac{40 \int \frac{e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx}{21 c^2}\\ &=-\frac{e^{3 \coth ^{-1}(a x)} (1-2 a x)}{9 a c^4 \left (1-a^2 x^2\right )^3}-\frac{10 e^{3 \coth ^{-1}(a x)} (3-4 a x)}{63 a c^4 \left (1-a^2 x^2\right )^2}+\frac{8 e^{3 \coth ^{-1}(a x)} (3-2 a x)}{21 a c^4 \left (1-a^2 x^2\right )}-\frac{16 \int \frac{e^{3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{21 c^3}\\ &=-\frac{16 e^{3 \coth ^{-1}(a x)}}{63 a c^4}-\frac{e^{3 \coth ^{-1}(a x)} (1-2 a x)}{9 a c^4 \left (1-a^2 x^2\right )^3}-\frac{10 e^{3 \coth ^{-1}(a x)} (3-4 a x)}{63 a c^4 \left (1-a^2 x^2\right )^2}+\frac{8 e^{3 \coth ^{-1}(a x)} (3-2 a x)}{21 a c^4 \left (1-a^2 x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.306899, size = 82, normalized size = 0.65 \[ -\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (16 a^6 x^6-48 a^5 x^5+24 a^4 x^4+56 a^3 x^3-66 a^2 x^2+6 a x+19\right )}{63 c^4 (a x-1)^5 (a x+1)^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.135, size = 81, normalized size = 0.6 \begin{align*} -{\frac{16\,{x}^{6}{a}^{6}-48\,{x}^{5}{a}^{5}+24\,{x}^{4}{a}^{4}+56\,{x}^{3}{a}^{3}-66\,{a}^{2}{x}^{2}+6\,ax+19}{63\,{c}^{4} \left ({a}^{2}{x}^{2}-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.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.21075, size = 177, normalized size = 1.39 \begin{align*} \frac{1}{4032} \, a{\left (\frac{21 \,{\left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 18 \, \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{2} c^{4}} + \frac{\frac{54 \,{\left (a x - 1\right )}}{a x + 1} - \frac{189 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{420 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac{945 \,{\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} - 7}{a^{2} c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{2}}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.53683, size = 263, normalized size = 2.07 \begin{align*} -\frac{{\left (16 \, a^{6} x^{6} - 48 \, a^{5} x^{5} + 24 \, a^{4} x^{4} + 56 \, a^{3} x^{3} - 66 \, a^{2} x^{2} + 6 \, a x + 19\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{63 \,{\left (a^{7} c^{4} x^{6} - 4 \, a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{3} c^{4} x^{2} + 4 \, a^{2} c^{4} x - a c^{4}\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 = 1.22066, size = 230, normalized size = 1.81 \begin{align*} \frac{1}{4032} \, a{\left (\frac{{\left (a x + 1\right )}^{4}{\left (\frac{54 \,{\left (a x - 1\right )}}{a x + 1} - \frac{189 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{420 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac{945 \,{\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} - 7\right )}}{{\left (a x - 1\right )}^{4} a^{2} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}} + \frac{21 \,{\left (\frac{{\left (a x - 1\right )} a^{4} c^{8} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} - 18 \, a^{4} c^{8} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{6} c^{12}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]