Optimal. Leaf size=114 \[ -\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (6 a-\frac {1}{x}\right )}{2 a^2}+c^4 x \left (1-\frac {1}{a^2 x^2}\right )^{3/2}-\frac {3 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}-\frac {c^4 \csc ^{-1}(a x)}{2 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.27, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6177, 1807, 1809, 815, 844, 216, 266, 63, 208} \[ -\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (6 a-\frac {1}{x}\right )}{2 a^2}+c^4 x \left (1-\frac {1}{a^2 x^2}\right )^{3/2}-\frac {3 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}-\frac {c^4 \csc ^{-1}(a x)}{2 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 208
Rule 216
Rule 266
Rule 815
Rule 844
Rule 1807
Rule 1809
Rule 6177
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx &=-\left (c \operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^3 \sqrt {1-\frac {x^2}{a^2}}}{x^2} \, dx,x,\frac {1}{x}\right )\right )\\ &=c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x+c \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}} \left (\frac {3 c^3}{a}-\frac {c^3 x}{a^2}+\frac {c^3 x^2}{a^3}\right )}{x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a}+c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x-\frac {1}{3} \left (a^2 c\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {9 c^3}{a^3}+\frac {3 c^3 x}{a^4}\right ) \sqrt {1-\frac {x^2}{a^2}}}{x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (6 a-\frac {1}{x}\right )}{2 a^2}+c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x+\frac {1}{6} \left (a^4 c\right ) \operatorname {Subst}\left (\int \frac {\frac {18 c^3}{a^5}-\frac {3 c^3 x}{a^6}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (6 a-\frac {1}{x}\right )}{2 a^2}+c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x-\frac {c^4 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 a^2}+\frac {\left (3 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (6 a-\frac {1}{x}\right )}{2 a^2}+c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x-\frac {c^4 \csc ^{-1}(a x)}{2 a}+\frac {\left (3 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 a}\\ &=-\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (6 a-\frac {1}{x}\right )}{2 a^2}+c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x-\frac {c^4 \csc ^{-1}(a x)}{2 a}-\left (3 a c^4\right ) \operatorname {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )\\ &=-\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (6 a-\frac {1}{x}\right )}{2 a^2}+c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x-\frac {c^4 \csc ^{-1}(a x)}{2 a}-\frac {3 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.20, size = 175, normalized size = 1.54 \[ \frac {c^4 \left (6 a^5 x^5+16 a^4 x^4-15 a^3 x^3-14 a^2 x^2+24 a^4 x^4 \sqrt {1-\frac {1}{a^2 x^2}} \sin ^{-1}\left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )+9 a^4 x^4 \sqrt {1-\frac {1}{a^2 x^2}} \sin ^{-1}\left (\frac {1}{a x}\right )-18 a^4 x^4 \sqrt {1-\frac {1}{a^2 x^2}} \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )+9 a x-2\right )}{6 a^5 x^4 \sqrt {1-\frac {1}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 156, normalized size = 1.37 \[ \frac {6 \, a^{3} c^{4} x^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - 18 \, a^{3} c^{4} x^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 18 \, a^{3} c^{4} x^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (6 \, a^{4} c^{4} x^{4} + 22 \, a^{3} c^{4} x^{3} + 7 \, a^{2} c^{4} x^{2} - 7 \, a c^{4} x + 2 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.18, size = 211, normalized size = 1.85 \[ \frac {1}{3} \, a c^{4} {\left (\frac {3 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {9 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} - \frac {6 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} {\left (\frac {a x - 1}{a x + 1} - 1\right )}} + \frac {\frac {28 \, {\left (a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} + \frac {27 \, {\left (a x - 1\right )}^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + 9 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} {\left (\frac {a x - 1}{a x + 1} + 1\right )}^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.05, size = 224, normalized size = 1.96 \[ -\frac {\left (a x -1\right ) c^{4} \left (-18 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{4} a^{4}+18 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{2} a^{2}+3 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{3} a^{3}+3 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+18 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{3} a^{4}-9 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x a +2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{6 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{4} x^{3} \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.41, size = 224, normalized size = 1.96 \[ \frac {1}{3} \, {\left (\frac {3 \, c^{4} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {9 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {9 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {21 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 17 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 37 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 15 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {2 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {2 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + a^{2}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.32, size = 183, normalized size = 1.61 \[ \frac {5\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}+\frac {37\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}+\frac {17\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{3}-7\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{a+\frac {2\,a\,\left (a\,x-1\right )}{a\,x+1}-\frac {2\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}}+\frac {c^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}-\frac {6\,c^4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {c^{4} \left (\int \frac {a^{4}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \frac {1}{x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {4 a}{x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx + \int \frac {6 a^{2}}{x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {4 a^{3}}{x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx\right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________