Optimal. Leaf size=84 \[ -\frac {c^5 (a x+1)^{11}}{11 a}+\frac {4 c^5 (a x+1)^{10}}{5 a}-\frac {8 c^5 (a x+1)^9}{3 a}+\frac {4 c^5 (a x+1)^8}{a}-\frac {16 c^5 (a x+1)^7}{7 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6167, 6140, 43} \[ -\frac {c^5 (a x+1)^{11}}{11 a}+\frac {4 c^5 (a x+1)^{10}}{5 a}-\frac {8 c^5 (a x+1)^9}{3 a}+\frac {4 c^5 (a x+1)^8}{a}-\frac {16 c^5 (a x+1)^7}{7 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 6140
Rule 6167
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx\\ &=-\left (c^5 \int (1-a x)^4 (1+a x)^6 \, dx\right )\\ &=-\left (c^5 \int \left (16 (1+a x)^6-32 (1+a x)^7+24 (1+a x)^8-8 (1+a x)^9+(1+a x)^{10}\right ) \, dx\right )\\ &=-\frac {16 c^5 (1+a x)^7}{7 a}+\frac {4 c^5 (1+a x)^8}{a}-\frac {8 c^5 (1+a x)^9}{3 a}+\frac {4 c^5 (1+a x)^{10}}{5 a}-\frac {c^5 (1+a x)^{11}}{11 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 47, normalized size = 0.56 \[ -\frac {c^5 (a x+1)^7 \left (105 a^4 x^4-504 a^3 x^3+938 a^2 x^2-812 a x+281\right )}{1155 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 113, normalized size = 1.35 \[ -\frac {1}{11} \, a^{10} c^{5} x^{11} - \frac {1}{5} \, a^{9} c^{5} x^{10} + \frac {1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} - \frac {2}{7} \, a^{6} c^{5} x^{7} - 2 \, a^{5} c^{5} x^{6} - \frac {2}{5} \, a^{4} c^{5} x^{5} + 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} - a c^{5} x^{2} - c^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.14, size = 113, normalized size = 1.35 \[ -\frac {1}{11} \, a^{10} c^{5} x^{11} - \frac {1}{5} \, a^{9} c^{5} x^{10} + \frac {1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} - \frac {2}{7} \, a^{6} c^{5} x^{7} - 2 \, a^{5} c^{5} x^{6} - \frac {2}{5} \, a^{4} c^{5} x^{5} + 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} - a c^{5} x^{2} - c^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 85, normalized size = 1.01 \[ c^{5} \left (-\frac {1}{11} a^{10} x^{11}-\frac {1}{5} a^{9} x^{10}+\frac {1}{3} x^{9} a^{8}+a^{7} x^{8}-\frac {2}{7} a^{6} x^{7}-2 x^{6} a^{5}-\frac {2}{5} a^{4} x^{5}+2 x^{4} a^{3}+x^{3} a^{2}-a \,x^{2}-x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 113, normalized size = 1.35 \[ -\frac {1}{11} \, a^{10} c^{5} x^{11} - \frac {1}{5} \, a^{9} c^{5} x^{10} + \frac {1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} - \frac {2}{7} \, a^{6} c^{5} x^{7} - 2 \, a^{5} c^{5} x^{6} - \frac {2}{5} \, a^{4} c^{5} x^{5} + 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} - a c^{5} x^{2} - c^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.25, size = 113, normalized size = 1.35 \[ -\frac {a^{10}\,c^5\,x^{11}}{11}-\frac {a^9\,c^5\,x^{10}}{5}+\frac {a^8\,c^5\,x^9}{3}+a^7\,c^5\,x^8-\frac {2\,a^6\,c^5\,x^7}{7}-2\,a^5\,c^5\,x^6-\frac {2\,a^4\,c^5\,x^5}{5}+2\,a^3\,c^5\,x^4+a^2\,c^5\,x^3-a\,c^5\,x^2-c^5\,x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.10, size = 119, normalized size = 1.42 \[ - \frac {a^{10} c^{5} x^{11}}{11} - \frac {a^{9} c^{5} x^{10}}{5} + \frac {a^{8} c^{5} x^{9}}{3} + a^{7} c^{5} x^{8} - \frac {2 a^{6} c^{5} x^{7}}{7} - 2 a^{5} c^{5} x^{6} - \frac {2 a^{4} c^{5} x^{5}}{5} + 2 a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} - a c^{5} x^{2} - c^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________