Optimal. Leaf size=137 \[ \frac {5 \sin ^{-1}(a x)}{2 a^7 c^2}+\frac {x^5 (a x+1)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {(15 a x+32) \sqrt {1-a^2 x^2}}{6 a^7 c^2}-\frac {8 x^2 \sqrt {1-a^2 x^2}}{3 a^5 c^2}-\frac {x^3 (6 a x+5)}{3 a^4 c^2 \sqrt {1-a^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6148, 819, 833, 780, 216} \[ \frac {x^5 (a x+1)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {x^3 (6 a x+5)}{3 a^4 c^2 \sqrt {1-a^2 x^2}}-\frac {8 x^2 \sqrt {1-a^2 x^2}}{3 a^5 c^2}-\frac {(15 a x+32) \sqrt {1-a^2 x^2}}{6 a^7 c^2}+\frac {5 \sin ^{-1}(a x)}{2 a^7 c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 216
Rule 780
Rule 819
Rule 833
Rule 6148
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^6}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac {\int \frac {x^6 (1+a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2}\\ &=\frac {x^5 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {\int \frac {x^4 (5+6 a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 a^2 c^2}\\ &=\frac {x^5 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {x^3 (5+6 a x)}{3 a^4 c^2 \sqrt {1-a^2 x^2}}+\frac {\int \frac {x^2 (15+24 a x)}{\sqrt {1-a^2 x^2}} \, dx}{3 a^4 c^2}\\ &=\frac {x^5 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {x^3 (5+6 a x)}{3 a^4 c^2 \sqrt {1-a^2 x^2}}-\frac {8 x^2 \sqrt {1-a^2 x^2}}{3 a^5 c^2}-\frac {\int \frac {x \left (-48 a-45 a^2 x\right )}{\sqrt {1-a^2 x^2}} \, dx}{9 a^6 c^2}\\ &=\frac {x^5 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {x^3 (5+6 a x)}{3 a^4 c^2 \sqrt {1-a^2 x^2}}-\frac {8 x^2 \sqrt {1-a^2 x^2}}{3 a^5 c^2}-\frac {(32+15 a x) \sqrt {1-a^2 x^2}}{6 a^7 c^2}+\frac {5 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{2 a^6 c^2}\\ &=\frac {x^5 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {x^3 (5+6 a x)}{3 a^4 c^2 \sqrt {1-a^2 x^2}}-\frac {8 x^2 \sqrt {1-a^2 x^2}}{3 a^5 c^2}-\frac {(32+15 a x) \sqrt {1-a^2 x^2}}{6 a^7 c^2}+\frac {5 \sin ^{-1}(a x)}{2 a^7 c^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 93, normalized size = 0.68 \[ \frac {2 a^5 x^5+a^4 x^4+11 a^3 x^3-31 a^2 x^2+15 (a x-1) \sqrt {1-a^2 x^2} \sin ^{-1}(a x)-17 a x+32}{6 a^7 c^2 (a x-1) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.56, size = 159, normalized size = 1.16 \[ -\frac {32 \, a^{3} x^{3} - 32 \, a^{2} x^{2} - 32 \, a x + 30 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (2 \, a^{5} x^{5} + a^{4} x^{4} + 11 \, a^{3} x^{3} - 31 \, a^{2} x^{2} - 17 \, a x + 32\right )} \sqrt {-a^{2} x^{2} + 1} + 32}{6 \, {\left (a^{10} c^{2} x^{3} - a^{9} c^{2} x^{2} - a^{8} c^{2} x + a^{7} c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )} x^{6}}{{\left (a^{2} c x^{2} - c\right )}^{2} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 225, normalized size = 1.64 \[ -\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{5} c^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{3 c^{2} a^{7}}-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 c^{2} a^{6}}+\frac {5 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 c^{2} a^{6} \sqrt {a^{2}}}+\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{6 c^{2} a^{9} \left (x -\frac {1}{a}\right )^{2}}+\frac {31 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{12 c^{2} a^{8} \left (x -\frac {1}{a}\right )}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 c^{2} a^{8} \left (x +\frac {1}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )} x^{6}}{{\left (a^{2} c x^{2} - c\right )}^{2} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.08, size = 241, normalized size = 1.76 \[ \frac {\sqrt {1-a^2\,x^2}}{6\,\left (a^9\,c^2\,x^2-2\,a^8\,c^2\,x+a^7\,c^2\right )}+\frac {\sqrt {1-a^2\,x^2}}{4\,\left (a^5\,c^2\,\sqrt {-a^2}+a^6\,c^2\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {31\,\sqrt {1-a^2\,x^2}}{12\,\left (a^5\,c^2\,\sqrt {-a^2}-a^6\,c^2\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {8\,\sqrt {1-a^2\,x^2}}{3\,a^7\,c^2}-\frac {x\,\sqrt {1-a^2\,x^2}}{2\,a^6\,c^2}+\frac {5\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a^6\,c^2\,\sqrt {-a^2}}-\frac {x^2\,\sqrt {1-a^2\,x^2}}{3\,a^5\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x^{6}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{7}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________