Optimal. Leaf size=89 \[ \frac {2 c (a x+1)^3 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}-\frac {c (a x+1)^4 \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 89, 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 = {6143, 6140, 43} \[ \frac {2 c (a x+1)^3 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}-\frac {c (a x+1)^4 \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 6140
Rule 6143
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx &=\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \int e^{\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{3/2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \int (1-a x) (1+a x)^2 \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \int \left (2 (1+a x)^2-(1+a x)^3\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {2 c (1+a x)^3 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}-\frac {c (1+a x)^4 \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 57, normalized size = 0.64 \[ -\frac {c x \left (3 a^3 x^3+4 a^2 x^2-6 a x-12\right ) \sqrt {c-a^2 c x^2}}{12 \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.81, size = 68, normalized size = 0.76 \[ \frac {{\left (3 \, a^{3} c x^{4} + 4 \, a^{2} c x^{3} - 6 \, a c x^{2} - 12 \, c x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{12 \, {\left (a^{2} x^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (a x + 1\right )}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 65, normalized size = 0.73 \[ \frac {x \left (3 x^{3} a^{3}+4 a^{2} x^{2}-6 a x -12\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{12 \left (a x -1\right ) \left (a x +1\right ) \sqrt {-a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 66, normalized size = 0.74 \[ -\frac {1}{3} \, a^{2} c^{\frac {3}{2}} x^{3} + c^{\frac {3}{2}} x + \frac {1}{4} \, {\left (a^{2} c^{\frac {3}{2}} x^{4} + 2 \, c^{\frac {3}{2}} x^{2} - \frac {4 \, \sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c} c}{a^{2}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.06, size = 55, normalized size = 0.62 \[ \frac {\sqrt {c-a^2\,c\,x^2}\,\left (-\frac {c\,a^3\,x^4}{4}-\frac {c\,a^2\,x^3}{3}+\frac {c\,a\,x^2}{2}+c\,x\right )}{\sqrt {1-a^2\,x^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________