Optimal. Leaf size=86 \[ -\frac {2^{p+\frac {1}{2}} (1-a x)^{p+\frac {3}{2}} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac {1}{2}-p,p+\frac {3}{2};p+\frac {5}{2};\frac {1}{2} (1-a x)\right )}{a (2 p+3)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6143, 6140, 69} \[ -\frac {2^{p+\frac {1}{2}} (1-a x)^{p+\frac {3}{2}} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac {1}{2}-p,p+\frac {3}{2};p+\frac {5}{2};\frac {1}{2} (1-a x)\right )}{a (2 p+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 69
Rule 6140
Rule 6143
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int e^{-\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^p \, dx\\ &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int (1-a x)^{\frac {1}{2}+p} (1+a x)^{-\frac {1}{2}+p} \, dx\\ &=-\frac {2^{\frac {1}{2}+p} (1-a x)^{\frac {3}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac {1}{2}-p,\frac {3}{2}+p;\frac {5}{2}+p;\frac {1}{2} (1-a x)\right )}{a (3+2 p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 83, normalized size = 0.97 \[ \frac {(a x-1) (2-2 a x)^{p+\frac {1}{2}} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac {1}{2}-p,p+\frac {3}{2};p+\frac {5}{2};\frac {1}{2} (1-a x)\right )}{a (2 p+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left (-a^{2} c x^{2} + c\right )}^{p}}{a x + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} {\left (-a^{2} c x^{2} + c\right )}^{p}}{a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} \sqrt {-a^{2} x^{2}+1}}{a x +1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} {\left (-a^{2} c x^{2} + c\right )}^{p}}{a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^p\,\sqrt {1-a^2\,x^2}}{a\,x+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p}}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________