Optimal. Leaf size=43 \[ \text {Int}\left (\frac {1}{\left (1-c^2 x^2\right ) \left (a+b \tan ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )},x\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \tan ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \tan ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )} \, dx &=\int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \tan ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \tan ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {1}{a c^{2} x^{2} + {\left (b c^{2} x^{2} - b\right )} \arctan \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) - a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {1}{{\left (c^{2} x^{2} - 1\right )} {\left (b \arctan \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.34, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-c^{2} x^{2}+1\right ) \left (a +b \arctan \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {1}{{\left (c^{2} x^{2} - 1\right )} {\left (b \arctan \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [A] time = 0.00, size = -1, normalized size = -0.02 \[ -\int \frac {1}{\left (a+b\,\mathrm {atan}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )\,\left (c^2\,x^2-1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {1}{a c^{2} x^{2} - a + b c^{2} x^{2} \operatorname {atan}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )} - b \operatorname {atan}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________