\(\int \frac {1}{(1-c^2 x^2) (a+b \cot ^{-1}(\frac {\sqrt {1-c x}}{\sqrt {1+c x}}))^2} \, dx\) [156]

Optimal result

Integrand size = 40, antiderivative size = 40 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx=\text {Int}\left (\frac {1}{\left (1-c^2 x^2\right ) \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2},x\right ) \]

[Out]

Unintegrable(1/(-c^2*x^2+1)/(a+b*arccot((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2,x)

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx=\int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx \]

[In]

Int[1/((1 - c^2*x^2)*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2),x]

[Out]

Defer[Int][1/((1 - c^2*x^2)*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx=\int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx \]

[In]

Integrate[1/((1 - c^2*x^2)*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2),x]

[Out]

Integrate[1/((1 - c^2*x^2)*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.91 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.90

\[\int \frac {1}{\left (-c^{2} x^{2}+1\right ) \left (a +b \,\operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )\right )^{2}}d x\]

[In]

int(1/(-c^2*x^2+1)/(a+b*arccot((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2,x)

[Out]

int(1/(-c^2*x^2+1)/(a+b*arccot((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2,x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 91, normalized size of antiderivative = 2.28 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx=\int { -\frac {1}{{\left (c^{2} x^{2} - 1\right )} {\left (b \operatorname {arccot}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(-c^2*x^2+1)/(a+b*arccot((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2,x, algorithm="fricas")

[Out]

integral(-1/(a^2*c^2*x^2 + (b^2*c^2*x^2 - b^2)*arccot(sqrt(-c*x + 1)/sqrt(c*x + 1))^2 - a^2 + 2*(a*b*c^2*x^2 -
a*b)*arccot(sqrt(-c*x + 1)/sqrt(c*x + 1))), x)

Sympy [N/A]

Not integrable

Time = 6.40 (sec) , antiderivative size = 126, normalized size of antiderivative = 3.15 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx=- \int \frac {1}{a^{2} c^{2} x^{2} - a^{2} + 2 a b c^{2} x^{2} \operatorname {acot}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )} - 2 a b \operatorname {acot}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )} + b^{2} c^{2} x^{2} \operatorname {acot}^{2}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )} - b^{2} \operatorname {acot}^{2}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}\, dx \]

[In]

integrate(1/(-c**2*x**2+1)/(a+b*acot((-c*x+1)**(1/2)/(c*x+1)**(1/2)))**2,x)

[Out]

-Integral(1/(a**2*c**2*x**2 - a**2 + 2*a*b*c**2*x**2*acot(sqrt(-c*x + 1)/sqrt(c*x + 1)) - 2*a*b*acot(sqrt(-c*x
+ 1)/sqrt(c*x + 1)) + b**2*c**2*x**2*acot(sqrt(-c*x + 1)/sqrt(c*x + 1))**2 - b**2*acot(sqrt(-c*x + 1)/sqrt(c*
x + 1))**2), x)

Maxima [N/A]

Not integrable

Time = 0.50 (sec) , antiderivative size = 163, normalized size of antiderivative = 4.08 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx=\int { -\frac {1}{{\left (c^{2} x^{2} - 1\right )} {\left (b \operatorname {arccot}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(-c^2*x^2+1)/(a+b*arccot((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2,x, algorithm="maxima")

[Out]

-2*(2*(b^2*c^2*arctan2(sqrt(c*x + 1), sqrt(-c*x + 1)) + a*b*c^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*integrate(1/2*x/
((a*b*c^2*x^2 - a*b + (b^2*c^2*x^2 - b^2)*arctan2(sqrt(c*x + 1), sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1)
), x) + 1)/((b^2*c*arctan2(sqrt(c*x + 1), sqrt(-c*x + 1)) + a*b*c)*sqrt(c*x + 1)*sqrt(-c*x + 1))

Giac [N/A]

Not integrable

Time = 0.53 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx=\int { -\frac {1}{{\left (c^{2} x^{2} - 1\right )} {\left (b \operatorname {arccot}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(-c^2*x^2+1)/(a+b*arccot((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2,x, algorithm="giac")

[Out]

integrate(-1/((c^2*x^2 - 1)*(b*arccot(sqrt(-c*x + 1)/sqrt(c*x + 1)) + a)^2), x)

Mupad [N/A]

Not integrable

Time = 1.69 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2} \, dx=-\int \frac {1}{{\left (a+b\,\mathrm {acot}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^2\,\left (c^2\,x^2-1\right )} \,d x \]

[In]

int(-1/((a + b*acot((1 - c*x)^(1/2)/(c*x + 1)^(1/2)))^2*(c^2*x^2 - 1)),x)

[Out]

-int(1/((a + b*acot((1 - c*x)^(1/2)/(c*x + 1)^(1/2)))^2*(c^2*x^2 - 1)), x)