\(\int \frac {(a+b \cot ^{-1}(\frac {\sqrt {1-c x}}{\sqrt {1+c x}}))^n}{1-c^2 x^2} \, dx\) [151]

Optimal result

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

[Out]

Unintegrable((a+b*arccot((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^n/(-c^2*x^2+1),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 = {} \[ \int \frac {\left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^n}{1-c^2 x^2} \, dx=\int \frac {\left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^n}{1-c^2 x^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^n}{1-c^2 x^2} \, dx=\int \frac {\left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^n}{1-c^2 x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.41 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.90

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^n}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {arccot}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{n}}{c^{2} x^{2} - 1} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 3.35 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^n}{1-c^2 x^2} \, dx=- \int \frac {\left (a + b \operatorname {acot}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}\right )^{n}}{c^{2} x^{2} - 1}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.88 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^n}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {arccot}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{n}}{c^{2} x^{2} - 1} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^n}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {arccot}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{n}}{c^{2} x^{2} - 1} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.20 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^n}{1-c^2 x^2} \, dx=-\int \frac {{\left (a+b\,\mathrm {acot}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^n}{c^2\,x^2-1} \,d x \]

[In]

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

[Out]

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