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

Optimal. Leaf size=43 \[ \text {Int}\left (\frac {1}{\left (1-c^2 x^2\right ) \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.05, 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 = {} \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \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 )} \, 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 )} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {1}{a c^{2} x^{2} + {\left (b c^{2} x^{2} - b\right )} \operatorname {arccot}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) - a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {1}{{\left (c^{2} x^{2} - 1\right )} {\left (b \operatorname {arccot}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 1.38, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-c^{2} x^{2}+1\right ) \left (a +b \,\mathrm {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {1}{{\left (c^{2} x^{2} - 1\right )} {\left (b \operatorname {arccot}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ -\int \frac {1}{\left (a+b\,\mathrm {acot}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )\,\left (c^2\,x^2-1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {1}{a c^{2} x^{2} - a + b c^{2} x^{2} \operatorname {acot}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )} - b \operatorname {acot}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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