3.470 \(\int \cot ^2(e+f x) (a+b (c \sec (e+f x))^n)^p \, dx\)

Optimal. Leaf size=28 \[ \text {Int}\left (\cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p,x\right ) \]

[Out]

Unintegrable(cot(f*x+e)^2*(a+b*(c*sec(f*x+e))^n)^p,x)

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Rubi [A]  time = 0.06, 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 = {} \[ \int \cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cot[e + f*x]^2*(a + b*(c*Sec[e + f*x])^n)^p,x]

[Out]

Defer[Int][Cot[e + f*x]^2*(a + b*(c*Sec[e + f*x])^n)^p, x]

Rubi steps

\begin {align*} \int \cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \, dx &=\int \cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \, dx\\ \end {align*}

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Mathematica [A]  time = 1.85, size = 0, normalized size = 0.00 \[ \int \cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Cot[e + f*x]^2*(a + b*(c*Sec[e + f*x])^n)^p,x]

[Out]

Integrate[Cot[e + f*x]^2*(a + b*(c*Sec[e + f*x])^n)^p, x]

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fricas [A]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \cot \left (f x + e\right )^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^2*(a+b*(c*sec(f*x+e))^n)^p,x, algorithm="fricas")

[Out]

integral(((c*sec(f*x + e))^n*b + a)^p*cot(f*x + e)^2, x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \cot \left (f x + e\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^2*(a+b*(c*sec(f*x+e))^n)^p,x, algorithm="giac")

[Out]

integrate(((c*sec(f*x + e))^n*b + a)^p*cot(f*x + e)^2, x)

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maple [A]  time = 1.76, size = 0, normalized size = 0.00 \[ \int \left (\cot ^{2}\left (f x +e \right )\right ) \left (a +b \left (c \sec \left (f x +e \right )\right )^{n}\right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^2*(a+b*(c*sec(f*x+e))^n)^p,x)

[Out]

int(cot(f*x+e)^2*(a+b*(c*sec(f*x+e))^n)^p,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \cot \left (f x + e\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^2*(a+b*(c*sec(f*x+e))^n)^p,x, algorithm="maxima")

[Out]

integrate(((c*sec(f*x + e))^n*b + a)^p*cot(f*x + e)^2, x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int {\mathrm {cot}\left (e+f\,x\right )}^2\,{\left (a+b\,{\left (\frac {c}{\cos \left (e+f\,x\right )}\right )}^n\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(e + f*x)^2*(a + b*(c/cos(e + f*x))^n)^p,x)

[Out]

int(cot(e + f*x)^2*(a + b*(c/cos(e + f*x))^n)^p, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \left (c \sec {\left (e + f x \right )}\right )^{n}\right )^{p} \cot ^{2}{\left (e + f x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)**2*(a+b*(c*sec(f*x+e))**n)**p,x)

[Out]

Integral((a + b*(c*sec(e + f*x))**n)**p*cot(e + f*x)**2, x)

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