3.424 \(\int (d \cot (e+f x))^m (a+b (c \tan (e+f x))^n)^p \, dx\)

Optimal. Leaf size=56 \[ \left (\frac{\tan (e+f x)}{d}\right )^m (d \cot (e+f x))^m \text{Unintegrable}\left (\left (\frac{\tan (e+f x)}{d}\right )^{-m} \left (a+b (c \tan (e+f x))^n\right )^p,x\right ) \]

[Out]

(d*Cot[e + f*x])^m*(Tan[e + f*x]/d)^m*Unintegrable[(a + b*(c*Tan[e + f*x])^n)^p/(Tan[e + f*x]/d)^m, x]

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Rubi [A]  time = 0.138833, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int (d \cot (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

Int[(d*Cot[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p,x]

[Out]

(d*Cot[e + f*x])^m*(Tan[e + f*x]/d)^m*Defer[Int][(a + b*(c*Tan[e + f*x])^n)^p/(Tan[e + f*x]/d)^m, x]

Rubi steps

\begin{align*} \int (d \cot (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx &=\left ((d \cot (e+f x))^m \left (\frac{\tan (e+f x)}{d}\right )^m\right ) \int \left (\frac{\tan (e+f x)}{d}\right )^{-m} \left (a+b (c \tan (e+f x))^n\right )^p \, dx\\ \end{align*}

Mathematica [A]  time = 9.02188, size = 0, normalized size = 0. \[ \int (d \cot (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(d*Cot[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p,x]

[Out]

Integrate[(d*Cot[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p, x]

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Maple [A]  time = 0.536, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cot \left ( fx+e \right ) \right ) ^{m} \left ( a+b \left ( c\tan \left ( fx+e \right ) \right ) ^{n} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cot(f*x+e))^m*(a+b*(c*tan(f*x+e))^n)^p,x)

[Out]

int((d*cot(f*x+e))^m*(a+b*(c*tan(f*x+e))^n)^p,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \left (d \cot \left (f x + e\right )\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((c*tan(f*x + e))^n*b + a)^p*(d*cot(f*x + e))^m, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \left (d \cot \left (f x + e\right )\right )^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(((c*tan(f*x + e))^n*b + a)^p*(d*cot(f*x + e))^m, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cot(f*x+e))**m*(a+b*(c*tan(f*x+e))**n)**p,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \left (d \cot \left (f x + e\right )\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((c*tan(f*x + e))^n*b + a)^p*(d*cot(f*x + e))^m, x)