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

Optimal. Leaf size=57 \[ \left (\frac {\tan (e+f x)}{d}\right )^m (d \cot (e+f x))^m \text {Int}\left (\left (\frac {\tan (e+f x)}{d}\right )^{-m} \left (a+b (c \tan (e+f x))^n\right )^p,x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.14, 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 = {} \[ \int (d \cot (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx \]

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 = 9.86, size = 0, normalized size = 0.00 \[ \int (d \cot (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx \]

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]

________________________________________________________________________________________

fricas [A]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \left (d \cot \left (f x + e\right )\right )^{m}, x\right ) \]

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

maple [A]  time = 2.00, size = 0, normalized size = 0.00 \[ \int \left (d \cot \left (f x +e \right )\right )^{m} \left (a +b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}\, dx \]

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^m\,{\left (a+b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

________________________________________________________________________________________