\(\int (d \csc (e+f x))^m (a+b (c \tan (e+f x))^n)^p \, dx\) [499]

Optimal result

Integrand size = 27, antiderivative size = 27 \[ \int (d \csc (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx=(d \csc (e+f x))^m \left (\frac {\sin (e+f x)}{d}\right )^m \text {Int}\left (\left (\frac {\sin (e+f x)}{d}\right )^{-m} \left (a+b (c \tan (e+f x))^n\right )^p,x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 27, 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 = {} \[ \int (d \csc (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int (d \csc (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 5.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int (d \csc (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int (d \csc (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.37 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00

\[\int \left (d \csc \left (f x +e \right )\right )^{m} \left (a +b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int (d \csc (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int { {\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \left (d \csc \left (f x + e\right )\right )^{m} \,d x } \]

[In]

integrate((d*csc(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*csc(f*x + e))^m, x)

Sympy [N/A]

Not integrable

Time = 166.56 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int (d \csc (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int \left (d \csc {\left (e + f x \right )}\right )^{m} \left (a + b \left (c \tan {\left (e + f x \right )}\right )^{n}\right )^{p}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 7.38 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int (d \csc (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int { {\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \left (d \csc \left (f x + e\right )\right )^{m} \,d x } \]

[In]

integrate((d*csc(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*csc(f*x + e))^m, x)

Giac [N/A]

Not integrable

Time = 1.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int (d \csc (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int { {\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \left (d \csc \left (f x + e\right )\right )^{m} \,d x } \]

[In]

integrate((d*csc(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*csc(f*x + e))^m, x)

Mupad [N/A]

Not integrable

Time = 12.44 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int (d \csc (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int {\left (a+b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p\,{\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^m \,d x \]

[In]

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

[Out]

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