\(\int (a+b \cos (e+f x))^m (g \tan (e+f x))^p \, dx\) [22]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int (a+b \cos (e+f x))^m (g \tan (e+f x))^p \, dx=(g \cot (e+f x))^p (g \tan (e+f x))^p \text {Int}\left ((a+b \cos (e+f x))^m (g \cot (e+f x))^{-p},x\right ) \]

[Out]

(g*cot(f*x+e))^p*(g*tan(f*x+e))^p*Unintegrable((a+b*cos(f*x+e))^m/((g*cot(f*x+e))^p),x)

Rubi [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 23, 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 = {} \[ \int (a+b \cos (e+f x))^m (g \tan (e+f x))^p \, dx=\int (a+b \cos (e+f x))^m (g \tan (e+f x))^p \, dx \]

[In]

Int[(a + b*Cos[e + f*x])^m*(g*Tan[e + f*x])^p,x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \left ((g \cot (e+f x))^p (g \tan (e+f x))^p\right ) \int (a+b \cos (e+f x))^m (g \cot (e+f x))^{-p} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 6.95 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (a+b \cos (e+f x))^m (g \tan (e+f x))^p \, dx=\int (a+b \cos (e+f x))^m (g \tan (e+f x))^p \, dx \]

[In]

Integrate[(a + b*Cos[e + f*x])^m*(g*Tan[e + f*x])^p,x]

[Out]

Integrate[(a + b*Cos[e + f*x])^m*(g*Tan[e + f*x])^p, x]

Maple [N/A] (verified)

Not integrable

Time = 1.62 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

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

[In]

int((a+b*cos(f*x+e))^m*(g*tan(f*x+e))^p,x)

[Out]

int((a+b*cos(f*x+e))^m*(g*tan(f*x+e))^p,x)

Fricas [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (a+b \cos (e+f x))^m (g \tan (e+f x))^p \, dx=\int { {\left (b \cos \left (f x + e\right ) + a\right )}^{m} \left (g \tan \left (f x + e\right )\right )^{p} \,d x } \]

[In]

integrate((a+b*cos(f*x+e))^m*(g*tan(f*x+e))^p,x, algorithm="fricas")

[Out]

integral((b*cos(f*x + e) + a)^m*(g*tan(f*x + e))^p, x)

Sympy [N/A]

Not integrable

Time = 73.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int (a+b \cos (e+f x))^m (g \tan (e+f x))^p \, dx=\int \left (g \tan {\left (e + f x \right )}\right )^{p} \left (a + b \cos {\left (e + f x \right )}\right )^{m}\, dx \]

[In]

integrate((a+b*cos(f*x+e))**m*(g*tan(f*x+e))**p,x)

[Out]

Integral((g*tan(e + f*x))**p*(a + b*cos(e + f*x))**m, x)

Maxima [N/A]

Not integrable

Time = 1.68 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (a+b \cos (e+f x))^m (g \tan (e+f x))^p \, dx=\int { {\left (b \cos \left (f x + e\right ) + a\right )}^{m} \left (g \tan \left (f x + e\right )\right )^{p} \,d x } \]

[In]

integrate((a+b*cos(f*x+e))^m*(g*tan(f*x+e))^p,x, algorithm="maxima")

[Out]

integrate((b*cos(f*x + e) + a)^m*(g*tan(f*x + e))^p, x)

Giac [N/A]

Not integrable

Time = 1.53 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (a+b \cos (e+f x))^m (g \tan (e+f x))^p \, dx=\int { {\left (b \cos \left (f x + e\right ) + a\right )}^{m} \left (g \tan \left (f x + e\right )\right )^{p} \,d x } \]

[In]

integrate((a+b*cos(f*x+e))^m*(g*tan(f*x+e))^p,x, algorithm="giac")

[Out]

integrate((b*cos(f*x + e) + a)^m*(g*tan(f*x + e))^p, x)

Mupad [N/A]

Not integrable

Time = 14.78 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (a+b \cos (e+f x))^m (g \tan (e+f x))^p \, dx=\int {\left (g\,\mathrm {tan}\left (e+f\,x\right )\right )}^p\,{\left (a+b\,\cos \left (e+f\,x\right )\right )}^m \,d x \]

[In]

int((g*tan(e + f*x))^p*(a + b*cos(e + f*x))^m,x)

[Out]

int((g*tan(e + f*x))^p*(a + b*cos(e + f*x))^m, x)