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

Optimal. Leaf size=49 \[ (g \cot (e+f x))^p (g \tan (e+f x))^p \text {Int}\left ((a+b \cos (e+f x))^m (g \cot (e+f x))^{-p},x\right ) \]

[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 [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 = {} \begin {gather*} \int (a+b \cos (e+f x))^m (g \tan (e+f x))^p \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \int (a+b \cos (e+f x))^m (g \tan (e+f x))^p \, dx &=\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 [A]
time = 2.56, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b \cos (e+f x))^m (g \tan (e+f x))^p \, dx \end {gather*}

Verification is not applicable to the result.

[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 [A]
time = 0.18, size = 0, normalized size = 0.00 \[\int \left (a +b \cos \left (f x +e \right )\right )^{m} \left (g \tan \left (f x +e \right )\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (g \tan {\left (e + f x \right )}\right )^{p} \left (a + b \cos {\left (e + f x \right )}\right )^{m}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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 [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________