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

Optimal. Leaf size=30 \[ \text {Int}\left ((d \tan (e+f x))^m \left (a+b (c \sec (e+f x))^n\right )^p,x\right ) \]

[Out]

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

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

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 3.19, size = 0, normalized size = 0.00 \[ \int \left (a+b (c \sec (e+f x))^n\right )^p (d \tan (e+f x))^m \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \left (d \tan \left (f x + e\right )\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 6.12, size = 0, normalized size = 0.00 \[ \int \left (a +b \left (c \sec \left (f x +e \right )\right )^{n}\right )^{p} \left (d \tan \left (f x +e \right )\right )^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^m\,{\left (a+b\,{\left (\frac {c}{\cos \left (e+f\,x\right )}\right )}^n\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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