3.3.36 \(\int (c (d \sec (e+f x))^p)^n (a+b \sec (e+f x))^m \, dx\) [236]

Optimal. Leaf size=56 \[ (d \sec (e+f x))^{-n p} \left (c (d \sec (e+f x))^p\right )^n \text {Int}\left ((d \sec (e+f x))^{n p} (a+b \sec (e+f x))^m,x\right ) \]

[Out]

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

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Rubi [A]
time = 0.08, 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 \left (c (d \sec (e+f x))^p\right )^n (a+b \sec (e+f x))^m \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 2.46, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c (d \sec (e+f x))^p\right )^n (a+b \sec (e+f x))^m \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.20, size = 0, normalized size = 0.00 \[\int \left (c \left (d \sec \left (f x +e \right )\right )^{p}\right )^{n} \left (a +b \sec \left (f x +e \right )\right )^{m}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (c\,{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^p\right )}^n\,{\left (a+\frac {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((c*(d/cos(e + f*x))^p)^n*(a + b/cos(e + f*x))^m,x)

[Out]

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

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