3.5.78 \(\int (c \sec (e+f x))^n (a+b \sec (e+f x))^m (A+B \sec (e+f x)) \, dx\) [478]

Optimal. Leaf size=36 \[ \text {Int}\left ((c \sec (e+f x))^n (a+b \sec (e+f x))^m (A+B \sec (e+f x)),x\right ) \]

[Out]

Unintegrable((c*sec(f*x+e))^n*(a+b*sec(f*x+e))^m*(A+B*sec(f*x+e)),x)

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Rubi [A]
time = 0.05, 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 (c \sec (e+f x))^n (a+b \sec (e+f x))^m (A+B \sec (e+f x)) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[Int][(c*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^m*(A + B*Sec[e + f*x]), x]

Rubi steps

\begin {align*} \int (c \sec (e+f x))^n (a+b \sec (e+f x))^m (A+B \sec (e+f x)) \, dx &=\int (c \sec (e+f x))^n (a+b \sec (e+f x))^m (A+B \sec (e+f x)) \, dx\\ \end {align*}

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Mathematica [A]
time = 4.46, size = 0, normalized size = 0.00 \begin {gather*} \int (c \sec (e+f x))^n (a+b \sec (e+f x))^m (A+B \sec (e+f x)) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.48, size = 0, normalized size = 0.00 \[\int \left (c \sec \left (f x +e \right )\right )^{n} \left (a +b \sec \left (f x +e \right )\right )^{m} \left (A +B \sec \left (f x +e \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((c*sec(f*x+e))^n*(a+b*sec(f*x+e))^m*(A+B*sec(f*x+e)),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*sec(f*x+e))^n*(a+b*sec(f*x+e))^m*(A+B*sec(f*x+e)),x, algorithm="maxima")

[Out]

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

[Out]

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

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

[Out]

Integral((c*sec(e + f*x))**n*(A + B*sec(e + f*x))*(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*sec(f*x+e))^n*(a+b*sec(f*x+e))^m*(A+B*sec(f*x+e)),x, algorithm="giac")

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \left (A+\frac {B}{\cos \left (e+f\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^m\,{\left (\frac {c}{\cos \left (e+f\,x\right )}\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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