3.787 \(\int (d \sec (e+f x))^n (a+b \sec (e+f x))^m \, dx\)

Optimal. Leaf size=26 \[ \text {Int}\left ((d \sec (e+f x))^n (a+b \sec (e+f x))^m,x\right ) \]

[Out]

Unintegrable((d*sec(f*x+e))^n*(a+b*sec(f*x+e))^m,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 = {} \[ \int (d \sec (e+f x))^n (a+b \sec (e+f x))^m \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.64, size = 0, normalized size = 0.00 \[ \int (d \sec (e+f x))^n (a+b \sec (e+f x))^m \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sec \left (f x + e\right ) + a\right )}^{m} \left (d \sec \left (f x + e\right )\right )^{n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 2.53, size = 0, normalized size = 0.00 \[ \int \left (d \sec \left (f x +e \right )\right )^{n} \left (a +b \sec \left (f x +e \right )\right )^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int {\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^m\,{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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