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

Optimal. Leaf size=25 \[ \text{Unintegrable}\left ((d \sec (e+f x))^n (a+b \sec (e+f x))^m,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0465105, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}

Mathematica [A]  time = 0.495998, size = 0, normalized size = 0. \[ \int (d \sec (e+f x))^n (a+b \sec (e+f x))^m \, dx \]

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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Maple [A]  time = 0.835, size = 0, normalized size = 0. \begin{align*} \int \left ( d\sec \left ( fx+e \right ) \right ) ^{n} \left ( a+b\sec \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sec \left (f x + e\right ) + a\right )}^{m} \left (d \sec \left (f x + e\right )\right )^{n}\,{d x} \end{align*}

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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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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

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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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sec \left (f x + e\right ) + a\right )}^{m} \left (d \sec \left (f x + e\right )\right )^{n}\,{d x} \end{align*}

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)