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

Optimal. Leaf size=96 \[ -\frac{\tan (e+f x) (1-\sec (e+f x))^{-m-\frac{1}{2}} (a-a \sec (e+f x))^m (d \sec (e+f x))^n F_1\left (n;\frac{1}{2}-m,\frac{1}{2};n+1;\sec (e+f x),-\sec (e+f x)\right )}{f n \sqrt{\sec (e+f x)+1}} \]

[Out]

-((AppellF1[n, 1/2 - m, 1/2, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(1 - Sec[e + f*x])^(-1/2 - m)*(d*Sec[e + f*x]
)^n*(a - a*Sec[e + f*x])^m*Tan[e + f*x])/(f*n*Sqrt[1 + Sec[e + f*x]]))

________________________________________________________________________________________

Rubi [A]  time = 0.118964, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3828, 3827, 133} \[ -\frac{\tan (e+f x) (1-\sec (e+f x))^{-m-\frac{1}{2}} (a-a \sec (e+f x))^m (d \sec (e+f x))^n F_1\left (n;\frac{1}{2}-m,\frac{1}{2};n+1;\sec (e+f x),-\sec (e+f x)\right )}{f n \sqrt{\sec (e+f x)+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((AppellF1[n, 1/2 - m, 1/2, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(1 - Sec[e + f*x])^(-1/2 - m)*(d*Sec[e + f*x]
)^n*(a - a*Sec[e + f*x])^m*Tan[e + f*x])/(f*n*Sqrt[1 + Sec[e + f*x]]))

Rule 3828

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Dist[(a^In
tPart[m]*(a + b*Csc[e + f*x])^FracPart[m])/(1 + (b*Csc[e + f*x])/a)^FracPart[m], Int[(1 + (b*Csc[e + f*x])/a)^
m*(d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[m] &&  !GtQ
[a, 0]

Rule 3827

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Dist[(a^2*
d*Cot[e + f*x])/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]]), Subst[Int[((d*x)^(n - 1)*(a + b*x)^(m -
1/2))/Sqrt[a - b*x], x], x, Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] &&  !In
tegerQ[m] && GtQ[a, 0]

Rule 133

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(c^n*e^p*(b*x)^(m +
1)*AppellF1[m + 1, -n, -p, m + 2, -((d*x)/c), -((f*x)/e)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rubi steps

\begin{align*} \int (d \sec (e+f x))^n (a-a \sec (e+f x))^m \, dx &=\left ((1-\sec (e+f x))^{-m} (a-a \sec (e+f x))^m\right ) \int (1-\sec (e+f x))^m (d \sec (e+f x))^n \, dx\\ &=-\frac{\left (d (1-\sec (e+f x))^{-\frac{1}{2}-m} (a-a \sec (e+f x))^m \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{(1-x)^{-\frac{1}{2}+m} (d x)^{-1+n}}{\sqrt{1+x}} \, dx,x,\sec (e+f x)\right )}{f \sqrt{1+\sec (e+f x)}}\\ &=-\frac{F_1\left (n;\frac{1}{2}-m,\frac{1}{2};1+n;\sec (e+f x),-\sec (e+f x)\right ) (1-\sec (e+f x))^{-\frac{1}{2}-m} (d \sec (e+f x))^n (a-a \sec (e+f x))^m \tan (e+f x)}{f n \sqrt{1+\sec (e+f x)}}\\ \end{align*}

Mathematica [F]  time = 0.206287, size = 0, normalized size = 0. \[ \int (d \sec (e+f x))^n (a-a \sec (e+f x))^m \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.786, size = 0, normalized size = 0. \begin{align*} \int \left ( d\sec \left ( fx+e \right ) \right ) ^{n} \left ( a-a\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-a*sec(f*x+e))^m,x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a \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-a*sec(f*x+e))^m,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (-a \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-a*sec(f*x+e))^m,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \sec{\left (e + f x \right )}\right )^{n} \left (- a \left (\sec{\left (e + f x \right )} - 1\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-a*sec(f*x+e))**m,x)

[Out]

Integral((d*sec(e + f*x))**n*(-a*(sec(e + f*x) - 1))**m, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a \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-a*sec(f*x+e))^m,x, algorithm="giac")

[Out]

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