3.167 \(\int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{2-m} \, dx\)

Optimal. Leaf size=101 \[ -\frac{c^2 2^{\frac{5}{2}-m} \tan (e+f x) (1-\sec (e+f x))^{m-\frac{1}{2}} (a \sec (e+f x)+a)^m (c-c \sec (e+f x))^{-m} \text{Hypergeometric2F1}\left (m-\frac{3}{2},m+\frac{1}{2},m+\frac{3}{2},\frac{1}{2} (\sec (e+f x)+1)\right )}{f (2 m+1)} \]

[Out]

-((2^(5/2 - m)*c^2*Hypergeometric2F1[-3/2 + m, 1/2 + m, 3/2 + m, (1 + Sec[e + f*x])/2]*(1 - Sec[e + f*x])^(-1/
2 + m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + 2*m)*(c - c*Sec[e + f*x])^m))

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Rubi [A]  time = 0.154416, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3961, 70, 69} \[ -\frac{c^2 2^{\frac{5}{2}-m} \tan (e+f x) (1-\sec (e+f x))^{m-\frac{1}{2}} (a \sec (e+f x)+a)^m (c-c \sec (e+f x))^{-m} \, _2F_1\left (m-\frac{3}{2},m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sec (e+f x)+1)\right )}{f (2 m+1)} \]

Antiderivative was successfully verified.

[In]

Int[Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(2 - m),x]

[Out]

-((2^(5/2 - m)*c^2*Hypergeometric2F1[-3/2 + m, 1/2 + m, 3/2 + m, (1 + Sec[e + f*x])/2]*(1 - Sec[e + f*x])^(-1/
2 + m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + 2*m)*(c - c*Sec[e + f*x])^m))

Rule 3961

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

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d)
)^IntPart[n]*((b*(c + d*x))/(b*c - a*d))^FracPart[n]), Int[(a + b*x)^m*Simp[(b*c)/(b*c - a*d) + (b*d*x)/(b*c -
a*d), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rubi steps

\begin{align*} \int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{2-m} \, dx &=-\frac{(a c \tan (e+f x)) \operatorname{Subst}\left (\int (a+a x)^{-\frac{1}{2}+m} (c-c x)^{\frac{3}{2}-m} \, dx,x,\sec (e+f x)\right )}{f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}\\ &=-\frac{\left (2^{\frac{3}{2}-m} a c^2 (c-c \sec (e+f x))^{-m} \left (\frac{c-c \sec (e+f x)}{c}\right )^{-\frac{1}{2}+m} \tan (e+f x)\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2}-\frac{x}{2}\right )^{\frac{3}{2}-m} (a+a x)^{-\frac{1}{2}+m} \, dx,x,\sec (e+f x)\right )}{f \sqrt{a+a \sec (e+f x)}}\\ &=-\frac{2^{\frac{5}{2}-m} c^2 \, _2F_1\left (-\frac{3}{2}+m,\frac{1}{2}+m;\frac{3}{2}+m;\frac{1}{2} (1+\sec (e+f x))\right ) (1-\sec (e+f x))^{-\frac{1}{2}+m} (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{-m} \tan (e+f x)}{f (1+2 m)}\\ \end{align*}

Mathematica [F]  time = 2.77688, size = 0, normalized size = 0. \[ \int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{2-m} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(2 - m),x]

[Out]

Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(2 - m), x]

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Maple [F]  time = 0.665, size = 0, normalized size = 0. \begin{align*} \int \sec \left ( fx+e \right ) \left ( a+a\sec \left ( fx+e \right ) \right ) ^{m} \left ( c-c\sec \left ( fx+e \right ) \right ) ^{2-m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)*(a+a*sec(f*x+e))^m*(c-c*sec(f*x+e))^(2-m),x)

[Out]

int(sec(f*x+e)*(a+a*sec(f*x+e))^m*(c-c*sec(f*x+e))^(2-m),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sec \left (f x + e\right ) + a\right )}^{m}{\left (-c \sec \left (f x + e\right ) + c\right )}^{-m + 2} \sec \left (f x + e\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))^m*(c-c*sec(f*x+e))^(2-m),x, algorithm="maxima")

[Out]

integrate((a*sec(f*x + e) + a)^m*(-c*sec(f*x + e) + c)^(-m + 2)*sec(f*x + e), x)

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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 (-c \sec \left (f x + e\right ) + c\right )}^{-m + 2} \sec \left (f x + e\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))^m*(c-c*sec(f*x+e))^(2-m),x, algorithm="fricas")

[Out]

integral((a*sec(f*x + e) + a)^m*(-c*sec(f*x + e) + c)^(-m + 2)*sec(f*x + e), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))**m*(c-c*sec(f*x+e))**(2-m),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sec \left (f x + e\right ) + a\right )}^{m}{\left (-c \sec \left (f x + e\right ) + c\right )}^{-m + 2} \sec \left (f x + e\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))^m*(c-c*sec(f*x+e))^(2-m),x, algorithm="giac")

[Out]

integrate((a*sec(f*x + e) + a)^m*(-c*sec(f*x + e) + c)^(-m + 2)*sec(f*x + e), x)