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

Optimal. Leaf size=101 \[ -\frac {c 2^{n+\frac {1}{2}} \tan (e+f x) (1-\sec (e+f x))^{\frac {1}{2}-n} (a \sec (e+f x)+a)^m (c-c \sec (e+f x))^{n-1} \, _2F_1\left (m+\frac {1}{2},\frac {1}{2}-n;m+\frac {3}{2};\frac {1}{2} (\sec (e+f x)+1)\right )}{f (2 m+1)} \]

[Out]

-2^(1/2+n)*c*hypergeom([1/2+m, 1/2-n],[3/2+m],1/2+1/2*sec(f*x+e))*(1-sec(f*x+e))^(1/2-n)*(a+a*sec(f*x+e))^m*(c
-c*sec(f*x+e))^(-1+n)*tan(f*x+e)/f/(1+2*m)

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Rubi [A]  time = 0.12, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {3961, 70, 69} \[ -\frac {c 2^{n+\frac {1}{2}} \tan (e+f x) (1-\sec (e+f x))^{\frac {1}{2}-n} (a \sec (e+f x)+a)^m (c-c \sec (e+f x))^{n-1} \, _2F_1\left (m+\frac {1}{2},\frac {1}{2}-n;m+\frac {3}{2};\frac {1}{2} (\sec (e+f x)+1)\right )}{f (2 m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]))

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 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]

Rubi steps

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

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Mathematica [F]  time = 1.35, size = 0, normalized size = 0.00 \[ \int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^n \, dx \]

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])^n,x]

[Out]

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

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fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sec \left (f x + e\right ) + a\right )}^{m} {\left (-c \sec \left (f x + e\right ) + c\right )}^{n} \sec \left (f x + e\right ), x\right ) \]

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))^n,x, algorithm="fricas")

[Out]

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

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

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))^n,x, algorithm="giac")

[Out]

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

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maple [F]  time = 2.96, size = 0, normalized size = 0.00 \[ \int \sec \left (f x +e \right ) \left (a +a \sec \left (f x +e \right )\right )^{m} \left (c -c \sec \left (f x +e \right )\right )^{n}\, dx \]

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))^n,x)

[Out]

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

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

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))^n,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^m\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^n}{\cos \left (e+f\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{m} \left (- c \left (\sec {\left (e + f x \right )} - 1\right )\right )^{n} \sec {\left (e + f x \right )}\, dx \]

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))**n,x)

[Out]

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

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