3.8.79 \(\int (d \sec (e+f x))^n (a+b \sec (e+f x)) \, dx\) [779]

Optimal. Leaf size=137 \[ -\frac {a d \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(e+f x)\right ) (d \sec (e+f x))^{-1+n} \sin (e+f x)}{f (1-n) \sqrt {\sin ^2(e+f x)}}+\frac {b \, _2F_1\left (\frac {1}{2},-\frac {n}{2};\frac {2-n}{2};\cos ^2(e+f x)\right ) (d \sec (e+f x))^n \sin (e+f x)}{f n \sqrt {\sin ^2(e+f x)}} \]

[Out]

-a*d*hypergeom([1/2, 1/2-1/2*n],[3/2-1/2*n],cos(f*x+e)^2)*(d*sec(f*x+e))^(-1+n)*sin(f*x+e)/f/(1-n)/(sin(f*x+e)
^2)^(1/2)+b*hypergeom([1/2, -1/2*n],[1-1/2*n],cos(f*x+e)^2)*(d*sec(f*x+e))^n*sin(f*x+e)/f/n/(sin(f*x+e)^2)^(1/
2)

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3872, 3857, 2722} \begin {gather*} \frac {b \sin (e+f x) (d \sec (e+f x))^n \, _2F_1\left (\frac {1}{2},-\frac {n}{2};\frac {2-n}{2};\cos ^2(e+f x)\right )}{f n \sqrt {\sin ^2(e+f x)}}-\frac {a d \sin (e+f x) (d \sec (e+f x))^{n-1} \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(e+f x)\right )}{f (1-n) \sqrt {\sin ^2(e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*d*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*(d*Sec[e + f*x])^(-1 + n)*Sin[e + f*x])/(f
*(1 - n)*Sqrt[Sin[e + f*x]^2])) + (b*Hypergeometric2F1[1/2, -1/2*n, (2 - n)/2, Cos[e + f*x]^2]*(d*Sec[e + f*x]
)^n*Sin[e + f*x])/(f*n*Sqrt[Sin[e + f*x]^2])

Rule 2722

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1
)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rule 3857

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x])^(n - 1)*((Sin[c + d*x]/b)^(n - 1)
*Int[1/(Sin[c + d*x]/b)^n, x]), x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rule 3872

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rubi steps

\begin {align*} \int (d \sec (e+f x))^n (a+b \sec (e+f x)) \, dx &=a \int (d \sec (e+f x))^n \, dx+\frac {b \int (d \sec (e+f x))^{1+n} \, dx}{d}\\ &=\left (a \left (\frac {\cos (e+f x)}{d}\right )^n (d \sec (e+f x))^n\right ) \int \left (\frac {\cos (e+f x)}{d}\right )^{-n} \, dx+\frac {\left (b \left (\frac {\cos (e+f x)}{d}\right )^n (d \sec (e+f x))^n\right ) \int \left (\frac {\cos (e+f x)}{d}\right )^{-1-n} \, dx}{d}\\ &=-\frac {a \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(e+f x)\right ) (d \sec (e+f x))^n \sin (e+f x)}{f (1-n) \sqrt {\sin ^2(e+f x)}}+\frac {b \, _2F_1\left (\frac {1}{2},-\frac {n}{2};\frac {2-n}{2};\cos ^2(e+f x)\right ) (d \sec (e+f x))^n \sin (e+f x)}{f n \sqrt {\sin ^2(e+f x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.10, size = 107, normalized size = 0.78 \begin {gather*} \frac {\csc (e+f x) \left (a (1+n) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {n}{2};\frac {2+n}{2};\sec ^2(e+f x)\right )+b n \, _2F_1\left (\frac {1}{2},\frac {1+n}{2};\frac {3+n}{2};\sec ^2(e+f x)\right )\right ) (d \sec (e+f x))^n \sqrt {-\tan ^2(e+f x)}}{f n (1+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[e + f*x]*(a*(1 + n)*Cos[e + f*x]*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Sec[e + f*x]^2] + b*n*Hypergeomet
ric2F1[1/2, (1 + n)/2, (3 + n)/2, Sec[e + f*x]^2])*(d*Sec[e + f*x])^n*Sqrt[-Tan[e + f*x]^2])/(f*n*(1 + n))

________________________________________________________________________________________

Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \left (d \sec \left (f x +e \right )\right )^{n} \left (a +b \sec \left (f x +e \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d \sec {\left (e + f x \right )}\right )^{n} \left (a + b \sec {\left (e + f x \right )}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )\,{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________