3.780 \(\int \frac{(d \sec (e+f x))^n}{a+b \sec (e+f x)} \, dx\)

Optimal. Leaf size=192 \[ \frac{a \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{n-1}{2}} (d \sec (e+f x))^n F_1\left (\frac{1}{2};\frac{n-1}{2},1;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right )}{f \left (a^2-b^2\right )}-\frac{b \sin (e+f x) \cos ^2(e+f x)^{n/2} (d \sec (e+f x))^n F_1\left (\frac{1}{2};\frac{n}{2},1;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right )}{f \left (a^2-b^2\right )} \]

[Out]

(a*AppellF1[1/2, (-1 + n)/2, 1, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*Cos[e + f*x]*(Cos[e + f
*x]^2)^((-1 + n)/2)*(d*Sec[e + f*x])^n*Sin[e + f*x])/((a^2 - b^2)*f) - (b*AppellF1[1/2, n/2, 1, 3/2, Sin[e + f
*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*(Cos[e + f*x]^2)^(n/2)*(d*Sec[e + f*x])^n*Sin[e + f*x])/((a^2 - b^2)*
f)

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Rubi [A]  time = 0.302176, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3869, 2823, 3189, 429} \[ \frac{a \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{n-1}{2}} (d \sec (e+f x))^n F_1\left (\frac{1}{2};\frac{n-1}{2},1;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right )}{f \left (a^2-b^2\right )}-\frac{b \sin (e+f x) \cos ^2(e+f x)^{n/2} (d \sec (e+f x))^n F_1\left (\frac{1}{2};\frac{n}{2},1;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right )}{f \left (a^2-b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*AppellF1[1/2, (-1 + n)/2, 1, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*Cos[e + f*x]*(Cos[e + f
*x]^2)^((-1 + n)/2)*(d*Sec[e + f*x])^n*Sin[e + f*x])/((a^2 - b^2)*f) - (b*AppellF1[1/2, n/2, 1, 3/2, Sin[e + f
*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*(Cos[e + f*x]^2)^(n/2)*(d*Sec[e + f*x])^n*Sin[e + f*x])/((a^2 - b^2)*
f)

Rule 3869

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

Rule 2823

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[a, Int[(d*
Sin[e + f*x])^n/(a^2 - b^2*Sin[e + f*x]^2), x], x] - Dist[b/d, Int[(d*Sin[e + f*x])^(n + 1)/(a^2 - b^2*Sin[e +
f*x]^2), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0]

Rule 3189

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff
= FreeFactors[Cos[e + f*x], x]}, -Dist[(ff*d^(2*IntPart[(m - 1)/2] + 1)*(d*Sin[e + f*x])^(2*FracPart[(m - 1)/
2]))/(f*(Sin[e + f*x]^2)^FracPart[(m - 1)/2]), Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x]
, x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] &&  !IntegerQ[m]

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \frac{(d \sec (e+f x))^n}{a+b \sec (e+f x)} \, dx &=\left (\cos ^n(e+f x) (d \sec (e+f x))^n\right ) \int \frac{\cos ^{1-n}(e+f x)}{b+a \cos (e+f x)} \, dx\\ &=-\left (\left (a \cos ^n(e+f x) (d \sec (e+f x))^n\right ) \int \frac{\cos ^{2-n}(e+f x)}{b^2-a^2 \cos ^2(e+f x)} \, dx\right )+\left (b \cos ^n(e+f x) (d \sec (e+f x))^n\right ) \int \frac{\cos ^{1-n}(e+f x)}{b^2-a^2 \cos ^2(e+f x)} \, dx\\ &=-\frac{\left (a \cos ^{2 \left (\frac{1}{2}-\frac{n}{2}\right )+n}(e+f x) \cos ^2(e+f x)^{-\frac{1}{2}+\frac{n}{2}} (d \sec (e+f x))^n\right ) \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^{\frac{1-n}{2}}}{-a^2+b^2+a^2 x^2} \, dx,x,\sin (e+f x)\right )}{f}+\frac{\left (b \cos ^2(e+f x)^{n/2} (d \sec (e+f x))^n\right ) \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^{-n/2}}{-a^2+b^2+a^2 x^2} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{a F_1\left (\frac{1}{2};\frac{1}{2} (-1+n),1;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right ) \cos (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (-1+n)} (d \sec (e+f x))^n \sin (e+f x)}{\left (a^2-b^2\right ) f}-\frac{b F_1\left (\frac{1}{2};\frac{n}{2},1;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right ) \cos ^2(e+f x)^{n/2} (d \sec (e+f x))^n \sin (e+f x)}{\left (a^2-b^2\right ) f}\\ \end{align*}

Mathematica [B]  time = 25.4514, size = 5280, normalized size = 27.5 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [F]  time = 0.73, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d\sec \left ( fx+e \right ) \right ) ^{n}}{a+b\sec \left ( fx+e \right ) }}\, 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)),x)

[Out]

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

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

[Out]

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

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

[Out]

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

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

[Out]

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

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

[Out]

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