3.30 \(\int (b \sec (c+d x))^n (A+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=113 \[ \frac{C \tan (c+d x) (b \sec (c+d x))^n}{d (n+1)}-\frac{b (A n+A+C n) \sin (c+d x) (b \sec (c+d x))^{n-1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1-n}{2},\frac{3-n}{2},\cos ^2(c+d x)\right )}{d (1-n) (n+1) \sqrt{\sin ^2(c+d x)}} \]

[Out]

-((b*(A + A*n + C*n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-1 + n)*Si
n[c + d*x])/(d*(1 - n)*(1 + n)*Sqrt[Sin[c + d*x]^2])) + (C*(b*Sec[c + d*x])^n*Tan[c + d*x])/(d*(1 + n))

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Rubi [A]  time = 0.0831302, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {4046, 3772, 2643} \[ \frac{C \tan (c+d x) (b \sec (c+d x))^n}{d (n+1)}-\frac{b (A n+A+C n) \sin (c+d x) (b \sec (c+d x))^{n-1} \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(c+d x)\right )}{d (1-n) (n+1) \sqrt{\sin ^2(c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]

[Out]

-((b*(A + A*n + C*n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-1 + n)*Si
n[c + d*x])/(d*(1 - n)*(1 + n)*Sqrt[Sin[c + d*x]^2])) + (C*(b*Sec[c + d*x])^n*Tan[c + d*x])/(d*(1 + n))

Rule 4046

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> -Simp[(C*Cot[
e + f*x]*(b*Csc[e + f*x])^m)/(f*(m + 1)), x] + Dist[(C*m + A*(m + 1))/(m + 1), Int[(b*Csc[e + f*x])^m, x], x]
/; FreeQ[{b, e, f, A, C, m}, x] && NeQ[C*m + A*(m + 1), 0] &&  !LeQ[m, -1]

Rule 3772

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 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps

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

Mathematica [C]  time = 5.61037, size = 273, normalized size = 2.42 \[ -\frac{i 2^{n+1} e^{-i (n+1) (c+d x)} \left (\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{n+1} \sec ^{-n-2}(c+d x) \left (A+C \sec ^2(c+d x)\right ) (b \sec (c+d x))^n \left (n e^{i (n+2) (c+d x)} \left (2 (n+4) (A+2 C) \text{Hypergeometric2F1}\left (1,-\frac{n}{2},\frac{n+4}{2},-e^{2 i (c+d x)}\right )+A (n+2) e^{2 i (c+d x)} \text{Hypergeometric2F1}\left (1,1-\frac{n}{2},\frac{n+6}{2},-e^{2 i (c+d x)}\right )\right )+A \left (n^2+6 n+8\right ) e^{i n (c+d x)} \text{Hypergeometric2F1}\left (1,-\frac{n}{2}-1,\frac{n+2}{2},-e^{2 i (c+d x)}\right )\right )}{d n (n+2) (n+4) (A \cos (2 c+2 d x)+A+2 C)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]

[Out]

((-I)*2^(1 + n)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1 + n)*(A*E^(I*n*(c + d*x))*(8 + 6*n + n^2)*Hyper
geometric2F1[1, -1 - n/2, (2 + n)/2, -E^((2*I)*(c + d*x))] + E^(I*(2 + n)*(c + d*x))*n*(A*E^((2*I)*(c + d*x))*
(2 + n)*Hypergeometric2F1[1, 1 - n/2, (6 + n)/2, -E^((2*I)*(c + d*x))] + 2*(A + 2*C)*(4 + n)*Hypergeometric2F1
[1, -n/2, (4 + n)/2, -E^((2*I)*(c + d*x))]))*Sec[c + d*x]^(-2 - n)*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2))/
(d*E^(I*(1 + n)*(c + d*x))*n*(2 + n)*(4 + n)*(A + 2*C + A*Cos[2*c + 2*d*x]))

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Maple [F]  time = 0.687, size = 0, normalized size = 0. \begin{align*} \int \left ( b\sec \left ( dx+c \right ) \right ) ^{n} \left ( A+C \left ( \sec \left ( dx+c \right ) \right ) ^{2} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*sec(d*x+c))^n*(A+C*sec(d*x+c)^2),x)

[Out]

int((b*sec(d*x+c))^n*(A+C*sec(d*x+c)^2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))^n*(A+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

integrate((C*sec(d*x + c)^2 + A)*(b*sec(d*x + c))^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))^n*(A+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

integral((C*sec(d*x + c)^2 + A)*(b*sec(d*x + c))^n, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec{\left (c + d x \right )}\right )^{n} \left (A + C \sec ^{2}{\left (c + d x \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))**n*(A+C*sec(d*x+c)**2),x)

[Out]

Integral((b*sec(c + d*x))**n*(A + C*sec(c + d*x)**2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))^n*(A+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + A)*(b*sec(d*x + c))^n, x)