3.38 \(\int \frac{(b \sec (c+d x))^n (A+C \sec ^2(c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=140 \[ -\frac{2 (-2 A n+A+C (3-2 n)) \sin (c+d x) (b \sec (c+d x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{4} (5-2 n),\frac{1}{4} (9-2 n),\cos ^2(c+d x)\right )}{d (1-2 n) (5-2 n) \sqrt{\sin ^2(c+d x)} \sec ^{\frac{5}{2}}(c+d x)}-\frac{2 C \sin (c+d x) (b \sec (c+d x))^n}{d (1-2 n) \sqrt{\sec (c+d x)}} \]

[Out]

(-2*C*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Sec[c + d*x]]) - (2*(A + C*(3 - 2*n) - 2*A*n)*Hyperge
ometric2F1[1/2, (5 - 2*n)/4, (9 - 2*n)/4, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*(5 - 2
*n)*Sec[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2])

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Rubi [A]  time = 0.125937, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {20, 4046, 3772, 2643} \[ -\frac{2 (-2 A n+A+C (3-2 n)) \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{4} (5-2 n);\frac{1}{4} (9-2 n);\cos ^2(c+d x)\right )}{d (1-2 n) (5-2 n) \sqrt{\sin ^2(c+d x)} \sec ^{\frac{5}{2}}(c+d x)}-\frac{2 C \sin (c+d x) (b \sec (c+d x))^n}{d (1-2 n) \sqrt{\sec (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[((b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]

[Out]

(-2*C*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Sec[c + d*x]]) - (2*(A + C*(3 - 2*n) - 2*A*n)*Hyperge
ometric2F1[1/2, (5 - 2*n)/4, (9 - 2*n)/4, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*(5 - 2
*n)*Sec[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2])

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + 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 \frac{(b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac{3}{2}}(c+d x)} \, dx &=\left (\sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{-\frac{3}{2}+n}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx\\ &=-\frac{2 C (b \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) \sqrt{\sec (c+d x)}}+\frac{\left (\left (C \left (-\frac{3}{2}+n\right )+A \left (-\frac{1}{2}+n\right )\right ) \sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{-\frac{3}{2}+n}(c+d x) \, dx}{-\frac{1}{2}+n}\\ &=-\frac{2 C (b \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) \sqrt{\sec (c+d x)}}+\frac{\left (\left (C \left (-\frac{3}{2}+n\right )+A \left (-\frac{1}{2}+n\right )\right ) \cos ^{\frac{1}{2}+n}(c+d x) \sqrt{\sec (c+d x)} (b \sec (c+d x))^n\right ) \int \cos ^{\frac{3}{2}-n}(c+d x) \, dx}{-\frac{1}{2}+n}\\ &=-\frac{2 C (b \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) \sqrt{\sec (c+d x)}}-\frac{2 (A (1-2 n)+C (3-2 n)) \, _2F_1\left (\frac{1}{2},\frac{1}{4} (5-2 n);\frac{1}{4} (9-2 n);\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) (5-2 n) \sec ^{\frac{5}{2}}(c+d x) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}

Mathematica [C]  time = 2.85872, size = 343, normalized size = 2.45 \[ -\frac{i 2^{n+\frac{1}{2}} e^{-\frac{1}{2} i (4 c+d (2 n+1) x)} \left (\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{n+\frac{1}{2}} \left (1+e^{2 i (c+d x)}\right )^{n+\frac{1}{2}} \sec ^{-n-2}(c+d x) \left (A+C \sec ^2(c+d x)\right ) (b \sec (c+d x))^n \left (\frac{e^{\frac{1}{2} i (4 c+d (2 n+1) x)} \left (2 (2 n+5) (A+2 C) \text{Hypergeometric2F1}\left (n+\frac{1}{2},\frac{1}{4} (2 n+1),\frac{1}{4} (2 n+5),-e^{2 i (c+d x)}\right )+A (2 n+1) e^{2 i (c+d x)} \text{Hypergeometric2F1}\left (n+\frac{1}{2},\frac{1}{4} (2 n+5),\frac{1}{4} (2 n+9),-e^{2 i (c+d x)}\right )\right )}{d (2 n+1) (2 n+5)}+\frac{A e^{\frac{1}{2} i d (2 n-3) x} \text{Hypergeometric2F1}\left (n+\frac{1}{2},\frac{1}{4} (2 n-3),\frac{1}{4} (2 n+1),-e^{2 i (c+d x)}\right )}{d (2 n-3)}\right )}{A \cos (2 c+2 d x)+A+2 C} \]

Antiderivative was successfully verified.

[In]

Integrate[((b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]

[Out]

((-I)*2^(1/2 + n)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1/2 + n)*(1 + E^((2*I)*(c + d*x)))^(1/2 + n)*((
A*E^((I/2)*d*(-3 + 2*n)*x)*Hypergeometric2F1[1/2 + n, (-3 + 2*n)/4, (1 + 2*n)/4, -E^((2*I)*(c + d*x))])/(d*(-3
+ 2*n)) + (E^((I/2)*(4*c + d*(1 + 2*n)*x))*(2*(A + 2*C)*(5 + 2*n)*Hypergeometric2F1[1/2 + n, (1 + 2*n)/4, (5
+ 2*n)/4, -E^((2*I)*(c + d*x))] + A*E^((2*I)*(c + d*x))*(1 + 2*n)*Hypergeometric2F1[1/2 + n, (5 + 2*n)/4, (9 +
2*n)/4, -E^((2*I)*(c + d*x))]))/(d*(1 + 2*n)*(5 + 2*n)))*Sec[c + d*x]^(-2 - n)*(b*Sec[c + d*x])^n*(A + C*Sec[
c + d*x]^2))/(E^((I/2)*(4*c + d*(1 + 2*n)*x))*(A + 2*C + A*Cos[2*c + 2*d*x]))

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Maple [F]  time = 0.222, 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 ) \left ( \sec \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}}\, 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)/sec(d*x+c)^(3/2),x)

[Out]

int((b*sec(d*x+c))^n*(A+C*sec(d*x+c)^2)/sec(d*x+c)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n}}{\sec \left (d x + c\right )^{\frac{3}{2}}}\,{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)/sec(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

integrate((C*sec(d*x + c)^2 + A)*(b*sec(d*x + c))^n/sec(d*x + c)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n}}{\sec \left (d x + c\right )^{\frac{3}{2}}}, 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)/sec(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

integral((C*sec(d*x + c)^2 + A)*(b*sec(d*x + c))^n/sec(d*x + c)^(3/2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (b \sec{\left (c + d x \right )}\right )^{n} \left (A + C \sec ^{2}{\left (c + d x \right )}\right )}{\sec ^{\frac{3}{2}}{\left (c + d x \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)/sec(d*x+c)**(3/2),x)

[Out]

Integral((b*sec(c + d*x))**n*(A + C*sec(c + d*x)**2)/sec(c + d*x)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n}}{\sec \left (d x + c\right )^{\frac{3}{2}}}\,{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)/sec(d*x+c)^(3/2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + A)*(b*sec(d*x + c))^n/sec(d*x + c)^(3/2), x)