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

Optimal. Leaf size=221 \[ -\frac {2 (-2 A n+A+C (3-2 n)) \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (5-2 n);\frac {1}{4} (9-2 n);\cos ^2(c+d x)\right )}{d (1-2 n) (5-2 n) \sqrt {\sin ^2(c+d x)} \sec ^{\frac {5}{2}}(c+d x)}-\frac {2 B \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (3-2 n);\frac {1}{4} (7-2 n);\cos ^2(c+d x)\right )}{d (3-2 n) \sqrt {\sin ^2(c+d x)} \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 C \sin (c+d x) (b \sec (c+d x))^n}{d (1-2 n) \sqrt {\sec (c+d x)}} \]

[Out]

-2*C*(b*sec(d*x+c))^n*sin(d*x+c)/d/(1-2*n)/sec(d*x+c)^(1/2)-2*(A+C*(3-2*n)-2*A*n)*hypergeom([1/2, 5/4-1/2*n],[
9/4-1/2*n],cos(d*x+c)^2)*(b*sec(d*x+c))^n*sin(d*x+c)/d/(4*n^2-12*n+5)/sec(d*x+c)^(5/2)/(sin(d*x+c)^2)^(1/2)-2*
B*hypergeom([1/2, 3/4-1/2*n],[7/4-1/2*n],cos(d*x+c)^2)*(b*sec(d*x+c))^n*sin(d*x+c)/d/(3-2*n)/sec(d*x+c)^(3/2)/
(sin(d*x+c)^2)^(1/2)

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Rubi [A]  time = 0.20, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {20, 4047, 3772, 2643, 4046} \[ -\frac {2 (-2 A n+A+C (3-2 n)) \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (5-2 n);\frac {1}{4} (9-2 n);\cos ^2(c+d x)\right )}{d (1-2 n) (5-2 n) \sqrt {\sin ^2(c+d x)} \sec ^{\frac {5}{2}}(c+d x)}-\frac {2 B \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (3-2 n);\frac {1}{4} (7-2 n);\cos ^2(c+d x)\right )}{d (3-2 n) \sqrt {\sin ^2(c+d x)} \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 C \sin (c+d x) (b \sec (c+d x))^n}{d (1-2 n) \sqrt {\sec (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[((b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + 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]) - (2*B*Hypergeometric2F1[1/2, (3 - 2*n)/4, (7 - 2*n)/4, Cos[c + d
*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*Sec[c + d*x]^(3/2)*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 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]

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 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 4047

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(
C_.)), x_Symbol] :> Dist[B/b, Int[(b*Csc[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x
]^2), x] /; FreeQ[{b, e, f, A, B, C, m}, x]

Rubi steps

\begin {align*} \int \frac {(b \sec (c+d x))^n \left (A+B \sec (c+d x)+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+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, 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+\left (B \sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{-\frac {1}{2}+n}(c+d x) \, dx\\ &=-\frac {2 C (b \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) \sqrt {\sec (c+d x)}}+\left (B \cos ^{\frac {1}{2}+n}(c+d x) \sqrt {\sec (c+d x)} (b \sec (c+d x))^n\right ) \int \cos ^{\frac {1}{2}-n}(c+d x) \, dx+\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 {2 B \, _2F_1\left (\frac {1}{2},\frac {1}{4} (3-2 n);\frac {1}{4} (7-2 n);\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d (3-2 n) \sec ^{\frac {3}{2}}(c+d x) \sqrt {\sin ^2(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)}}-\frac {2 B \, _2F_1\left (\frac {1}{2},\frac {1}{4} (3-2 n);\frac {1}{4} (7-2 n);\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d (3-2 n) \sec ^{\frac {3}{2}}(c+d x) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}

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Mathematica [C]  time = 9.04, size = 502, normalized size = 2.27 \[ -\frac {i 2^{n+\frac {1}{2}} e^{-\frac {1}{2} i (4 c+d (2 n+1) x)} \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{n+\frac {1}{2}} \left (1+e^{2 i (c+d x)}\right )^{n+\frac {1}{2}} \sec ^{-n-2}(c+d x) (b \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (e^{2 i c} \left (\frac {e^{\frac {1}{2} i (2 c+d (2 n+3) x)} \left (A (2 n+3) e^{i (c+d x)} \, _2F_1\left (n+\frac {1}{2},\frac {1}{4} (2 n+5);\frac {1}{4} (2 n+9);-e^{2 i (c+d x)}\right )+2 B (2 n+5) \, _2F_1\left (n+\frac {1}{2},\frac {1}{4} (2 n+3);\frac {1}{4} (2 n+7);-e^{2 i (c+d x)}\right )\right )}{d (2 n+3) (2 n+5)}+\frac {2 (A+2 C) e^{\frac {1}{2} i d (2 n+1) x} \, _2F_1\left (n+\frac {1}{2},\frac {1}{4} (2 n+1);\frac {1}{4} (2 n+5);-e^{2 i (c+d x)}\right )}{2 d n+d}\right )+\frac {A e^{\frac {1}{2} i d (2 n-3) x} \, _2F_1\left (n+\frac {1}{2},\frac {1}{4} (2 n-3);\frac {1}{4} (2 n+1);-e^{2 i (c+d x)}\right )}{d (2 n-3)}+\frac {2 B e^{\frac {1}{2} i (2 c+d (2 n-1) x)} \, _2F_1\left (n+\frac {1}{2},\frac {1}{4} (2 n-1);\frac {1}{4} (2 n+3);-e^{2 i (c+d x)}\right )}{d (2 n-1)}\right )}{A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + 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)) + (2*B*E^((I/2)*(2*c + d*(-1 + 2*n)*x))*Hypergeometric2F1[1/2 + n, (-1 + 2*n)/4, (3 + 2*n)/4, -E^((2*
I)*(c + d*x))])/(d*(-1 + 2*n)) + E^((2*I)*c)*((2*(A + 2*C)*E^((I/2)*d*(1 + 2*n)*x)*Hypergeometric2F1[1/2 + n,
(1 + 2*n)/4, (5 + 2*n)/4, -E^((2*I)*(c + d*x))])/(d + 2*d*n) + (E^((I/2)*(2*c + d*(3 + 2*n)*x))*(2*B*(5 + 2*n)
*Hypergeometric2F1[1/2 + n, (3 + 2*n)/4, (7 + 2*n)/4, -E^((2*I)*(c + d*x))] + A*E^(I*(c + d*x))*(3 + 2*n)*Hype
rgeometric2F1[1/2 + n, (5 + 2*n)/4, (9 + 2*n)/4, -E^((2*I)*(c + d*x))]))/(d*(3 + 2*n)*(5 + 2*n))))*Sec[c + d*x
]^(-2 - n)*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(E^((I/2)*(4*c + d*(1 + 2*n)*x))*(A + 2
*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))

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fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{n}}{\sec \left (d x + c\right )^{\frac {3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))^n*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{n}}{\sec \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))^n*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(3/2),x, algorithm="giac")

[Out]

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

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maple [F]  time = 1.60, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \sec \left (d x +c \right )\right )^{n} \left (A +B \sec \left (d x +c \right )+C \left (\sec ^{2}\left (d x +c \right )\right )\right )}{\sec \left (d x +c \right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{n}}{\sec \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(d*x+c))^n*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^n\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b/cos(c + d*x))^n*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/cos(c + d*x))^(3/2),x)

[Out]

int(((b/cos(c + d*x))^n*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/cos(c + d*x))^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \sec {\left (c + d x \right )}\right )^{n} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right )}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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