3.1299 \(\int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=250 \[ \frac{2 \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (5 a^2 B+10 a A b+14 a b C+7 b^2 B\right )}{21 d}+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right )}{15 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (a^2 (7 A+9 C)+18 a b B+4 A b^2\right )}{45 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (5 a^2 B+10 a A b+14 a b C+7 b^2 B\right )}{21 d}+\frac{2 a (9 a B+4 A b) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^2}{9 d} \]

[Out]

(2*(18*a*b*B + 3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(10*a*A*b + 5*a^2*B
+ 7*b^2*B + 14*a*b*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(10*a*A*b + 5*a^2*B + 7*b^2*B + 14*a*b*C)*Sqrt[C
os[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(4*A*b^2 + 18*a*b*B + a^2*(7*A + 9*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])
/(45*d) + (2*a*(4*A*b + 9*a*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*A*Cos[c + d*x]^(3/2)*(b + a*Cos[c
+ d*x])^2*Sin[c + d*x])/(9*d)

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Rubi [A]  time = 0.603223, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.186, Rules used = {4112, 3049, 3033, 3023, 2748, 2639, 2635, 2641} \[ \frac{2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (5 a^2 B+10 a A b+14 a b C+7 b^2 B\right )}{21 d}+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right )}{15 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (a^2 (7 A+9 C)+18 a b B+4 A b^2\right )}{45 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (5 a^2 B+10 a A b+14 a b C+7 b^2 B\right )}{21 d}+\frac{2 a (9 a B+4 A b) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^2}{9 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(2*(18*a*b*B + 3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(10*a*A*b + 5*a^2*B
+ 7*b^2*B + 14*a*b*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(10*a*A*b + 5*a^2*B + 7*b^2*B + 14*a*b*C)*Sqrt[C
os[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(4*A*b^2 + 18*a*b*B + a^2*(7*A + 9*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])
/(45*d) + (2*a*(4*A*b + 9*a*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*A*Cos[c + d*x]^(3/2)*(b + a*Cos[c
+ d*x])^2*Sin[c + d*x])/(9*d)

Rule 4112

Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sec[(e_.)
+ (f_.)*(x_)] + (C_.)*sec[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[d^(m + 2), Int[(b + a*Cos[e + f*x])^m*(d*
Cos[e + f*x])^(n - m - 2)*(C + B*Cos[e + f*x] + A*Cos[e + f*x]^2), x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}
, x] &&  !IntegerQ[n] && IntegerQ[m]

Rule 3049

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)
*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e +
f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n + 2)), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x]
)^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2)
- C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2
, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

Rule 3033

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*d*Cos[e + f*x]*Sin[e + f*x]*(a + b
*Sin[e + f*x])^(m + 1))/(b*f*(m + 3)), x] + Dist[1/(b*(m + 3)), Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*c*
(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e +
f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &&
!LtQ[m, -1]

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 2748

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin{align*} \int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \sqrt{\cos (c+d x)} (b+a \cos (c+d x))^2 \left (C+B \cos (c+d x)+A \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 A \cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac{2}{9} \int \sqrt{\cos (c+d x)} (b+a \cos (c+d x)) \left (\frac{3}{2} b (A+3 C)+\frac{1}{2} (7 a A+9 b B+9 a C) \cos (c+d x)+\frac{1}{2} (4 A b+9 a B) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 a (4 A b+9 a B) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 A \cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac{4}{63} \int \sqrt{\cos (c+d x)} \left (\frac{21}{4} b^2 (A+3 C)+\frac{9}{4} \left (10 a A b+5 a^2 B+7 b^2 B+14 a b C\right ) \cos (c+d x)+\frac{7}{4} \left (4 A b^2+18 a b B+a^2 (7 A+9 C)\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 \left (4 A b^2+18 a b B+a^2 (7 A+9 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 a (4 A b+9 a B) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 A \cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac{8}{315} \int \sqrt{\cos (c+d x)} \left (\frac{21}{8} \left (18 a b B+3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right )+\frac{45}{8} \left (10 a A b+5 a^2 B+7 b^2 B+14 a b C\right ) \cos (c+d x)\right ) \, dx\\ &=\frac{2 \left (4 A b^2+18 a b B+a^2 (7 A+9 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 a (4 A b+9 a B) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 A \cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac{1}{7} \left (10 a A b+5 a^2 B+7 b^2 B+14 a b C\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx+\frac{1}{15} \left (18 a b B+3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 \left (18 a b B+3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (10 a A b+5 a^2 B+7 b^2 B+14 a b C\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 \left (4 A b^2+18 a b B+a^2 (7 A+9 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 a (4 A b+9 a B) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 A \cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac{1}{21} \left (10 a A b+5 a^2 B+7 b^2 B+14 a b C\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (18 a b B+3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (10 a A b+5 a^2 B+7 b^2 B+14 a b C\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 \left (10 a A b+5 a^2 B+7 b^2 B+14 a b C\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 \left (4 A b^2+18 a b B+a^2 (7 A+9 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 a (4 A b+9 a B) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 A \cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x))^2 \sin (c+d x)}{9 d}\\ \end{align*}

Mathematica [A]  time = 1.33587, size = 194, normalized size = 0.78 \[ \frac{60 \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (5 a^2 B+2 a b (5 A+7 C)+7 b^2 B\right )+84 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right )+\sin (c+d x) \sqrt{\cos (c+d x)} \left (7 \cos (c+d x) \left (a^2 (43 A+36 C)+72 a b B+36 A b^2\right )+5 \left (7 a^2 A \cos (3 (c+d x))+78 a^2 B+18 a (a B+2 A b) \cos (2 (c+d x))+156 a A b+168 a b C+84 b^2 B\right )\right )}{630 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(84*(18*a*b*B + 3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2] + 60*(5*a^2*B + 7*b^2*B + 2*a*b
*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*(36*A*b^2 + 72*a*b*B + a^2*(43*A + 36*C))*Cos[
c + d*x] + 5*(156*a*A*b + 78*a^2*B + 84*b^2*B + 168*a*b*C + 18*a*(2*A*b + a*B)*Cos[2*(c + d*x)] + 7*a^2*A*Cos[
3*(c + d*x)]))*Sin[c + d*x])/(630*d)

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Maple [B]  time = 2.587, size = 784, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)

[Out]

-2/315*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-1120*a^2*A*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2
*c)^10+(2240*A*a^2+1440*A*a*b+720*B*a^2)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-2072*A*a^2-2160*A*a*b-504*A
*b^2-1080*B*a^2-1008*B*a*b-504*C*a^2)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(952*A*a^2+1680*A*a*b+504*A*b^2+
840*B*a^2+1008*B*a*b+420*B*b^2+504*C*a^2+840*C*a*b)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-168*A*a^2-480*A*
a*b-126*A*b^2-240*B*a^2-252*B*a*b-210*B*b^2-126*C*a^2-420*C*a*b)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+150*A
*a*b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-147*A
*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^2-189*A
*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*b^2+75*B*
a^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+105*B*
b^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-378*B*
(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a*b+210*a*
b*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-189*C*
(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^2-315*C*
(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*b^2)/(-2*s
in(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C b^{2} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{4} +{\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{3} + A a^{2} \cos \left (d x + c\right )^{4} +{\left (C a^{2} + 2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{2} +{\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )\right )} \sqrt{\cos \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

integral((C*b^2*cos(d*x + c)^4*sec(d*x + c)^4 + (2*C*a*b + B*b^2)*cos(d*x + c)^4*sec(d*x + c)^3 + A*a^2*cos(d*
x + c)^4 + (C*a^2 + 2*B*a*b + A*b^2)*cos(d*x + c)^4*sec(d*x + c)^2 + (B*a^2 + 2*A*a*b)*cos(d*x + c)^4*sec(d*x
+ c))*sqrt(cos(d*x + c)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**(9/2)*(a+b*sec(d*x+c))**2*(A+B*sec(d*x+c)+C*sec(d*x+c)**2),x)

[Out]

Timed out

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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} + B \sec \left (d x + c\right ) + A\right )}{\left (b \sec \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac{9}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^2*cos(d*x + c)^(9/2), x)