3.1292 \(\int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=190 \[ \frac{2 \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) (5 a B+5 A b+7 b C)}{21 d}+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (7 a A+9 a C+9 b B)}{15 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (7 a A+9 a C+9 b B)}{45 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} (5 a B+5 A b+7 b C)}{21 d}+\frac{2 (a B+A b) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d} \]

[Out]

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

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Rubi [A]  time = 0.297042, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {4112, 3033, 3023, 2748, 2635, 2641, 2639} \[ \frac{2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (5 a B+5 A b+7 b C)}{21 d}+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (7 a A+9 a C+9 b B)}{15 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (7 a A+9 a C+9 b B)}{45 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} (5 a B+5 A b+7 b C)}{21 d}+\frac{2 (a B+A b) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rubi steps

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

Mathematica [A]  time = 1.01293, size = 143, normalized size = 0.75 \[ \frac{60 \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) (5 a B+5 A b+7 b C)+84 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (7 a A+9 a C+9 b B)+\sin (c+d x) \sqrt{\cos (c+d x)} (7 \cos (c+d x) (43 a A+36 a C+36 b B)+5 (18 (a B+A b) \cos (2 (c+d x))+7 a A \cos (3 (c+d x))+78 a B+78 A b+84 b C))}{630 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 2.467, size = 565, normalized size = 3. \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))*(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*a*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c
)^10+(2240*A*a+720*A*b+720*B*a)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-2072*A*a-1080*A*b-1080*B*a-504*B*b-5
04*C*a)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(952*A*a+840*A*b+840*B*a+504*B*b+504*C*a+420*C*b)*sin(1/2*d*x+
1/2*c)^4*cos(1/2*d*x+1/2*c)+(-168*A*a-240*A*b-240*B*a-126*B*b-126*C*a-210*C*b)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*
x+1/2*c)+75*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+75*B*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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))
-189*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))*b+1
05*C*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))-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*
sin(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))*(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 \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{3} +{\left (C a + B b\right )} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{2} + A a \cos \left (d x + c\right )^{4} +{\left (B 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))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

integral((C*b*cos(d*x + c)^4*sec(d*x + c)^3 + (C*a + B*b)*cos(d*x + c)^4*sec(d*x + c)^2 + A*a*cos(d*x + c)^4 +
(B*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))*(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 )} \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))*(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)*cos(d*x + c)^(9/2), x)