3.1174 \(\int \frac{(a+a \cos (c+d x))^2 (A+C \cos ^2(c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=270 \[ \frac{4 a^2 (9 A+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (99 A+89 C) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a^2 (33 A+25 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{8 a^2 (33 A+25 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{4 a^2 (9 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{8 C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{99 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{11 d \sec ^{\frac{5}{2}}(c+d x)} \]

[Out]

(4*a^2*(9*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (8*a^2*(33*A + 25
*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a^2*(99*A + 89*C)*Sin[c + d*
x])/(693*d*Sec[c + d*x]^(5/2)) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(5/2)) + (8*C*(a
^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(99*d*Sec[c + d*x]^(5/2)) + (4*a^2*(9*A + 7*C)*Sin[c + d*x])/(45*d*Sec[c
+ d*x]^(3/2)) + (8*a^2*(33*A + 25*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])

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Rubi [A]  time = 0.587857, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.257, Rules used = {4221, 3046, 2976, 2968, 3023, 2748, 2635, 2641, 2639} \[ \frac{4 a^2 (9 A+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (99 A+89 C) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a^2 (33 A+25 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{8 a^2 (33 A+25 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{4 a^2 (9 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{8 C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{99 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{11 d \sec ^{\frac{5}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a^2*(9*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (8*a^2*(33*A + 25
*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a^2*(99*A + 89*C)*Sin[c + d*
x])/(693*d*Sec[c + d*x]^(5/2)) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(5/2)) + (8*C*(a
^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(99*d*Sec[c + d*x]^(5/2)) + (4*a^2*(9*A + 7*C)*Sin[c + d*x])/(45*d*Sec[c
+ d*x]^(3/2)) + (8*a^2*(33*A + 25*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])

Rule 4221

Int[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Dist[(c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSineIntegrandQ[u,
x]

Rule 3046

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (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/(b*d*(m + n + 2)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*Simp
[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b
, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[m, -2^(-
1)] && NeQ[m + n + 2, 0]

Rule 2976

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

Rule 2968

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

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 \frac{(a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac{3}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2 \left (\frac{1}{2} a (11 A+5 C)+2 a C \cos (c+d x)\right ) \, dx}{11 a}\\ &=\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 C \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{99 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x)) \left (\frac{1}{4} a^2 (99 A+65 C)+\frac{1}{4} a^2 (99 A+89 C) \cos (c+d x)\right ) \, dx}{99 a}\\ &=\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 C \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{99 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) \left (\frac{1}{4} a^3 (99 A+65 C)+\left (\frac{1}{4} a^3 (99 A+65 C)+\frac{1}{4} a^3 (99 A+89 C)\right ) \cos (c+d x)+\frac{1}{4} a^3 (99 A+89 C) \cos ^2(c+d x)\right ) \, dx}{99 a}\\ &=\frac{2 a^2 (99 A+89 C) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 C \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{99 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) \left (\frac{9}{2} a^3 (33 A+25 C)+\frac{77}{4} a^3 (9 A+7 C) \cos (c+d x)\right ) \, dx}{693 a}\\ &=\frac{2 a^2 (99 A+89 C) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 C \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{99 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{1}{9} \left (2 a^2 (9 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{5}{2}}(c+d x) \, dx+\frac{1}{77} \left (4 a^2 (33 A+25 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 a^2 (99 A+89 C) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 C \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{99 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2 (9 A+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 a^2 (33 A+25 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{1}{15} \left (2 a^2 (9 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{231} \left (4 a^2 (33 A+25 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{4 a^2 (9 A+7 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{15 d}+\frac{8 a^2 (33 A+25 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{231 d}+\frac{2 a^2 (99 A+89 C) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 C \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{99 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2 (9 A+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 a^2 (33 A+25 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 2.90494, size = 228, normalized size = 0.84 \[ \frac{a^2 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (-2464 i (9 A+7 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right )+\cos (c+d x) (30 (1122 A+941 C) \sin (c+d x)+616 (18 A+19 C) \sin (2 (c+d x))+1980 A \sin (3 (c+d x))+66528 i A+4545 C \sin (3 (c+d x))+1540 C \sin (4 (c+d x))+315 C \sin (5 (c+d x))+51744 i C)+960 (33 A+25 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{27720 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(960*(33*A + 25*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2
] - (2464*I)*(9*A + 7*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2
*I)*(c + d*x))] + Cos[c + d*x]*((66528*I)*A + (51744*I)*C + 30*(1122*A + 941*C)*Sin[c + d*x] + 616*(18*A + 19*
C)*Sin[2*(c + d*x)] + 1980*A*Sin[3*(c + d*x)] + 4545*C*Sin[3*(c + d*x)] + 1540*C*Sin[4*(c + d*x)] + 315*C*Sin[
5*(c + d*x)])))/(27720*d*E^(I*d*x))

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Maple [A]  time = 0.916, size = 436, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3465*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^2*(10080*C*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/
2*c)^12-37520*C*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(3960*A+57040*C)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2
*c)+(-11484*A-46192*C)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(12474*A+22022*C)*sin(1/2*d*x+1/2*c)^4*cos(1/2*
d*x+1/2*c)+(-3861*A-4563*C)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+990*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))-2079*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))+750*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))-1617*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)))/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/si
n(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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{2}}{\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((a+a*cos(d*x+c))^2*(A+C*cos(d*x+c)^2)/sec(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

integrate((C*cos(d*x + c)^2 + A)*(a*cos(d*x + c) + a)^2/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{C a^{2} \cos \left (d x + c\right )^{4} + 2 \, C a^{2} \cos \left (d x + c\right )^{3} +{\left (A + C\right )} a^{2} \cos \left (d x + c\right )^{2} + 2 \, A a^{2} \cos \left (d x + c\right ) + A a^{2}}{\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((a+a*cos(d*x+c))^2*(A+C*cos(d*x+c)^2)/sec(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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

[Out]

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