3.382 \(\int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=243 \[ \frac{2 a^2 (99 A+110 B+84 C) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (429 A+374 B+336 C) \sin (c+d x)}{495 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (429 A+374 B+336 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 d}-\frac{4 a (429 A+374 B+336 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3465 d}+\frac{2 a (11 B+3 C) \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{99 d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d} \]

[Out]

(2*a^2*(429*A + 374*B + 336*C)*Sin[c + d*x])/(495*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(99*A + 110*B + 84*C)*C
os[c + d*x]^3*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) - (4*a*(429*A + 374*B + 336*C)*Sqrt[a + a*Cos[c +
d*x]]*Sin[c + d*x])/(3465*d) + (2*a*(11*B + 3*C)*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(99*d)
+ (2*(429*A + 374*B + 336*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(1155*d) + (2*C*Cos[c + d*x]^3*(a + a*C
os[c + d*x])^(3/2)*Sin[c + d*x])/(11*d)

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Rubi [A]  time = 0.700156, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.14, Rules used = {3045, 2976, 2981, 2759, 2751, 2646} \[ \frac{2 a^2 (99 A+110 B+84 C) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (429 A+374 B+336 C) \sin (c+d x)}{495 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (429 A+374 B+336 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 d}-\frac{4 a (429 A+374 B+336 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3465 d}+\frac{2 a (11 B+3 C) \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{99 d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*(429*A + 374*B + 336*C)*Sin[c + d*x])/(495*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(99*A + 110*B + 84*C)*C
os[c + d*x]^3*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) - (4*a*(429*A + 374*B + 336*C)*Sqrt[a + a*Cos[c +
d*x]]*Sin[c + d*x])/(3465*d) + (2*a*(11*B + 3*C)*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(99*d)
+ (2*(429*A + 374*B + 336*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(1155*d) + (2*C*Cos[c + d*x]^3*(a + a*C
os[c + d*x])^(3/2)*Sin[c + d*x])/(11*d)

Rule 3045

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/(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)) + b*B*
d*(m + n + 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, 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 2981

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-2*b*B*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(2*n + 3)*Sqr
t[a + b*Sin[e + f*x]]), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b*d*(2*n + 3)), Int[Sqrt[a + b*
Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, 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] &&  !LtQ[n, -1]

Rule 2759

Int[sin[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(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*(b*(m + 1) - a*
Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] &&  !LtQ[m, -2^(-1)]

Rule 2751

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

Rule 2646

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(-2*b*Cos[c + d*x])/(d*Sqrt[a + b*Sin[c + d*
x]]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac{2 \int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left (\frac{1}{2} a (11 A+6 C)+\frac{1}{2} a (11 B+3 C) \cos (c+d x)\right ) \, dx}{11 a}\\ &=\frac{2 a (11 B+3 C) \cos ^3(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{99 d}+\frac{2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac{4 \int \cos ^2(c+d x) \sqrt{a+a \cos (c+d x)} \left (\frac{3}{4} a^2 (33 A+22 B+24 C)+\frac{1}{4} a^2 (99 A+110 B+84 C) \cos (c+d x)\right ) \, dx}{99 a}\\ &=\frac{2 a^2 (99 A+110 B+84 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a (11 B+3 C) \cos ^3(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{99 d}+\frac{2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac{1}{231} (a (429 A+374 B+336 C)) \int \cos ^2(c+d x) \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{2 a^2 (99 A+110 B+84 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a (11 B+3 C) \cos ^3(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{99 d}+\frac{2 (429 A+374 B+336 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 d}+\frac{2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac{(2 (429 A+374 B+336 C)) \int \left (\frac{3 a}{2}-a \cos (c+d x)\right ) \sqrt{a+a \cos (c+d x)} \, dx}{1155}\\ &=\frac{2 a^2 (99 A+110 B+84 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt{a+a \cos (c+d x)}}-\frac{4 a (429 A+374 B+336 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{3465 d}+\frac{2 a (11 B+3 C) \cos ^3(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{99 d}+\frac{2 (429 A+374 B+336 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 d}+\frac{2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac{1}{495} (a (429 A+374 B+336 C)) \int \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{2 a^2 (429 A+374 B+336 C) \sin (c+d x)}{495 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (99 A+110 B+84 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt{a+a \cos (c+d x)}}-\frac{4 a (429 A+374 B+336 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{3465 d}+\frac{2 a (11 B+3 C) \cos ^3(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{99 d}+\frac{2 (429 A+374 B+336 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 d}+\frac{2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d}\\ \end{align*}

Mathematica [A]  time = 1.2803, size = 145, normalized size = 0.6 \[ \frac{a \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} ((33396 A+35156 B+34734 C) \cos (c+d x)+8 (1287 A+1507 B+1743 C) \cos (2 (c+d x))+1980 A \cos (3 (c+d x))+65208 A+3740 B \cos (3 (c+d x))+770 B \cos (4 (c+d x))+59158 B+4935 C \cos (3 (c+d x))+1470 C \cos (4 (c+d x))+315 C \cos (5 (c+d x))+55482 C)}{27720 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sqrt[a*(1 + Cos[c + d*x])]*(65208*A + 59158*B + 55482*C + (33396*A + 35156*B + 34734*C)*Cos[c + d*x] + 8*(1
287*A + 1507*B + 1743*C)*Cos[2*(c + d*x)] + 1980*A*Cos[3*(c + d*x)] + 3740*B*Cos[3*(c + d*x)] + 4935*C*Cos[3*(
c + d*x)] + 770*B*Cos[4*(c + d*x)] + 1470*C*Cos[4*(c + d*x)] + 315*C*Cos[5*(c + d*x)])*Tan[(c + d*x)/2])/(2772
0*d)

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Maple [A]  time = 0.067, size = 154, normalized size = 0.6 \begin{align*}{\frac{4\,{a}^{2}\sqrt{2}}{3465\,d}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( -5040\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{10}+ \left ( 3080\,B+18480\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{8}+ \left ( -1980\,A-9900\,B-27720\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}+ \left ( 5544\,A+12474\,B+22176\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+ \left ( -5775\,A-8085\,B-10395\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+3465\,A+3465\,B+3465\,C \right ){\frac{1}{\sqrt{a \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3465*cos(1/2*d*x+1/2*c)*a^2*sin(1/2*d*x+1/2*c)*(-5040*C*sin(1/2*d*x+1/2*c)^10+(3080*B+18480*C)*sin(1/2*d*x+1
/2*c)^8+(-1980*A-9900*B-27720*C)*sin(1/2*d*x+1/2*c)^6+(5544*A+12474*B+22176*C)*sin(1/2*d*x+1/2*c)^4+(-5775*A-8
085*B-10395*C)*sin(1/2*d*x+1/2*c)^2+3465*A+3465*B+3465*C)*2^(1/2)/(a*cos(1/2*d*x+1/2*c)^2)^(1/2)/d

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Maxima [A]  time = 2.23586, size = 340, normalized size = 1.4 \begin{align*} \frac{132 \,{\left (15 \, \sqrt{2} a \sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 63 \, \sqrt{2} a \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 175 \, \sqrt{2} a \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 735 \, \sqrt{2} a \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} A \sqrt{a} + 22 \,{\left (35 \, \sqrt{2} a \sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ) + 135 \, \sqrt{2} a \sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 378 \, \sqrt{2} a \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 1050 \, \sqrt{2} a \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 3780 \, \sqrt{2} a \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} B \sqrt{a} + 21 \,{\left (15 \, \sqrt{2} a \sin \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right ) + 55 \, \sqrt{2} a \sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ) + 165 \, \sqrt{2} a \sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 429 \, \sqrt{2} a \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 990 \, \sqrt{2} a \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 3630 \, \sqrt{2} a \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} C \sqrt{a}}{55440 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/55440*(132*(15*sqrt(2)*a*sin(7/2*d*x + 7/2*c) + 63*sqrt(2)*a*sin(5/2*d*x + 5/2*c) + 175*sqrt(2)*a*sin(3/2*d*
x + 3/2*c) + 735*sqrt(2)*a*sin(1/2*d*x + 1/2*c))*A*sqrt(a) + 22*(35*sqrt(2)*a*sin(9/2*d*x + 9/2*c) + 135*sqrt(
2)*a*sin(7/2*d*x + 7/2*c) + 378*sqrt(2)*a*sin(5/2*d*x + 5/2*c) + 1050*sqrt(2)*a*sin(3/2*d*x + 3/2*c) + 3780*sq
rt(2)*a*sin(1/2*d*x + 1/2*c))*B*sqrt(a) + 21*(15*sqrt(2)*a*sin(11/2*d*x + 11/2*c) + 55*sqrt(2)*a*sin(9/2*d*x +
9/2*c) + 165*sqrt(2)*a*sin(7/2*d*x + 7/2*c) + 429*sqrt(2)*a*sin(5/2*d*x + 5/2*c) + 990*sqrt(2)*a*sin(3/2*d*x
+ 3/2*c) + 3630*sqrt(2)*a*sin(1/2*d*x + 1/2*c))*C*sqrt(a))/d

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Fricas [A]  time = 2.18888, size = 392, normalized size = 1.61 \begin{align*} \frac{2 \,{\left (315 \, C a \cos \left (d x + c\right )^{5} + 35 \,{\left (11 \, B + 21 \, C\right )} a \cos \left (d x + c\right )^{4} + 5 \,{\left (99 \, A + 187 \, B + 168 \, C\right )} a \cos \left (d x + c\right )^{3} + 3 \,{\left (429 \, A + 374 \, B + 336 \, C\right )} a \cos \left (d x + c\right )^{2} + 4 \,{\left (429 \, A + 374 \, B + 336 \, C\right )} a \cos \left (d x + c\right ) + 8 \,{\left (429 \, A + 374 \, B + 336 \, C\right )} a\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{3465 \,{\left (d \cos \left (d x + c\right ) + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(315*C*a*cos(d*x + c)^5 + 35*(11*B + 21*C)*a*cos(d*x + c)^4 + 5*(99*A + 187*B + 168*C)*a*cos(d*x + c)^3
+ 3*(429*A + 374*B + 336*C)*a*cos(d*x + c)^2 + 4*(429*A + 374*B + 336*C)*a*cos(d*x + c) + 8*(429*A + 374*B +
336*C)*a)*sqrt(a*cos(d*x + c) + a)*sin(d*x + c)/(d*cos(d*x + c) + d)

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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)**2*(a+a*cos(d*x+c))**(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2),x)

[Out]

Timed out

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Giac [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)^2*(a+a*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="giac")

[Out]

Timed out