3.93 \(\int \cos (c+d x) (a+a \cos (c+d x))^{5/2} (A+C \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=211 \[ \frac{16 a^2 (33 A+25 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{693 d}+\frac{64 a^3 (33 A+25 C) \sin (c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (99 A+26 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{693 d}+\frac{2 a (33 A+25 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{231 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d}+\frac{10 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{99 a d} \]

[Out]

(64*a^3*(33*A + 25*C)*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(33*A + 25*C)*Sqrt[a + a*Cos[c
+ d*x]]*Sin[c + d*x])/(693*d) + (2*a*(33*A + 25*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(231*d) + (2*(99*A
+ 26*C)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(693*d) + (2*C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(5/2)*Sin
[c + d*x])/(11*d) + (10*C*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(99*a*d)

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Rubi [A]  time = 0.410216, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3046, 2968, 3023, 2751, 2647, 2646} \[ \frac{16 a^2 (33 A+25 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{693 d}+\frac{64 a^3 (33 A+25 C) \sin (c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (99 A+26 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{693 d}+\frac{2 a (33 A+25 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{231 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d}+\frac{10 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{99 a d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2),x]

[Out]

(64*a^3*(33*A + 25*C)*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(33*A + 25*C)*Sqrt[a + a*Cos[c
+ d*x]]*Sin[c + d*x])/(693*d) + (2*a*(33*A + 25*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(231*d) + (2*(99*A
+ 26*C)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(693*d) + (2*C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(5/2)*Sin
[c + d*x])/(11*d) + (10*C*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(99*a*d)

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 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 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 2647

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n -
1))/(d*n), x] + Dist[(a*(2*n - 1))/n, Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && Eq
Q[a^2 - b^2, 0] && IGtQ[n - 1/2, 0]

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 (c+d x) (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d}+\frac{2 \int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \left (\frac{1}{2} a (11 A+4 C)+\frac{5}{2} a C \cos (c+d x)\right ) \, dx}{11 a}\\ &=\frac{2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d}+\frac{2 \int (a+a \cos (c+d x))^{5/2} \left (\frac{1}{2} a (11 A+4 C) \cos (c+d x)+\frac{5}{2} a C \cos ^2(c+d x)\right ) \, dx}{11 a}\\ &=\frac{2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d}+\frac{10 C (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{99 a d}+\frac{4 \int (a+a \cos (c+d x))^{5/2} \left (\frac{35 a^2 C}{4}+\frac{1}{4} a^2 (99 A+26 C) \cos (c+d x)\right ) \, dx}{99 a^2}\\ &=\frac{2 (99 A+26 C) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{693 d}+\frac{2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d}+\frac{10 C (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{99 a d}+\frac{1}{231} (5 (33 A+25 C)) \int (a+a \cos (c+d x))^{5/2} \, dx\\ &=\frac{2 a (33 A+25 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{231 d}+\frac{2 (99 A+26 C) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{693 d}+\frac{2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d}+\frac{10 C (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{99 a d}+\frac{1}{231} (8 a (33 A+25 C)) \int (a+a \cos (c+d x))^{3/2} \, dx\\ &=\frac{16 a^2 (33 A+25 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{693 d}+\frac{2 a (33 A+25 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{231 d}+\frac{2 (99 A+26 C) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{693 d}+\frac{2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d}+\frac{10 C (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{99 a d}+\frac{1}{693} \left (32 a^2 (33 A+25 C)\right ) \int \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{64 a^3 (33 A+25 C) \sin (c+d x)}{693 d \sqrt{a+a \cos (c+d x)}}+\frac{16 a^2 (33 A+25 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{693 d}+\frac{2 a (33 A+25 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{231 d}+\frac{2 (99 A+26 C) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{693 d}+\frac{2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d}+\frac{10 C (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{99 a d}\\ \end{align*}

Mathematica [A]  time = 0.864536, size = 117, normalized size = 0.55 \[ \frac{a^2 \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} (2 (6666 A+6989 C) \cos (c+d x)+16 (198 A+325 C) \cos (2 (c+d x))+396 A \cos (3 (c+d x))+27456 A+1735 C \cos (3 (c+d x))+448 C \cos (4 (c+d x))+63 C \cos (5 (c+d x))+22928 C)}{5544 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2),x]

[Out]

(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(27456*A + 22928*C + 2*(6666*A + 6989*C)*Cos[c + d*x] + 16*(198*A + 325*C)*Cos
[2*(c + d*x)] + 396*A*Cos[3*(c + d*x)] + 1735*C*Cos[3*(c + d*x)] + 448*C*Cos[4*(c + d*x)] + 63*C*Cos[5*(c + d*
x)])*Tan[(c + d*x)/2])/(5544*d)

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Maple [A]  time = 0.043, size = 137, normalized size = 0.7 \begin{align*}{\frac{8\,{a}^{3}\sqrt{2}}{693\,d}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( -504\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{10}+2156\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}+ \left ( -198\,A-3762\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}+ \left ( 693\,A+3465\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+ \left ( -924\,A-1848\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+693\,A+693\,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)*(a+a*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2),x)

[Out]

8/693*cos(1/2*d*x+1/2*c)*a^3*sin(1/2*d*x+1/2*c)*(-504*C*sin(1/2*d*x+1/2*c)^10+2156*C*sin(1/2*d*x+1/2*c)^8+(-19
8*A-3762*C)*sin(1/2*d*x+1/2*c)^6+(693*A+3465*C)*sin(1/2*d*x+1/2*c)^4+(-924*A-1848*C)*sin(1/2*d*x+1/2*c)^2+693*
A+693*C)*2^(1/2)/(a*cos(1/2*d*x+1/2*c)^2)^(1/2)/d

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Maxima [A]  time = 2.07976, size = 255, normalized size = 1.21 \begin{align*} \frac{132 \,{\left (3 \, \sqrt{2} a^{2} \sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 21 \, \sqrt{2} a^{2} \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 77 \, \sqrt{2} a^{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 315 \, \sqrt{2} a^{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} A \sqrt{a} +{\left (63 \, \sqrt{2} a^{2} \sin \left (\frac{11}{2} \, d x + \frac{11}{2} \, c\right ) + 385 \, \sqrt{2} a^{2} \sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ) + 1287 \, \sqrt{2} a^{2} \sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 3465 \, \sqrt{2} a^{2} \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 8778 \, \sqrt{2} a^{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 31878 \, \sqrt{2} a^{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} C \sqrt{a}}{11088 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(a+a*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2),x, algorithm="maxima")

[Out]

1/11088*(132*(3*sqrt(2)*a^2*sin(7/2*d*x + 7/2*c) + 21*sqrt(2)*a^2*sin(5/2*d*x + 5/2*c) + 77*sqrt(2)*a^2*sin(3/
2*d*x + 3/2*c) + 315*sqrt(2)*a^2*sin(1/2*d*x + 1/2*c))*A*sqrt(a) + (63*sqrt(2)*a^2*sin(11/2*d*x + 11/2*c) + 38
5*sqrt(2)*a^2*sin(9/2*d*x + 9/2*c) + 1287*sqrt(2)*a^2*sin(7/2*d*x + 7/2*c) + 3465*sqrt(2)*a^2*sin(5/2*d*x + 5/
2*c) + 8778*sqrt(2)*a^2*sin(3/2*d*x + 3/2*c) + 31878*sqrt(2)*a^2*sin(1/2*d*x + 1/2*c))*C*sqrt(a))/d

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Fricas [A]  time = 1.63788, size = 339, normalized size = 1.61 \begin{align*} \frac{2 \,{\left (63 \, C a^{2} \cos \left (d x + c\right )^{5} + 224 \, C a^{2} \cos \left (d x + c\right )^{4} +{\left (99 \, A + 355 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 6 \,{\left (66 \, A + 71 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} +{\left (759 \, A + 568 \, C\right )} a^{2} \cos \left (d x + c\right ) + 2 \,{\left (759 \, A + 568 \, C\right )} a^{2}\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{693 \,{\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)*(a+a*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2),x, algorithm="fricas")

[Out]

2/693*(63*C*a^2*cos(d*x + c)^5 + 224*C*a^2*cos(d*x + c)^4 + (99*A + 355*C)*a^2*cos(d*x + c)^3 + 6*(66*A + 71*C
)*a^2*cos(d*x + c)^2 + (759*A + 568*C)*a^2*cos(d*x + c) + 2*(759*A + 568*C)*a^2)*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)*(a+a*cos(d*x+c))**(5/2)*(A+C*cos(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 \cos \left (d x + c\right )^{2} + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cos \left (d x + c\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(a+a*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2),x, algorithm="giac")

[Out]

integrate((C*cos(d*x + c)^2 + A)*(a*cos(d*x + c) + a)^(5/2)*cos(d*x + c), x)