3.674 \(\int \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x)) (A+C \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=196 \[ \frac {2 a (9 A+7 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a (9 A+7 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{45 d}+\frac {2 a C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}+\frac {10 b (11 A+9 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {2 b (11 A+9 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{77 d}+\frac {10 b (11 A+9 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{231 d}+\frac {2 b C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d} \]

[Out]

2/15*a*(9*A+7*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+10/23
1*b*(11*A+9*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))/d+2/45*a*
(9*A+7*C)*cos(d*x+c)^(3/2)*sin(d*x+c)/d+2/77*b*(11*A+9*C)*cos(d*x+c)^(5/2)*sin(d*x+c)/d+2/9*a*C*cos(d*x+c)^(7/
2)*sin(d*x+c)/d+2/11*b*C*cos(d*x+c)^(9/2)*sin(d*x+c)/d+10/231*b*(11*A+9*C)*sin(d*x+c)*cos(d*x+c)^(1/2)/d

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Rubi [A]  time = 0.24, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3034, 3023, 2748, 2635, 2639, 2641} \[ \frac {2 a (9 A+7 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a (9 A+7 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{45 d}+\frac {2 a C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}+\frac {10 b (11 A+9 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {2 b (11 A+9 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{77 d}+\frac {10 b (11 A+9 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{231 d}+\frac {2 b C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (10*b*(11*A + 9*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (
10*b*(11*A + 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*a*(9*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])
/(45*d) + (2*b*(11*A + 9*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (2*a*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/
(9*d) + (2*b*C*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(11*d)

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 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 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 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 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 3034

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

Rubi steps

\begin {align*} \int \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {2 b C \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {2}{11} \int \cos ^{\frac {5}{2}}(c+d x) \left (\frac {11 a A}{2}+\frac {1}{2} b (11 A+9 C) \cos (c+d x)+\frac {11}{2} a C \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 a C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {2 b C \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {4}{99} \int \cos ^{\frac {5}{2}}(c+d x) \left (\frac {11}{4} a (9 A+7 C)+\frac {9}{4} b (11 A+9 C) \cos (c+d x)\right ) \, dx\\ &=\frac {2 a C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {2 b C \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {1}{9} (a (9 A+7 C)) \int \cos ^{\frac {5}{2}}(c+d x) \, dx+\frac {1}{11} (b (11 A+9 C)) \int \cos ^{\frac {7}{2}}(c+d x) \, dx\\ &=\frac {2 a (9 A+7 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 b (11 A+9 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac {2 a C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {2 b C \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {1}{15} (a (9 A+7 C)) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{77} (5 b (11 A+9 C)) \int \cos ^{\frac {3}{2}}(c+d x) \, dx\\ &=\frac {2 a (9 A+7 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {10 b (11 A+9 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 a (9 A+7 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 b (11 A+9 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac {2 a C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {2 b C \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {1}{231} (5 b (11 A+9 C)) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 a (9 A+7 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {10 b (11 A+9 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {10 b (11 A+9 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 a (9 A+7 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 b (11 A+9 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac {2 a C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {2 b C \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{11 d}\\ \end {align*}

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Mathematica [A]  time = 1.65, size = 134, normalized size = 0.68 \[ \frac {\sin (c+d x) \sqrt {\cos (c+d x)} (154 a (36 A+43 C) \cos (c+d x)+770 a C \cos (3 (c+d x))+180 b (11 A+16 C) \cos (2 (c+d x))+8580 A b+315 b C \cos (4 (c+d x))+7965 b C)+1848 a (9 A+7 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+600 b (11 A+9 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{13860 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1848*a*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2] + 600*b*(11*A + 9*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*
x]]*(8580*A*b + 7965*b*C + 154*a*(36*A + 43*C)*Cos[c + d*x] + 180*b*(11*A + 16*C)*Cos[2*(c + d*x)] + 770*a*C*C
os[3*(c + d*x)] + 315*b*C*Cos[4*(c + d*x)])*Sin[c + d*x])/(13860*d)

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fricas [F]  time = 1.30, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b \cos \left (d x + c\right )^{5} + C a \cos \left (d x + c\right )^{4} + A b \cos \left (d x + c\right )^{3} + A a \cos \left (d x + c\right )^{2}\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((C*b*cos(d*x + c)^5 + C*a*cos(d*x + c)^4 + A*b*cos(d*x + c)^3 + A*a*cos(d*x + c)^2)*sqrt(cos(d*x + c)
), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 2.21, size = 481, normalized size = 2.45 \[ -\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (20160 C b \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-12320 a C -50400 C b \right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (7920 A b +24640 a C +56880 C b \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-5544 a A -11880 A b -22792 a C -34920 C b \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (5544 a A +9240 A b +10472 a C +13860 C b \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-1386 a A -2640 A b -1848 a C -2790 C b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-2079 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a +825 A b \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-1617 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a +675 C b \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{3465 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3465*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(20160*C*b*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*
c)^12+(-12320*C*a-50400*C*b)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(7920*A*b+24640*C*a+56880*C*b)*sin(1/2*d
*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-5544*A*a-11880*A*b-22792*C*a-34920*C*b)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*
c)+(5544*A*a+9240*A*b+10472*C*a+13860*C*b)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-1386*A*a-2640*A*b-1848*C*
a-2790*C*b)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-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))*a+825*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))-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))*a+675*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)))/(-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]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.67, size = 177, normalized size = 0.90 \[ -\frac {2\,A\,a\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,A\,b\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,a\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,b\,{\cos \left (c+d\,x\right )}^{13/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {13}{4};\ \frac {17}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{13\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (2*A*a*cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7*d*(sin(c + d*x)^2)^(1
/2)) - (2*A*b*cos(c + d*x)^(9/2)*sin(c + d*x)*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(9*d*(sin(c + d*x)^
2)^(1/2)) - (2*C*a*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeom([1/2, 11/4], 15/4, cos(c + d*x)^2))/(11*d*(sin(c
+ d*x)^2)^(1/2)) - (2*C*b*cos(c + d*x)^(13/2)*sin(c + d*x)*hypergeom([1/2, 13/4], 17/4, cos(c + d*x)^2))/(13*
d*(sin(c + d*x)^2)^(1/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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