3.869 \(\int \frac{(a+b \cos (c+d x))^3 (B \cos (c+d x)+C \cos ^2(c+d x))}{\sqrt{\cos (c+d x)}} \, dx\)

Optimal. Leaf size=255 \[ \frac{2 \left (21 a^2 b B+7 a^3 C+15 a b^2 C+5 b^3 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 \left (27 a^2 b C+15 a^3 B+27 a b^2 B+7 b^3 C\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 b \left (22 a^2 C+27 a b B+7 b^2 C\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 \left (21 a^2 b B+7 a^3 C+15 a b^2 C+5 b^3 B\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b^2 (13 a C+9 b B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d} \]

[Out]

(2*(15*a^3*B + 27*a*b^2*B + 27*a^2*b*C + 7*b^3*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(21*a^2*b*B + 5*b^3*B
+ 7*a^3*C + 15*a*b^2*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(21*a^2*b*B + 5*b^3*B + 7*a^3*C + 15*a*b^2*C)*
Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b*(27*a*b*B + 22*a^2*C + 7*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x]
)/(45*d) + (2*b^2*(9*b*B + 13*a*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*b*C*Cos[c + d*x]^(3/2)*(a + b*
Cos[c + d*x])^2*Sin[c + d*x])/(9*d)

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Rubi [A]  time = 0.581075, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3029, 2990, 3033, 3023, 2748, 2639, 2635, 2641} \[ \frac{2 \left (21 a^2 b B+7 a^3 C+15 a b^2 C+5 b^3 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 \left (27 a^2 b C+15 a^3 B+27 a b^2 B+7 b^3 C\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 b \left (22 a^2 C+27 a b B+7 b^2 C\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 \left (21 a^2 b B+7 a^3 C+15 a b^2 C+5 b^3 B\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b^2 (13 a C+9 b B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(15*a^3*B + 27*a*b^2*B + 27*a^2*b*C + 7*b^3*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(21*a^2*b*B + 5*b^3*B
+ 7*a^3*C + 15*a*b^2*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(21*a^2*b*B + 5*b^3*B + 7*a^3*C + 15*a*b^2*C)*
Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b*(27*a*b*B + 22*a^2*C + 7*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x]
)/(45*d) + (2*b^2*(9*b*B + 13*a*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*b*C*Cos[c + d*x]^(3/2)*(a + b*
Cos[c + d*x])^2*Sin[c + d*x])/(9*d)

Rule 3029

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] :> Dist[1/b^2, Int[(a + b*Sin[e + f*x])
^(m + 1)*(c + d*Sin[e + f*x])^n*(b*B - a*C + b*C*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m,
n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 0]

Rule 2990

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 - 2)*(c + d*Sin[e + f*x
])^n*Simp[a^2*A*d*(m + n + 1) + b*B*(b*c*(m - 1) + a*d*(n + 1)) + (a*d*(2*A*b + a*B)*(m + n + 1) - b*B*(a*c -
b*d*(m + n)))*Sin[e + f*x] + b*(A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x]^2, x], x], x] /; F
reeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m,
1] &&  !(IGtQ[n, 1] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

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 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 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]

Rubi steps

\begin{align*} \int \frac{(a+b \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx &=\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3 (B+C \cos (c+d x)) \, dx\\ &=\frac{2 b C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac{2}{9} \int \sqrt{\cos (c+d x)} (a+b \cos (c+d x)) \left (\frac{3}{2} a (3 a B+b C)+\frac{1}{2} \left (7 b^2 C+9 a (2 b B+a C)\right ) \cos (c+d x)+\frac{1}{2} b (9 b B+13 a C) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 b^2 (9 b B+13 a C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac{4}{63} \int \sqrt{\cos (c+d x)} \left (\frac{21}{4} a^2 (3 a B+b C)+\frac{9}{4} \left (21 a^2 b B+5 b^3 B+7 a^3 C+15 a b^2 C\right ) \cos (c+d x)+\frac{7}{4} b \left (27 a b B+22 a^2 C+7 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 b \left (27 a b B+22 a^2 C+7 b^2 C\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 b^2 (9 b B+13 a C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac{8}{315} \int \sqrt{\cos (c+d x)} \left (\frac{21}{8} \left (15 a^3 B+27 a b^2 B+27 a^2 b C+7 b^3 C\right )+\frac{45}{8} \left (21 a^2 b B+5 b^3 B+7 a^3 C+15 a b^2 C\right ) \cos (c+d x)\right ) \, dx\\ &=\frac{2 b \left (27 a b B+22 a^2 C+7 b^2 C\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 b^2 (9 b B+13 a C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac{1}{7} \left (21 a^2 b B+5 b^3 B+7 a^3 C+15 a b^2 C\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx+\frac{1}{15} \left (15 a^3 B+27 a b^2 B+27 a^2 b C+7 b^3 C\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 \left (15 a^3 B+27 a b^2 B+27 a^2 b C+7 b^3 C\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (21 a^2 b B+5 b^3 B+7 a^3 C+15 a b^2 C\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 b \left (27 a b B+22 a^2 C+7 b^2 C\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 b^2 (9 b B+13 a C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac{1}{21} \left (21 a^2 b B+5 b^3 B+7 a^3 C+15 a b^2 C\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (15 a^3 B+27 a b^2 B+27 a^2 b C+7 b^3 C\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (21 a^2 b B+5 b^3 B+7 a^3 C+15 a b^2 C\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 \left (21 a^2 b B+5 b^3 B+7 a^3 C+15 a b^2 C\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 b \left (27 a b B+22 a^2 C+7 b^2 C\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 b^2 (9 b B+13 a C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac{2 b C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d}\\ \end{align*}

Mathematica [A]  time = 1.15429, size = 197, normalized size = 0.77 \[ \frac{60 \left (21 a^2 b B+7 a^3 C+15 a b^2 C+5 b^3 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+84 \left (27 a^2 b C+15 a^3 B+27 a b^2 B+7 b^3 C\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\sin (c+d x) \sqrt{\cos (c+d x)} \left (7 b \left (108 a^2 C+108 a b B+43 b^2 C\right ) \cos (c+d x)+5 \left (252 a^2 b B+84 a^3 C+18 b^2 (3 a C+b B) \cos (2 (c+d x))+234 a b^2 C+78 b^3 B+7 b^3 C \cos (3 (c+d x))\right )\right )}{630 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(84*(15*a^3*B + 27*a*b^2*B + 27*a^2*b*C + 7*b^3*C)*EllipticE[(c + d*x)/2, 2] + 60*(21*a^2*b*B + 5*b^3*B + 7*a^
3*C + 15*a*b^2*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*b*(108*a*b*B + 108*a^2*C + 43*b^2*C)*Cos[c
+ d*x] + 5*(252*a^2*b*B + 78*b^3*B + 84*a^3*C + 234*a*b^2*C + 18*b^2*(b*B + 3*a*C)*Cos[2*(c + d*x)] + 7*b^3*C
*Cos[3*(c + d*x)]))*Sin[c + d*x])/(630*d)

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Maple [B]  time = 0.661, size = 745, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cos(d*x+c))^3*(B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(1/2),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*C*b^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2
*c)^10+(720*B*b^3+2160*C*a*b^2+2240*C*b^3)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-1512*B*a*b^2-1080*B*b^3-1
512*C*a^2*b-3240*C*a*b^2-2072*C*b^3)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(1260*B*a^2*b+1512*B*a*b^2+840*B*
b^3+420*C*a^3+1512*C*a^2*b+2520*C*a*b^2+952*C*b^3)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-630*B*a^2*b-378*B
*a*b^2-240*B*b^3-210*C*a^3-378*C*a^2*b-720*C*a*b^2-168*C*b^3)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+315*a^2*
b*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))+75*b^3
*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))-315*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))*a^3-567*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))*a*b^2+105*a
^3*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))+225*C
*a*b^2*(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))-567
*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*b-1
47*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))*b^3)/
(-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., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sqrt{\cos \left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^3*(B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c))*(b*cos(d*x + c) + a)^3/sqrt(cos(d*x + c)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C b^{3} \cos \left (d x + c\right )^{4} + B a^{3} +{\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (C a^{2} b + B a b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (C a^{3} + 3 \, B a^{2} b\right )} \cos \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((a+b*cos(d*x+c))^3*(B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

integral((C*b^3*cos(d*x + c)^4 + B*a^3 + (3*C*a*b^2 + B*b^3)*cos(d*x + c)^3 + 3*(C*a^2*b + B*a*b^2)*cos(d*x +
c)^2 + (C*a^3 + 3*B*a^2*b)*cos(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((a+b*cos(d*x+c))**3*(B*cos(d*x+c)+C*cos(d*x+c)**2)/cos(d*x+c)**(1/2),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} + B \cos \left (d x + c\right )\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sqrt{\cos \left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^3*(B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(1/2),x, algorithm="giac")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c))*(b*cos(d*x + c) + a)^3/sqrt(cos(d*x + c)), x)