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

Optimal. Leaf size=473 \[ -\frac{2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (8 a^2 C+7 A b^2-b^2 C\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{7 b^2 d \left (a^2-b^2\right )}-\frac{2 a \left (48 a^2 C+35 A b^2-13 b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{35 b^3 d \left (a^2-b^2\right )}+\frac{2 \left (2 a^2 b^2 (70 A-31 C)+192 a^4 C-5 b^4 (7 A+5 C)\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b^4 d \left (a^2-b^2\right )}+\frac{2 \left (4 a^2 b^2 (70 A+29 C)+384 a^4 C+5 b^4 (7 A+5 C)\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{105 b^5 d \sqrt{a+b \cos (c+d x)}}-\frac{2 a \left (4 a^2 b^2 (70 A-43 C)+384 a^4 C-b^4 (175 A+107 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{105 b^5 d \left (a^2-b^2\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}} \]

[Out]

(-2*a*(4*a^2*b^2*(70*A - 43*C) + 384*a^4*C - b^4*(175*A + 107*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)
/2, (2*b)/(a + b)])/(105*b^5*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(384*a^4*C + 5*b^4*(7*A +
5*C) + 4*a^2*b^2*(70*A + 29*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(105
*b^5*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 + a^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b
*Cos[c + d*x]]) + (2*(2*a^2*b^2*(70*A - 31*C) + 192*a^4*C - 5*b^4*(7*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c
+ d*x])/(105*b^4*(a^2 - b^2)*d) - (2*a*(35*A*b^2 + 48*a^2*C - 13*b^2*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*
Sin[c + d*x])/(35*b^3*(a^2 - b^2)*d) + (2*(7*A*b^2 + 8*a^2*C - b^2*C)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*
Sin[c + d*x])/(7*b^2*(a^2 - b^2)*d)

________________________________________________________________________________________

Rubi [A]  time = 1.12987, antiderivative size = 473, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {3048, 3049, 3023, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (8 a^2 C+7 A b^2-b^2 C\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{7 b^2 d \left (a^2-b^2\right )}-\frac{2 a \left (48 a^2 C+35 A b^2-13 b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{35 b^3 d \left (a^2-b^2\right )}+\frac{2 \left (2 a^2 b^2 (70 A-31 C)+192 a^4 C-5 b^4 (7 A+5 C)\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b^4 d \left (a^2-b^2\right )}+\frac{2 \left (4 a^2 b^2 (70 A+29 C)+384 a^4 C+5 b^4 (7 A+5 C)\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{105 b^5 d \sqrt{a+b \cos (c+d x)}}-\frac{2 a \left (4 a^2 b^2 (70 A-43 C)+384 a^4 C-b^4 (175 A+107 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{105 b^5 d \left (a^2-b^2\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*(4*a^2*b^2*(70*A - 43*C) + 384*a^4*C - b^4*(175*A + 107*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)
/2, (2*b)/(a + b)])/(105*b^5*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(384*a^4*C + 5*b^4*(7*A +
5*C) + 4*a^2*b^2*(70*A + 29*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(105
*b^5*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 + a^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b
*Cos[c + d*x]]) + (2*(2*a^2*b^2*(70*A - 31*C) + 192*a^4*C - 5*b^4*(7*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c
+ d*x])/(105*b^4*(a^2 - b^2)*d) - (2*a*(35*A*b^2 + 48*a^2*C - 13*b^2*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*
Sin[c + d*x])/(35*b^3*(a^2 - b^2)*d) + (2*(7*A*b^2 + 8*a^2*C - b^2*C)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*
Sin[c + d*x])/(7*b^2*(a^2 - b^2)*d)

Rule 3048

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

Rule 3049

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/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x]
)^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2)
- C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2
, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 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 2752

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

Rule 2663

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2655

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

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rubi steps

\begin{align*} \int \frac{\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx &=-\frac{2 \left (A b^2+a^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt{a+b \cos (c+d x)}}-\frac{2 \int \frac{\cos ^2(c+d x) \left (3 \left (A b^2+a^2 C\right )-\frac{1}{2} a b (A+C) \cos (c+d x)-\frac{1}{2} \left (7 A b^2+8 a^2 C-b^2 C\right ) \cos ^2(c+d x)\right )}{\sqrt{a+b \cos (c+d x)}} \, dx}{b \left (a^2-b^2\right )}\\ &=-\frac{2 \left (A b^2+a^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (7 A b^2+8 a^2 C-b^2 C\right ) \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{7 b^2 \left (a^2-b^2\right ) d}-\frac{4 \int \frac{\cos (c+d x) \left (-a \left (7 A b^2+\left (8 a^2-b^2\right ) C\right )+\frac{1}{4} b \left (7 A b^2+2 a^2 C+5 b^2 C\right ) \cos (c+d x)+\frac{1}{4} a \left (35 A b^2+48 a^2 C-13 b^2 C\right ) \cos ^2(c+d x)\right )}{\sqrt{a+b \cos (c+d x)}} \, dx}{7 b^2 \left (a^2-b^2\right )}\\ &=-\frac{2 \left (A b^2+a^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt{a+b \cos (c+d x)}}-\frac{2 a \left (35 A b^2+48 a^2 C-13 b^2 C\right ) \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{35 b^3 \left (a^2-b^2\right ) d}+\frac{2 \left (7 A b^2+8 a^2 C-b^2 C\right ) \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{7 b^2 \left (a^2-b^2\right ) d}-\frac{8 \int \frac{\frac{1}{4} a^2 \left (35 A b^2+48 a^2 C-13 b^2 C\right )-\frac{1}{8} a b \left (35 A b^2+16 a^2 C+19 b^2 C\right ) \cos (c+d x)-\frac{1}{8} \left (2 a^2 b^2 (70 A-31 C)+192 a^4 C-5 b^4 (7 A+5 C)\right ) \cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{35 b^3 \left (a^2-b^2\right )}\\ &=-\frac{2 \left (A b^2+a^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (2 a^2 b^2 (70 A-31 C)+192 a^4 C-5 b^4 (7 A+5 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{105 b^4 \left (a^2-b^2\right ) d}-\frac{2 a \left (35 A b^2+48 a^2 C-13 b^2 C\right ) \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{35 b^3 \left (a^2-b^2\right ) d}+\frac{2 \left (7 A b^2+8 a^2 C-b^2 C\right ) \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{7 b^2 \left (a^2-b^2\right ) d}-\frac{16 \int \frac{\frac{1}{16} b \left (2 a^2 b^2 (35 A-8 C)+96 a^4 C+5 b^4 (7 A+5 C)\right )+\frac{1}{16} a \left (4 a^2 b^2 (70 A-43 C)+384 a^4 C-b^4 (175 A+107 C)\right ) \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{105 b^4 \left (a^2-b^2\right )}\\ &=-\frac{2 \left (A b^2+a^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (2 a^2 b^2 (70 A-31 C)+192 a^4 C-5 b^4 (7 A+5 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{105 b^4 \left (a^2-b^2\right ) d}-\frac{2 a \left (35 A b^2+48 a^2 C-13 b^2 C\right ) \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{35 b^3 \left (a^2-b^2\right ) d}+\frac{2 \left (7 A b^2+8 a^2 C-b^2 C\right ) \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{7 b^2 \left (a^2-b^2\right ) d}+\frac{\left (384 a^4 C+5 b^4 (7 A+5 C)+4 a^2 b^2 (70 A+29 C)\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx}{105 b^5}-\frac{\left (a \left (4 a^2 b^2 (70 A-43 C)+384 a^4 C-b^4 (175 A+107 C)\right )\right ) \int \sqrt{a+b \cos (c+d x)} \, dx}{105 b^5 \left (a^2-b^2\right )}\\ &=-\frac{2 \left (A b^2+a^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (2 a^2 b^2 (70 A-31 C)+192 a^4 C-5 b^4 (7 A+5 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{105 b^4 \left (a^2-b^2\right ) d}-\frac{2 a \left (35 A b^2+48 a^2 C-13 b^2 C\right ) \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{35 b^3 \left (a^2-b^2\right ) d}+\frac{2 \left (7 A b^2+8 a^2 C-b^2 C\right ) \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{7 b^2 \left (a^2-b^2\right ) d}-\frac{\left (a \left (4 a^2 b^2 (70 A-43 C)+384 a^4 C-b^4 (175 A+107 C)\right ) \sqrt{a+b \cos (c+d x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \cos (c+d x)}{a+b}} \, dx}{105 b^5 \left (a^2-b^2\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left (\left (384 a^4 C+5 b^4 (7 A+5 C)+4 a^2 b^2 (70 A+29 C)\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}\right ) \int \frac{1}{\sqrt{\frac{a}{a+b}+\frac{b \cos (c+d x)}{a+b}}} \, dx}{105 b^5 \sqrt{a+b \cos (c+d x)}}\\ &=-\frac{2 a \left (4 a^2 b^2 (70 A-43 C)+384 a^4 C-b^4 (175 A+107 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{105 b^5 \left (a^2-b^2\right ) d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \left (384 a^4 C+5 b^4 (7 A+5 C)+4 a^2 b^2 (70 A+29 C)\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{105 b^5 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (A b^2+a^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (2 a^2 b^2 (70 A-31 C)+192 a^4 C-5 b^4 (7 A+5 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{105 b^4 \left (a^2-b^2\right ) d}-\frac{2 a \left (35 A b^2+48 a^2 C-13 b^2 C\right ) \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{35 b^3 \left (a^2-b^2\right ) d}+\frac{2 \left (7 A b^2+8 a^2 C-b^2 C\right ) \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{7 b^2 \left (a^2-b^2\right ) d}\\ \end{align*}

Mathematica [A]  time = 1.6127, size = 358, normalized size = 0.76 \[ \frac{b (a-b) (a+b) \left (420 a^3 \left (a^2 C+A b^2\right ) \sin (c+d x)+\left (a^2-b^2\right ) \left (348 a^2 C+140 A b^2+115 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))-78 a b C \left (a^2-b^2\right ) \sin (2 (c+d x)) (a+b \cos (c+d x))+15 b^2 C \left (a^2-b^2\right ) \sin (3 (c+d x)) (a+b \cos (c+d x))\right )-4 \left (a^2-b^2\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left (b \left (2 a^2 b^3 (35 A-8 C)+96 a^4 b C+5 b^5 (7 A+5 C)\right ) F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )+a \left (4 a^2 b^2 (70 A-43 C)+384 a^4 C-b^4 (175 A+107 C)\right ) \left ((a+b) E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )-a F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )\right )\right )}{210 b^5 d (a-b) (a+b) \left (a^2-b^2\right ) \sqrt{a+b \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b*(2*a^2*b^3*(35*A - 8*C) + 96*a^4*b*C + 5*b^5*(7*A + 5*C)
)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + a*(4*a^2*b^2*(70*A - 43*C) + 384*a^4*C - b^4*(175*A + 107*C))*((a +
b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + (a - b)*b*(a + b)*(420*
a^3*(A*b^2 + a^2*C)*Sin[c + d*x] + (a^2 - b^2)*(140*A*b^2 + 348*a^2*C + 115*b^2*C)*(a + b*Cos[c + d*x])*Sin[c
+ d*x] - 78*a*b*(a^2 - b^2)*C*(a + b*Cos[c + d*x])*Sin[2*(c + d*x)] + 15*b^2*(a^2 - b^2)*C*(a + b*Cos[c + d*x]
)*Sin[3*(c + d*x)]))/(210*(a - b)*b^5*(a + b)*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])

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Maple [B]  time = 1.828, size = 1788, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-(-2*cos(1/2*d*x+1/2*c)^2*b-a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(32*C/b*(-1/14/b*cos(1/2*d*x+1/2*c)^5*(-2*b*sin
(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)-1/140/b^2*(-6*a+18*b)*cos(1/2*d*x+1/2*c)^3*(-2*b*sin(1/2*d
*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)-1/420*(12*a^2-47*a*b+83*b^2)/b^3*cos(1/2*d*x+1/2*c)*(-2*b*sin(1/
2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)+1/420*(12*a^2-47*a*b+83*b^2)/b^3*(a-b)*(sin(1/2*d*x+1/2*c)^2)
^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/
2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-1/210*(-6*a^3+28*a^2*b-58*a*b^2+84*b^3)/b^4*(a-b)*(sin(1/2
*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x
+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(
1/2))))-16*C/b^2*(a+4*b)*(-1/10/b*cos(1/2*d*x+1/2*c)^3*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^
(1/2)-1/60/b^2*(-4*a+12*b)*cos(1/2*d*x+1/2*c)*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)+1/6
0/b^2*(-4*a+12*b)*(a-b)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/
2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-1/60*(4*a^2-
15*a*b+27*b^2)/b^3*(a-b)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1
/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-EllipticE(
cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))))+8/b^3*(A*b^2+C*a^2+3*C*a*b+6*C*b^2)*(-1/6/b*cos(1/2*d*x+1/2*c)*(-2*b*
sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)+1/6*(a-b)/b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*
x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*
d*x+1/2*c),(-2*b/(a-b))^(1/2))-1/12/b^2*(-2*a+6*b)*(a-b)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2
*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c
),(-2*b/(a-b))^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))))+2/b^5*(A*a*b^2+2*A*b^3+C*a^3+2*C*a^2*
b+3*C*a*b^2+4*C*b^3)*(a-b)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin
(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-Elliptic
E(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2)))+2*(A*a^2*b^2+A*a*b^3+A*b^4+C*a^4+C*a^3*b+C*a^2*b^2+C*a*b^3+C*b^4)/b^
5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*s
in(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-2*a^3*(A*b^2+C*a^2)/b^5/sin(1/2*d*
x+1/2*c)^2/(-2*sin(1/2*d*x+1/2*c)^2*b+a+b)/(a^2-b^2)*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1
/2)*((sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/
2*c),(-2*b/(a-b))^(1/2))*a-(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*El
lipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b+2*b*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2))/sin(1/2*d*x+1/2
*c)/(-2*sin(1/2*d*x+1/2*c)^2*b+a+b)^(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 )} \cos \left (d x + c\right )^{3}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C \cos \left (d x + c\right )^{5} + A \cos \left (d x + c\right )^{3}\right )} \sqrt{b \cos \left (d x + c\right ) + a}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((C*cos(d*x + c)^5 + A*cos(d*x + c)^3)*sqrt(b*cos(d*x + c) + a)/(b^2*cos(d*x + c)^2 + 2*a*b*cos(d*x +
c) + a^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(cos(d*x+c)**3*(A+C*cos(d*x+c)**2)/(a+b*cos(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 )} \cos \left (d x + c\right )^{3}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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