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

Optimal. Leaf size=132 \[ \frac{8 a^2 (35 A+19 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (35 A+19 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 a d}-\frac{4 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d} \]

[Out]

(8*a^2*(35*A + 19*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(35*A + 19*C)*Sqrt[a + a*Cos[c + d*
x]]*Sin[c + d*x])/(105*d) - (4*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*C*(a + a*Cos[c + d*x])^(
5/2)*Sin[c + d*x])/(7*a*d)

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Rubi [A]  time = 0.174563, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {3024, 2751, 2647, 2646} \[ \frac{8 a^2 (35 A+19 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (35 A+19 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 a d}-\frac{4 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*a^2*(35*A + 19*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(35*A + 19*C)*Sqrt[a + a*Cos[c + d*
x]]*Sin[c + d*x])/(105*d) - (4*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*C*(a + a*Cos[c + d*x])^(
5/2)*Sin[c + d*x])/(7*a*d)

Rule 3024

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (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) - a*C*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, 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 (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 a d}+\frac{2 \int (a+a \cos (c+d x))^{3/2} \left (\frac{1}{2} a (7 A+5 C)-a C \cos (c+d x)\right ) \, dx}{7 a}\\ &=-\frac{4 C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac{2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 a d}+\frac{1}{35} (35 A+19 C) \int (a+a \cos (c+d x))^{3/2} \, dx\\ &=\frac{2 a (35 A+19 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{105 d}-\frac{4 C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac{2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 a d}+\frac{1}{105} (4 a (35 A+19 C)) \int \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{8 a^2 (35 A+19 C) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a (35 A+19 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{105 d}-\frac{4 C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac{2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 a d}\\ \end{align*}

Mathematica [A]  time = 0.243845, size = 75, normalized size = 0.57 \[ \frac{a \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} ((140 A+253 C) \cos (c+d x)+700 A+78 C \cos (2 (c+d x))+15 C \cos (3 (c+d x))+494 C)}{210 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sqrt[a*(1 + Cos[c + d*x])]*(700*A + 494*C + (140*A + 253*C)*Cos[c + d*x] + 78*C*Cos[2*(c + d*x)] + 15*C*Cos
[3*(c + d*x)])*Tan[(c + d*x)/2])/(210*d)

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \text{hanged} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.90594, size = 146, normalized size = 1.11 \begin{align*} \frac{140 \,{\left (\sqrt{2} a \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 9 \, \sqrt{2} a \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} A \sqrt{a} +{\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 )} C \sqrt{a}}{420 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/420*(140*(sqrt(2)*a*sin(3/2*d*x + 3/2*c) + 9*sqrt(2)*a*sin(1/2*d*x + 1/2*c))*A*sqrt(a) + (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))*C*sqrt(a))/d

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Fricas [A]  time = 1.3654, size = 223, normalized size = 1.69 \begin{align*} \frac{2 \,{\left (15 \, C a \cos \left (d x + c\right )^{3} + 39 \, C a \cos \left (d x + c\right )^{2} +{\left (35 \, A + 52 \, C\right )} a \cos \left (d x + c\right ) +{\left (175 \, A + 104 \, C\right )} a\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{105 \,{\left (d \cos \left (d x + c\right ) + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*C*a*cos(d*x + c)^3 + 39*C*a*cos(d*x + c)^2 + (35*A + 52*C)*a*cos(d*x + c) + (175*A + 104*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((a+a*cos(d*x+c))**(3/2)*(A+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((a+a*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2),x, algorithm="giac")

[Out]

Timed out