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

Optimal. Leaf size=116 \[ \frac{64 a^3 \sin (c+d x)}{21 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{21 d}+\frac{2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}+\frac{2 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d} \]

[Out]

(64*a^3*Sin[c + d*x])/(21*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(21*d)
+ (2*a*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d) + (2*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)

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Rubi [A]  time = 0.0869676, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2751, 2647, 2646} \[ \frac{64 a^3 \sin (c+d x)}{21 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{21 d}+\frac{2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}+\frac{2 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(64*a^3*Sin[c + d*x])/(21*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(21*d)
+ (2*a*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d) + (2*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)

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} \, dx &=\frac{2 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{5}{7} \int (a+a \cos (c+d x))^{5/2} \, dx\\ &=\frac{2 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac{2 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{1}{7} (8 a) \int (a+a \cos (c+d x))^{3/2} \, dx\\ &=\frac{16 a^2 \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac{2 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{1}{21} \left (32 a^2\right ) \int \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{64 a^3 \sin (c+d x)}{21 d \sqrt{a+a \cos (c+d x)}}+\frac{16 a^2 \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac{2 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}\\ \end{align*}

Mathematica [A]  time = 0.225013, size = 84, normalized size = 0.72 \[ \frac{a^2 \left (315 \sin \left (\frac{1}{2} (c+d x)\right )+77 \sin \left (\frac{3}{2} (c+d x)\right )+3 \left (7 \sin \left (\frac{5}{2} (c+d x)\right )+\sin \left (\frac{7}{2} (c+d x)\right )\right )\right ) \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)}}{84 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(315*Sin[(c + d*x)/2] + 77*Sin[(3*(c + d*x))/2] + 3*(7*Sin[(5
*(c + d*x))/2] + Sin[(7*(c + d*x))/2])))/(84*d)

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Maple [A]  time = 0.694, size = 86, normalized size = 0.7 \begin{align*}{\frac{8\,{a}^{3}\sqrt{2}}{21\,d}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 6\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}+3\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+4\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+8 \right ){\frac{1}{\sqrt{ \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/21*cos(1/2*d*x+1/2*c)*a^3*sin(1/2*d*x+1/2*c)*(6*cos(1/2*d*x+1/2*c)^6+3*cos(1/2*d*x+1/2*c)^4+4*cos(1/2*d*x+1/
2*c)^2+8)*2^(1/2)/(cos(1/2*d*x+1/2*c)^2*a)^(1/2)/d

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Maxima [A]  time = 1.94901, size = 104, normalized size = 0.9 \begin{align*} \frac{{\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 )} \sqrt{a}}{84 \, 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),x, algorithm="maxima")

[Out]

1/84*(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))*sqrt(a)/d

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Fricas [A]  time = 1.56016, size = 193, normalized size = 1.66 \begin{align*} \frac{2 \,{\left (3 \, a^{2} \cos \left (d x + c\right )^{3} + 12 \, a^{2} \cos \left (d x + c\right )^{2} + 23 \, a^{2} \cos \left (d x + c\right ) + 46 \, a^{2}\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{21 \,{\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),x, algorithm="fricas")

[Out]

2/21*(3*a^2*cos(d*x + c)^3 + 12*a^2*cos(d*x + c)^2 + 23*a^2*cos(d*x + c) + 46*a^2)*sqrt(a*cos(d*x + c) + a)*si
n(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),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(cos(d*x+c)*(a+a*cos(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Timed out