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

Optimal. Leaf size=89 \[ \frac{64 a^3 \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d} \]

[Out]

(64*a^3*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d)
+ (2*a*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)

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Rubi [A]  time = 0.0491412, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2647, 2646} \[ \frac{64 a^3 \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cos[c + d*x])^(5/2),x]

[Out]

(64*a^3*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d)
+ (2*a*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)

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))^{5/2} \, dx &=\frac{2 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac{1}{5} (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)}{15 d}+\frac{2 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac{1}{15} \left (32 a^2\right ) \int \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{64 a^3 \sin (c+d x)}{15 d \sqrt{a+a \cos (c+d x)}}+\frac{16 a^2 \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}\\ \end{align*}

Mathematica [A]  time = 0.116834, size = 71, normalized size = 0.8 \[ \frac{a^2 \left (150 \sin \left (\frac{1}{2} (c+d x)\right )+25 \sin \left (\frac{3}{2} (c+d x)\right )+3 \sin \left (\frac{5}{2} (c+d x)\right )\right ) \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)}}{30 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(150*Sin[(c + d*x)/2] + 25*Sin[(3*(c + d*x))/2] + 3*Sin[(5*(c
+ d*x))/2]))/(30*d)

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Maple [A]  time = 0.711, size = 73, normalized size = 0.8 \begin{align*}{\frac{8\,{a}^{3}\sqrt{2}}{15\,d}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 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((a+cos(d*x+c)*a)^(5/2),x)

[Out]

8/15*a^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)*(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 = 2.10232, size = 81, normalized size = 0.91 \begin{align*} \frac{{\left (3 \, \sqrt{2} a^{2} \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 25 \, \sqrt{2} a^{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 150 \, \sqrt{2} a^{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} \sqrt{a}}{30 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30*(3*sqrt(2)*a^2*sin(5/2*d*x + 5/2*c) + 25*sqrt(2)*a^2*sin(3/2*d*x + 3/2*c) + 150*sqrt(2)*a^2*sin(1/2*d*x +
1/2*c))*sqrt(a)/d

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Fricas [A]  time = 1.56616, size = 161, normalized size = 1.81 \begin{align*} \frac{2 \,{\left (3 \, a^{2} \cos \left (d x + c\right )^{2} + 14 \, a^{2} \cos \left (d x + c\right ) + 43 \, a^{2}\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{15 \,{\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))^(5/2),x, algorithm="fricas")

[Out]

2/15*(3*a^2*cos(d*x + c)^2 + 14*a^2*cos(d*x + c) + 43*a^2)*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))**(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((a+a*cos(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Timed out