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

Optimal. Leaf size=146 \[ \frac{208 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{832 a^3 \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}-\frac{4 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{63 d}+\frac{26 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d} \]

[Out]

(832*a^3*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (208*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315
*d) + (26*a*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*d) - (4*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63
*d) + (2*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*a*d)

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Rubi [A]  time = 0.160045, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2759, 2751, 2647, 2646} \[ \frac{208 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{832 a^3 \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}-\frac{4 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{63 d}+\frac{26 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(832*a^3*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (208*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315
*d) + (26*a*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*d) - (4*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63
*d) + (2*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*a*d)

Rule 2759

Int[sin[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(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*(b*(m + 1) - a*
Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] &&  !LtQ[m, -2^(-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 \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \, dx &=\frac{2 (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d}+\frac{2 \int \left (\frac{7 a}{2}-a \cos (c+d x)\right ) (a+a \cos (c+d x))^{5/2} \, dx}{9 a}\\ &=-\frac{4 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{63 d}+\frac{2 (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d}+\frac{13}{21} \int (a+a \cos (c+d x))^{5/2} \, dx\\ &=\frac{26 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 d}-\frac{4 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{63 d}+\frac{2 (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d}+\frac{1}{105} (104 a) \int (a+a \cos (c+d x))^{3/2} \, dx\\ &=\frac{208 a^2 \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac{26 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 d}-\frac{4 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{63 d}+\frac{2 (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d}+\frac{1}{315} \left (416 a^2\right ) \int \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{832 a^3 \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}+\frac{208 a^2 \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac{26 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 d}-\frac{4 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{63 d}+\frac{2 (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d}\\ \end{align*}

Mathematica [A]  time = 0.283529, size = 95, normalized size = 0.65 \[ \frac{a^2 \left (8190 \sin \left (\frac{1}{2} (c+d x)\right )+2100 \sin \left (\frac{3}{2} (c+d x)\right )+756 \sin \left (\frac{5}{2} (c+d x)\right )+225 \sin \left (\frac{7}{2} (c+d x)\right )+35 \sin \left (\frac{9}{2} (c+d x)\right )\right ) \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)}}{2520 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(8190*Sin[(c + d*x)/2] + 2100*Sin[(3*(c + d*x))/2] + 756*Sin[
(5*(c + d*x))/2] + 225*Sin[(7*(c + d*x))/2] + 35*Sin[(9*(c + d*x))/2]))/(2520*d)

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Maple [A]  time = 0.766, size = 99, normalized size = 0.7 \begin{align*}{\frac{8\,{a}^{3}\sqrt{2}}{315\,d}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 140\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}-20\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}+39\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+52\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+104 \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)^2*(a+cos(d*x+c)*a)^(5/2),x)

[Out]

8/315*cos(1/2*d*x+1/2*c)*a^3*sin(1/2*d*x+1/2*c)*(140*cos(1/2*d*x+1/2*c)^8-20*cos(1/2*d*x+1/2*c)^6+39*cos(1/2*d
*x+1/2*c)^4+52*cos(1/2*d*x+1/2*c)^2+104)*2^(1/2)/(cos(1/2*d*x+1/2*c)^2*a)^(1/2)/d

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Maxima [A]  time = 2.00699, size = 127, normalized size = 0.87 \begin{align*} \frac{{\left (35 \, \sqrt{2} a^{2} \sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ) + 225 \, \sqrt{2} a^{2} \sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 756 \, \sqrt{2} a^{2} \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 2100 \, \sqrt{2} a^{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 8190 \, \sqrt{2} a^{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} \sqrt{a}}{2520 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2520*(35*sqrt(2)*a^2*sin(9/2*d*x + 9/2*c) + 225*sqrt(2)*a^2*sin(7/2*d*x + 7/2*c) + 756*sqrt(2)*a^2*sin(5/2*d
*x + 5/2*c) + 2100*sqrt(2)*a^2*sin(3/2*d*x + 3/2*c) + 8190*sqrt(2)*a^2*sin(1/2*d*x + 1/2*c))*sqrt(a)/d

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Fricas [A]  time = 1.51386, size = 234, normalized size = 1.6 \begin{align*} \frac{2 \,{\left (35 \, a^{2} \cos \left (d x + c\right )^{4} + 130 \, a^{2} \cos \left (d x + c\right )^{3} + 219 \, a^{2} \cos \left (d x + c\right )^{2} + 292 \, a^{2} \cos \left (d x + c\right ) + 584 \, a^{2}\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{315 \,{\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)^2*(a+a*cos(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

2/315*(35*a^2*cos(d*x + c)^4 + 130*a^2*cos(d*x + c)^3 + 219*a^2*cos(d*x + c)^2 + 292*a^2*cos(d*x + c) + 584*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(cos(d*x+c)**2*(a+a*cos(d*x+c))**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cos \left (d x + c\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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