3.96 \(\int \cos ^3(c+d x) \sqrt{a+a \cos (c+d x)} \, dx\)

Optimal. Leaf size=122 \[ \frac{2 a \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}+\frac{12 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 a d}-\frac{8 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{35 d}+\frac{4 a \sin (c+d x)}{5 d \sqrt{a \cos (c+d x)+a}} \]

[Out]

(4*a*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c +
d*x]]) - (8*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(35*d) + (12*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*a
*d)

________________________________________________________________________________________

Rubi [A]  time = 0.175556, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2770, 2759, 2751, 2646} \[ \frac{2 a \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}+\frac{12 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 a d}-\frac{8 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{35 d}+\frac{4 a \sin (c+d x)}{5 d \sqrt{a \cos (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]],x]

[Out]

(4*a*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c +
d*x]]) - (8*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(35*d) + (12*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*a
*d)

Rule 2770

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

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 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 ^3(c+d x) \sqrt{a+a \cos (c+d x)} \, dx &=\frac{2 a \cos ^3(c+d x) \sin (c+d x)}{7 d \sqrt{a+a \cos (c+d x)}}+\frac{6}{7} \int \cos ^2(c+d x) \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{2 a \cos ^3(c+d x) \sin (c+d x)}{7 d \sqrt{a+a \cos (c+d x)}}+\frac{12 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 a d}+\frac{12 \int \left (\frac{3 a}{2}-a \cos (c+d x)\right ) \sqrt{a+a \cos (c+d x)} \, dx}{35 a}\\ &=\frac{2 a \cos ^3(c+d x) \sin (c+d x)}{7 d \sqrt{a+a \cos (c+d x)}}-\frac{8 \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{35 d}+\frac{12 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 a d}+\frac{2}{5} \int \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{4 a \sin (c+d x)}{5 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a \cos ^3(c+d x) \sin (c+d x)}{7 d \sqrt{a+a \cos (c+d x)}}-\frac{8 \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{35 d}+\frac{12 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 a d}\\ \end{align*}

Mathematica [A]  time = 0.156356, size = 80, normalized size = 0.66 \[ \frac{\left (105 \sin \left (\frac{1}{2} (c+d x)\right )+35 \sin \left (\frac{3}{2} (c+d x)\right )+7 \sin \left (\frac{5}{2} (c+d x)\right )+5 \sin \left (\frac{7}{2} (c+d x)\right )\right ) \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)}}{140 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]],x]

[Out]

(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(105*Sin[(c + d*x)/2] + 35*Sin[(3*(c + d*x))/2] + 7*Sin[(5*(c + d
*x))/2] + 5*Sin[(7*(c + d*x))/2]))/(140*d)

________________________________________________________________________________________

Maple [A]  time = 0.694, size = 84, normalized size = 0.7 \begin{align*}{\frac{2\,a\sqrt{2}}{35\,d}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 40\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}-36\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+22\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+9 \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)^3*(a+cos(d*x+c)*a)^(1/2),x)

[Out]

2/35*cos(1/2*d*x+1/2*c)*a*sin(1/2*d*x+1/2*c)*(40*cos(1/2*d*x+1/2*c)^6-36*cos(1/2*d*x+1/2*c)^4+22*cos(1/2*d*x+1
/2*c)^2+9)*2^(1/2)/(cos(1/2*d*x+1/2*c)^2*a)^(1/2)/d

________________________________________________________________________________________

Maxima [A]  time = 1.9278, size = 88, normalized size = 0.72 \begin{align*} \frac{{\left (5 \, \sqrt{2} \sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 7 \, \sqrt{2} \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 35 \, \sqrt{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 105 \, \sqrt{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} \sqrt{a}}{140 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+a*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

1/140*(5*sqrt(2)*sin(7/2*d*x + 7/2*c) + 7*sqrt(2)*sin(5/2*d*x + 5/2*c) + 35*sqrt(2)*sin(3/2*d*x + 3/2*c) + 105
*sqrt(2)*sin(1/2*d*x + 1/2*c))*sqrt(a)/d

________________________________________________________________________________________

Fricas [A]  time = 1.55646, size = 169, normalized size = 1.39 \begin{align*} \frac{2 \,{\left (5 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right )^{2} + 8 \, \cos \left (d x + c\right ) + 16\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{35 \,{\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)^3*(a+a*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

2/35*(5*cos(d*x + c)^3 + 6*cos(d*x + c)^2 + 8*cos(d*x + c) + 16)*sqrt(a*cos(d*x + c) + a)*sin(d*x + c)/(d*cos(
d*x + c) + d)

________________________________________________________________________________________

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+a*cos(d*x+c))**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

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)^3*(a+a*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

Timed out