\(\int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \, dx\) [106]

Optimal result

Integrand size = 21, antiderivative size = 86 \[ \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \, dx=\frac {8 a^2 \sin (c+d x)}{5 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{5 d}+\frac {2 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d} \]

[Out]

2/5*(a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/d+8/5*a^2*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+2/5*a*sin(d*x+c)*(a+a*cos(
d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2830, 2726, 2725} \[ \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \, dx=\frac {8 a^2 \sin (c+d x)}{5 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{5 d}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d} \]

[In]

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

[Out]

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

Rule 2725

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]

Rule 2726

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] &&
EqQ[a^2 - b^2, 0] && IGtQ[n - 1/2, 0]

Rule 2830

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*S
in[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)]

Rubi steps \begin{align*} \text {integral}& = \frac {2 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {3}{5} \int (a+a \cos (c+d x))^{3/2} \, dx \\ & = \frac {2 a \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{5 d}+\frac {2 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {1}{5} (4 a) \int \sqrt {a+a \cos (c+d x)} \, dx \\ & = \frac {8 a^2 \sin (c+d x)}{5 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{5 d}+\frac {2 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.78 \[ \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \, dx=\frac {a \sqrt {a (1+\cos (c+d x))} \sec \left (\frac {1}{2} (c+d x)\right ) \left (20 \sin \left (\frac {1}{2} (c+d x)\right )+5 \sin \left (\frac {3}{2} (c+d x)\right )+\sin \left (\frac {5}{2} (c+d x)\right )\right )}{10 d} \]

[In]

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

[Out]

(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(20*Sin[(c + d*x)/2] + 5*Sin[(3*(c + d*x))/2] + Sin[(5*(c + d*x
))/2]))/(10*d)

Maple [A] (verified)

Time = 0.97 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.83

method	result	size
default	\(\frac {4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (2 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+2\right ) \sqrt {2}}{5 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\)	\(71\)

[In]

int(cos(d*x+c)*(a+cos(d*x+c)*a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

4/5*cos(1/2*d*x+1/2*c)*a^2*sin(1/2*d*x+1/2*c)*(2*cos(1/2*d*x+1/2*c)^4+cos(1/2*d*x+1/2*c)^2+2)*2^(1/2)/(a*cos(1
/2*d*x+1/2*c)^2)^(1/2)/d

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.64 \[ \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \, dx=\frac {2 \, {\left (a \cos \left (d x + c\right )^{2} + 3 \, a \cos \left (d x + c\right ) + 6 \, a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{5 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \, dx=\int \left (a \left (\cos {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}} \cos {\left (c + d x \right )}\, dx \]

[In]

integrate(cos(d*x+c)*(a+a*cos(d*x+c))**(3/2),x)

[Out]

Integral((a*(cos(c + d*x) + 1))**(3/2)*cos(c + d*x), x)

Maxima [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.62 \[ \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \, dx=\frac {{\left (\sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 5 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 20 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{10 \, d} \]

[In]

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

[Out]

1/10*(sqrt(2)*a*sin(5/2*d*x + 5/2*c) + 5*sqrt(2)*a*sin(3/2*d*x + 3/2*c) + 20*sqrt(2)*a*sin(1/2*d*x + 1/2*c))*s
qrt(a)/d

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.90 \[ \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \, dx=\frac {\sqrt {2} {\left (a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 5 \, a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 20 \, a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{10 \, d} \]

[In]

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

[Out]

1/10*sqrt(2)*(a*sgn(cos(1/2*d*x + 1/2*c))*sin(5/2*d*x + 5/2*c) + 5*a*sgn(cos(1/2*d*x + 1/2*c))*sin(3/2*d*x + 3
/2*c) + 20*a*sgn(cos(1/2*d*x + 1/2*c))*sin(1/2*d*x + 1/2*c))*sqrt(a)/d

Mupad [F(-1)]

Timed out. \[ \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \, dx=\int \cos \left (c+d\,x\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2} \,d x \]

[In]

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

[Out]

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