\(\int \sin (c+d x) (a+a \sin (c+d x))^{2/3} \, dx\) [94]

Optimal result

Integrand size = 21, antiderivative size = 96 \[ \int \sin (c+d x) (a+a \sin (c+d x))^{2/3} \, dx=-\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{5 d}-\frac {4 \sqrt [6]{2} \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (a+a \sin (c+d x))^{2/3}}{5 d (1+\sin (c+d x))^{7/6}} \] Output:

-3/5*cos(d*x+c)*(a+a*sin(d*x+c))^(2/3)/d-4/5*cos(d*x+c)*hypergeom([-1/6, 1 
/2],[3/2],1/2-1/2*sin(d*x+c))*(a+a*sin(d*x+c))^(2/3)*2^(1/6)/d/(1+sin(d*x+ 
c))^(7/6)
 

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.44 \[ \int \sin (c+d x) (a+a \sin (c+d x))^{2/3} \, dx=-\frac {3 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a (1+\sin (c+d x)))^{2/3} \left (-\sqrt {2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\sin ^2\left (\frac {1}{4} (2 c+\pi +2 d x)\right )\right )+\sqrt {1-\sin (c+d x)} (2+\sin (c+d x))\right )}{5 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {1-\sin (c+d x)}} \] Input:

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

Output:

(-3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(a*(1 + Sin[c + d*x]))^(2/3)*(-( 
Sqrt[2]*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[(2*c + Pi + 2*d*x)/4]^2]) + S 
qrt[1 - Sin[c + d*x]]*(2 + Sin[c + d*x])))/(5*d*(Cos[(c + d*x)/2] + Sin[(c 
 + d*x)/2])*Sqrt[1 - Sin[c + d*x]])
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 3230, 3042, 3131, 3042, 3130}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \sin (c+d x) (a \sin (c+d x)+a)^{2/3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sin (c+d x) (a \sin (c+d x)+a)^{2/3}dx\)

\(\Big \downarrow \) 3230

\(\displaystyle \frac {2}{5} \int (\sin (c+d x) a+a)^{2/3}dx-\frac {3 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{5 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2}{5} \int (\sin (c+d x) a+a)^{2/3}dx-\frac {3 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{5 d}\)

\(\Big \downarrow \) 3131

\(\displaystyle \frac {2 (a \sin (c+d x)+a)^{2/3} \int (\sin (c+d x)+1)^{2/3}dx}{5 (\sin (c+d x)+1)^{2/3}}-\frac {3 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{5 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 (a \sin (c+d x)+a)^{2/3} \int (\sin (c+d x)+1)^{2/3}dx}{5 (\sin (c+d x)+1)^{2/3}}-\frac {3 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{5 d}\)

\(\Big \downarrow \) 3130

\(\displaystyle -\frac {4 \sqrt [6]{2} \cos (c+d x) (a \sin (c+d x)+a)^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{5 d (\sin (c+d x)+1)^{7/6}}-\frac {3 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{5 d}\)

Input:

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

Output:

(-3*Cos[c + d*x]*(a + a*Sin[c + d*x])^(2/3))/(5*d) - (4*2^(1/6)*Cos[c + d* 
x]*Hypergeometric2F1[-1/6, 1/2, 3/2, (1 - Sin[c + d*x])/2]*(a + a*Sin[c + 
d*x])^(2/3))/(5*d*(1 + Sin[c + d*x])^(7/6))
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3130

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-2^(n + 
 1/2))*a^(n - 1/2)*b*(Cos[c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]]))*Hypergeome 
tric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - b*(Sin[c + d*x]/a))], x] /; FreeQ[{a, 
 b, c, d, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] && GtQ[a, 0]
 

rule 3131

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[a^IntPar 
t[n]*((a + b*Sin[c + d*x])^FracPart[n]/(1 + (b/a)*Sin[c + d*x])^FracPart[n] 
)   Int[(1 + (b/a)*Sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && 
EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] &&  !GtQ[a, 0]
 

rule 3230

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] + Simp[(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)]
 

Maple [F]

\[\int \sin \left (d x +c \right ) \left (a +a \sin \left (d x +c \right )\right )^{\frac {2}{3}}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((a*sin(d*x + c) + a)^(2/3)*sin(d*x + c), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

a**(2/3)*int((sin(c + d*x) + 1)**(2/3)*sin(c + d*x),x)