\(\int \frac {(e \cos (c+d x))^{7/3}}{\sqrt {a+a \sin (c+d x)}} \, dx\) [321]

Optimal result

Integrand size = 27, antiderivative size = 77 \[ \int \frac {(e \cos (c+d x))^{7/3}}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {3 \sqrt [6]{2} (e \cos (c+d x))^{10/3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {5}{3},\frac {8}{3},\frac {1}{2} (1-\sin (c+d x))\right )}{5 d e (1+\sin (c+d x))^{7/6} \sqrt {a+a \sin (c+d x)}} \] Output:

-3/5*2^(1/6)*(e*cos(d*x+c))^(10/3)*hypergeom([-1/6, 5/3],[8/3],1/2-1/2*sin 
(d*x+c))/d/e/(1+sin(d*x+c))^(7/6)/(a+a*sin(d*x+c))^(1/2)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00 \[ \int \frac {(e \cos (c+d x))^{7/3}}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {3 \sqrt [6]{2} (e \cos (c+d x))^{10/3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {5}{3},\frac {8}{3},\frac {1}{2} (1-\sin (c+d x))\right )}{5 d e (1+\sin (c+d x))^{7/6} \sqrt {a (1+\sin (c+d x))}} \] Input:

Integrate[(e*Cos[c + d*x])^(7/3)/Sqrt[a + a*Sin[c + d*x]],x]
 

Output:

(-3*2^(1/6)*(e*Cos[c + d*x])^(10/3)*Hypergeometric2F1[-1/6, 5/3, 8/3, (1 - 
 Sin[c + d*x])/2])/(5*d*e*(1 + Sin[c + d*x])^(7/6)*Sqrt[a*(1 + Sin[c + d*x 
])])
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3042, 3168, 80, 27, 79}

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.

\(\displaystyle \int \frac {(e \cos (c+d x))^{7/3}}{\sqrt {a \sin (c+d x)+a}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(e \cos (c+d x))^{7/3}}{\sqrt {a \sin (c+d x)+a}}dx\)

\(\Big \downarrow \) 3168

\(\displaystyle \frac {a^2 (e \cos (c+d x))^{10/3} \int (a-a \sin (c+d x))^{2/3} \sqrt [6]{\sin (c+d x) a+a}d\sin (c+d x)}{d e (a-a \sin (c+d x))^{5/3} (a \sin (c+d x)+a)^{5/3}}\)

\(\Big \downarrow \) 80

\(\displaystyle \frac {\sqrt [6]{2} a^2 (e \cos (c+d x))^{10/3} \int \frac {\sqrt [6]{\sin (c+d x)+1} (a-a \sin (c+d x))^{2/3}}{\sqrt [6]{2}}d\sin (c+d x)}{d e \sqrt [6]{\sin (c+d x)+1} (a-a \sin (c+d x))^{5/3} (a \sin (c+d x)+a)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a^2 (e \cos (c+d x))^{10/3} \int \sqrt [6]{\sin (c+d x)+1} (a-a \sin (c+d x))^{2/3}d\sin (c+d x)}{d e \sqrt [6]{\sin (c+d x)+1} (a-a \sin (c+d x))^{5/3} (a \sin (c+d x)+a)^{3/2}}\)

\(\Big \downarrow \) 79

\(\displaystyle -\frac {3 \sqrt [6]{2} a (e \cos (c+d x))^{10/3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {5}{3},\frac {8}{3},\frac {1}{2} (1-\sin (c+d x))\right )}{5 d e \sqrt [6]{\sin (c+d x)+1} (a \sin (c+d x)+a)^{3/2}}\)

Input:

Int[(e*Cos[c + d*x])^(7/3)/Sqrt[a + a*Sin[c + d*x]],x]
 

Output:

(-3*2^(1/6)*a*(e*Cos[c + d*x])^(10/3)*Hypergeometric2F1[-1/6, 5/3, 8/3, (1 
 - Sin[c + d*x])/2])/(5*d*e*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^ 
(3/2))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 80

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c 
 + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) 
^FracPart[n])   Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) 
), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integ 
erQ[n] && (RationalQ[m] ||  !SimplerQ[n + 1, m + 1])
 

rule 3042

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

rule 3168

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_.), x_Symbol] :> Simp[a^2*((g*Cos[e + f*x])^(p + 1)/(f*g*(a + b*Sin 
[e + f*x])^((p + 1)/2)*(a - b*Sin[e + f*x])^((p + 1)/2)))   Subst[Int[(a + 
b*x)^(m + (p - 1)/2)*(a - b*x)^((p - 1)/2), x], x, Sin[e + f*x]], x] /; Fre 
eQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[m]
 

Maple [F]

\[\int \frac {\left (e \cos \left (d x +c \right )\right )^{\frac {7}{3}}}{\sqrt {a +a \sin \left (d x +c \right )}}d x\]

Input:

int((e*cos(d*x+c))^(7/3)/(a+a*sin(d*x+c))^(1/2),x)
 

Output:

int((e*cos(d*x+c))^(7/3)/(a+a*sin(d*x+c))^(1/2),x)
 

Fricas [F]

\[ \int \frac {(e \cos (c+d x))^{7/3}}{\sqrt {a+a \sin (c+d x)}} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {7}{3}}}{\sqrt {a \sin \left (d x + c\right ) + a}} \,d x } \] Input:

integrate((e*cos(d*x+c))^(7/3)/(a+a*sin(d*x+c))^(1/2),x, algorithm="fricas 
")
 

Output:

integral((e*cos(d*x + c))^(1/3)*e^2*cos(d*x + c)^2/sqrt(a*sin(d*x + c) + a 
), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e \cos (c+d x))^{7/3}}{\sqrt {a+a \sin (c+d x)}} \, dx=\text {Timed out} \] Input:

integrate((e*cos(d*x+c))**(7/3)/(a+a*sin(d*x+c))**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(e \cos (c+d x))^{7/3}}{\sqrt {a+a \sin (c+d x)}} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {7}{3}}}{\sqrt {a \sin \left (d x + c\right ) + a}} \,d x } \] Input:

integrate((e*cos(d*x+c))^(7/3)/(a+a*sin(d*x+c))^(1/2),x, algorithm="maxima 
")
 

Output:

integrate((e*cos(d*x + c))^(7/3)/sqrt(a*sin(d*x + c) + a), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {(e \cos (c+d x))^{7/3}}{\sqrt {a+a \sin (c+d x)}} \, dx=\text {Timed out} \] Input:

integrate((e*cos(d*x+c))^(7/3)/(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e \cos (c+d x))^{7/3}}{\sqrt {a+a \sin (c+d x)}} \, dx=\int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/3}}{\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \] Input:

int((e*cos(c + d*x))^(7/3)/(a + a*sin(c + d*x))^(1/2),x)
 

Output:

int((e*cos(c + d*x))^(7/3)/(a + a*sin(c + d*x))^(1/2), x)
 

Reduce [F]

\[ \int \frac {(e \cos (c+d x))^{7/3}}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {e^{\frac {7}{3}} \sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{\frac {7}{3}}}{\sin \left (d x +c \right )+1}d x \right )}{a} \] Input:

int((e*cos(d*x+c))^(7/3)/(a+a*sin(d*x+c))^(1/2),x)
 

Output:

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