\(\int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx\) [100]

Optimal result

Integrand size = 42, antiderivative size = 180 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {14 a^2 (g \cos (e+f x))^{5/2}}{15 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {14 a^2 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{5 f g \sqrt {c-c \sin (e+f x)}} \] Output:

-14/15*a^2*(g*cos(f*x+e))^(5/2)/f/g/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e) 
)^(1/2)+14/5*a^2*g*cos(f*x+e)^(1/2)*(g*cos(f*x+e))^(1/2)*EllipticE(sin(1/2 
*f*x+1/2*e),2^(1/2))/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)-2/5*a 
*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(1/2)/f/g/(c-c*sin(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 6.94 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.82 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {(g \cos (e+f x))^{3/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (a (1+\sin (e+f x)))^{3/2} \left (-42 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )+\sqrt {\cos (e+f x)} (20 \cos (e+f x)+3 \sin (2 (e+f x)))\right )}{15 f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \sqrt {c-c \sin (e+f x)}} \] Input:

Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2))/Sqrt[c - c*S 
in[e + f*x]],x]
 

Output:

-1/15*((g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 
+ Sin[e + f*x]))^(3/2)*(-42*EllipticE[(e + f*x)/2, 2] + Sqrt[Cos[e + f*x]] 
*(20*Cos[e + f*x] + 3*Sin[2*(e + f*x)])))/(f*Cos[e + f*x]^(3/2)*(Cos[(e + 
f*x)/2] + Sin[(e + f*x)/2])^3*Sqrt[c - c*Sin[e + f*x]])
 

Rubi [A] (verified)

Time = 1.39 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 3330, 3042, 3330, 3042, 3321, 3042, 3121, 3042, 3119}

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.

\(\displaystyle \int \frac {(a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}}dx\)

\(\Big \downarrow \) 3330

\(\displaystyle \frac {7}{5} a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {7}{5} a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\)

\(\Big \downarrow \) 3330

\(\displaystyle \frac {7}{5} a \left (a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {7}{5} a \left (a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\)

\(\Big \downarrow \) 3321

\(\displaystyle \frac {7}{5} a \left (\frac {a g \cos (e+f x) \int \sqrt {g \cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {7}{5} a \left (\frac {a g \cos (e+f x) \int \sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\)

\(\Big \downarrow \) 3121

\(\displaystyle \frac {7}{5} a \left (\frac {a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {7}{5} a \left (\frac {a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {7}{5} a \left (\frac {2 a g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\)

Input:

Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2))/Sqrt[c - c*Sin[e + 
 f*x]],x]
 

Output:

(-2*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(5*f*g*Sqrt[c - c*S 
in[e + f*x]]) + (7*a*((-2*a*(g*Cos[e + f*x])^(5/2))/(3*f*g*Sqrt[a + a*Sin[ 
e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*a*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos 
[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - 
 c*Sin[e + f*x]])))/5
 

Defintions of rubi rules used

rule 3042

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

rule 3119

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* 
(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
 

rule 3121

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) 
^n/Sin[c + d*x]^n   Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt 
Q[-1, n, 1] && IntegerQ[2*n]
 

rule 3321

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_ 
.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[g* 
(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]))   Int[(g 
*Cos[e + f*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ 
[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
 

rule 3330

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(- 
b)*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f* 
x])^n/(f*g*(m + n + p))), x] + Simp[a*((2*m + p - 1)/(m + n + p))   Int[(g* 
Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] 
 /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - 
 b^2, 0] && GtQ[m, 0] && NeQ[m + n + p, 0] &&  !LtQ[0, n, m] && IntegersQ[2 
*m, 2*n, 2*p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(760\) vs. \(2(156)=312\).

Time = 10.24 (sec) , antiderivative size = 761, normalized size of antiderivative = 4.23

method	result	size
default	\(\text {Expression too large to display}\)	\(761\)

Input:

int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(1/2),x,m 
ethod=_RETURNVERBOSE)
 

Output:

1/15*g*a/f/(1+2^(1/2))/(2*2^(1/2)-3)*((6*(4*cos(1/2*f*x+1/2*e)^4+4*cos(1/2 
*f*x+1/2*e)^3-2*cos(1/2*f*x+1/2*e)^2-2*cos(1/2*f*x+1/2*e)-7)*(-1+2*cos(1/2 
*f*x+1/2*e)^2)*sin(1/2*f*x+1/2*e)+20*(2*cos(1/2*f*x+1/2*e)^3+2*cos(1/2*f*x 
+1/2*e)^2-cos(1/2*f*x+1/2*e)-1)*(-1+2*cos(1/2*f*x+1/2*e)^2))*2^(1/2)-6*(4* 
cos(1/2*f*x+1/2*e)^4+4*cos(1/2*f*x+1/2*e)^3-2*cos(1/2*f*x+1/2*e)^2-2*cos(1 
/2*f*x+1/2*e)-7)*(-1+2*cos(1/2*f*x+1/2*e)^2)*sin(1/2*f*x+1/2*e)-20*(2*cos( 
1/2*f*x+1/2*e)^3+2*cos(1/2*f*x+1/2*e)^2-cos(1/2*f*x+1/2*e)-1)*(-1+2*cos(1/ 
2*f*x+1/2*e)^2)+(-21*cos(1/2*f*x+1/2*e)^2-42*cos(1/2*f*x+1/2*e)-21)*2^(1/2 
)*EllipticE((1+2^(1/2))*(csc(1/2*f*x+1/2*e)-cot(1/2*f*x+1/2*e)),-2*2^(1/2) 
+3)*((2^(1/2)*cos(1/2*f*x+1/2*e)-2^(1/2)+2*cos(1/2*f*x+1/2*e)-1)/(cos(1/2* 
f*x+1/2*e)+1))^(1/2)*(-2*(2^(1/2)*cos(1/2*f*x+1/2*e)-2^(1/2)-2*cos(1/2*f*x 
+1/2*e)+1)/(cos(1/2*f*x+1/2*e)+1))^(1/2)+((-42*cos(1/2*f*x+1/2*e)^2-84*cos 
(1/2*f*x+1/2*e)-42)*2^(1/2)+84*cos(1/2*f*x+1/2*e)^2+168*cos(1/2*f*x+1/2*e) 
+84)*(-2*(2^(1/2)*cos(1/2*f*x+1/2*e)-2^(1/2)-2*cos(1/2*f*x+1/2*e)+1)/(cos( 
1/2*f*x+1/2*e)+1))^(1/2)*((2^(1/2)*cos(1/2*f*x+1/2*e)-2^(1/2)+2*cos(1/2*f* 
x+1/2*e)-1)/(cos(1/2*f*x+1/2*e)+1))^(1/2)*EllipticF((1+2^(1/2))*(csc(1/2*f 
*x+1/2*e)-cot(1/2*f*x+1/2*e)),-2*2^(1/2)+3))*((2*cos(1/2*f*x+1/2*e)*sin(1/ 
2*f*x+1/2*e)+1)*a)^(1/2)*(g*(-1+2*cos(1/2*f*x+1/2*e)^2))^(1/2)/(2*cos(1/2* 
f*x+1/2*e)*(cos(1/2*f*x+1/2*e)+1)*sin(1/2*f*x+1/2*e)+cos(1/2*f*x+1/2*e)+1) 
/(-(2*cos(1/2*f*x+1/2*e)*sin(1/2*f*x+1/2*e)-1)*c)^(1/2)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.71 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {2 \, {\left (21 i \, \sqrt {\frac {1}{2}} \sqrt {a c g} a g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - 21 i \, \sqrt {\frac {1}{2}} \sqrt {a c g} a g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + {\left (3 \, a g \sin \left (f x + e\right ) + 10 \, a g\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}\right )}}{15 \, c f} \] Input:

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

Output:

-2/15*(21*I*sqrt(1/2)*sqrt(a*c*g)*a*g*weierstrassZeta(-4, 0, weierstrassPI 
nverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) - 21*I*sqrt(1/2)*sqrt(a*c*g) 
*a*g*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*si 
n(f*x + e))) + (3*a*g*sin(f*x + e) + 10*a*g)*sqrt(g*cos(f*x + e))*sqrt(a*s 
in(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c))/(c*f)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=\text {Timed out} \] Input:

integrate((g*cos(f*x+e))**(3/2)*(a+a*sin(f*x+e))**(3/2)/(c-c*sin(f*x+e))** 
(1/2),x)
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

integrate((g*cos(f*x + e))^(3/2)*(a*sin(f*x + e) + a)^(3/2)/sqrt(-c*sin(f* 
x + e) + c), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Mupad [F(-1)]

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

int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(3/2))/(c - c*sin(e + f*x 
))^(1/2),x)
 

Output:

int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(3/2))/(c - c*sin(e + f*x 
))^(1/2), x)
 

Reduce [F]

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

int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(1/2),x)
 

Output:

( - sqrt(g)*sqrt(c)*sqrt(a)*a*g*(int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e 
 + f*x) + 1)*sqrt(cos(e + f*x))*cos(e + f*x)*sin(e + f*x))/(sin(e + f*x) - 
 1),x) + int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e 
+ f*x))*cos(e + f*x))/(sin(e + f*x) - 1),x)))/c