\(\int \frac {(c+d \sin (e+f x))^{5/2}}{a+a \sin (e+f x)} \, dx\) [512]

Optimal result

Integrand size = 27, antiderivative size = 246 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{a+a \sin (e+f x)} \, dx=\frac {(3 c-5 d) d \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 a f}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{f (a+a \sin (e+f x))}-\frac {\left (3 c^2-20 c d+9 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 a f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(3 c-5 d) \left (c^2-d^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 a f \sqrt {c+d \sin (e+f x)}} \] Output:

1/3*(3*c-5*d)*d*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/a/f-(c-d)*cos(f*x+e)*(c+ 
d*sin(f*x+e))^(3/2)/f/(a+a*sin(f*x+e))+1/3*(3*c^2-20*c*d+9*d^2)*EllipticE( 
cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(f*x+e))^(1/2)/ 
a/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+1/3*(3*c-5*d)*(c^2-d^2)*InverseJacobiAM 
(1/2*e-1/4*Pi+1/2*f*x,2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1 
/2)/a/f/(c+d*sin(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 1.17 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.21 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{a+a \sin (e+f x)} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \left (-3 (c-d)^2 (c+d \sin (e+f x))-2 d^2 \cos (e+f x) (c+d \sin (e+f x))+\frac {6 (c-d)^2 \sin \left (\frac {1}{2} (e+f x)\right ) (c+d \sin (e+f x))}{\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )}-d \left (15 c^2-12 c d+5 d^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+\left (3 c^2-20 c d+9 d^2\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right )}{3 a f (1+\sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \] Input:

Integrate[(c + d*Sin[e + f*x])^(5/2)/(a + a*Sin[e + f*x]),x]
 

Output:

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*(-3*(c - d)^2*(c + d*Sin[e + f*x] 
) - 2*d^2*Cos[e + f*x]*(c + d*Sin[e + f*x]) + (6*(c - d)^2*Sin[(e + f*x)/2 
]*(c + d*Sin[e + f*x]))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - d*(15*c^2 
- 12*c*d + 5*d^2)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c 
+ d*Sin[e + f*x])/(c + d)] + (3*c^2 - 20*c*d + 9*d^2)*((c + d)*EllipticE[( 
-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, 
(2*d)/(c + d)])*Sqrt[(c + d*Sin[e + f*x])/(c + d)]))/(3*a*f*(1 + Sin[e + f 
*x])*Sqrt[c + d*Sin[e + f*x]])
 

Rubi [A] (verified)

Time = 1.27 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 3246, 27, 3042, 3232, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}

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.

\(\displaystyle \int \frac {(c+d \sin (e+f x))^{5/2}}{a \sin (e+f x)+a} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(c+d \sin (e+f x))^{5/2}}{a \sin (e+f x)+a}dx\)

\(\Big \downarrow \) 3246

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3232

\(\displaystyle \frac {d \left (\frac {2}{3} \int \frac {a \left (15 c^2-12 d c+5 d^2\right )-a \left (3 c^2-20 d c+9 d^2\right ) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx+\frac {2 a (3 c-5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )}{2 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{f (a \sin (e+f x)+a)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \left (\frac {1}{3} \int \frac {a \left (15 c^2-12 d c+5 d^2\right )-a \left (3 c^2-20 d c+9 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx+\frac {2 a (3 c-5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )}{2 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{f (a \sin (e+f x)+a)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {d \left (\frac {1}{3} \int \frac {a \left (15 c^2-12 d c+5 d^2\right )-a \left (3 c^2-20 d c+9 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx+\frac {2 a (3 c-5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )}{2 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{f (a \sin (e+f x)+a)}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {d \left (\frac {1}{3} \left (\frac {a (3 c-5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a \left (3 c^2-20 c d+9 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}\right )+\frac {2 a (3 c-5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )}{2 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{f (a \sin (e+f x)+a)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {d \left (\frac {1}{3} \left (\frac {a (3 c-5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a \left (3 c^2-20 c d+9 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}\right )+\frac {2 a (3 c-5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )}{2 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{f (a \sin (e+f x)+a)}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {d \left (\frac {1}{3} \left (\frac {a (3 c-5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a \left (3 c^2-20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a (3 c-5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )}{2 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{f (a \sin (e+f x)+a)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {d \left (\frac {1}{3} \left (\frac {a (3 c-5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a \left (3 c^2-20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a (3 c-5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )}{2 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{f (a \sin (e+f x)+a)}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {d \left (\frac {1}{3} \left (\frac {a (3 c-5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {2 a \left (3 c^2-20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a (3 c-5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )}{2 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{f (a \sin (e+f x)+a)}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {d \left (\frac {1}{3} \left (\frac {a (3 c-5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 a \left (3 c^2-20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a (3 c-5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )}{2 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{f (a \sin (e+f x)+a)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {d \left (\frac {1}{3} \left (\frac {a (3 c-5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 a \left (3 c^2-20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a (3 c-5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )}{2 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{f (a \sin (e+f x)+a)}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {d \left (\frac {1}{3} \left (\frac {2 a (3 c-5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}-\frac {2 a \left (3 c^2-20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a (3 c-5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )}{2 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{f (a \sin (e+f x)+a)}\)

Input:

Int[(c + d*Sin[e + f*x])^(5/2)/(a + a*Sin[e + f*x]),x]
 

Output:

-(((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(f*(a + a*Sin[e + f*x] 
))) + (d*((2*a*(3*c - 5*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*f) + 
((-2*a*(3*c^2 - 20*c*d + 9*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d 
)]*Sqrt[c + d*Sin[e + f*x]])/(d*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2 
*a*(3*c - 5*d)*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sq 
rt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]]))/3))/(2*a 
^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 3042

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

rule 3132

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a 
 + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, 
b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3134

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + 
b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)]   Int[Sqrt[a/(a + b) + ( 
b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 
, 0] &&  !GtQ[a + b, 0]
 

rule 3140

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S 
qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ 
{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3142

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

rule 3231

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

rule 3232

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[1/(m + 1)   Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[b* 
d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ 
[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 
 0] && IntegerQ[2*m]
 

rule 3246

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1371\) vs. \(2(233)=466\).

Time = 2.80 (sec) , antiderivative size = 1372, normalized size of antiderivative = 5.58

method	result	size
default	\(\text {Expression too large to display}\)	\(1372\)

Input:

int((c+d*sin(f*x+e))^(5/2)/(a+sin(f*x+e)*a),x,method=_RETURNVERBOSE)
 

Output:

1/3*(cos(f*x+e)^2*sin(f*x+e)*d+cos(f*x+e)^2*c)^(1/2)*(3*(d/(c-d)*sin(f*x+e 
)+1/(c-d)*c)^(1/2)*(-d/(c+d)*sin(f*x+e)+d/(c+d))^(1/2)*(-d/(c-d)*sin(f*x+e 
)-d/(c-d))^(1/2)*EllipticE((d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2),((c-d)/(c+ 
d))^(1/2))*c^4-20*(d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2)*(-d/(c+d)*sin(f*x+e 
)+d/(c+d))^(1/2)*(-d/(c-d)*sin(f*x+e)-d/(c-d))^(1/2)*EllipticE((d/(c-d)*si 
n(f*x+e)+1/(c-d)*c)^(1/2),((c-d)/(c+d))^(1/2))*c^3*d+6*(d/(c-d)*sin(f*x+e) 
+1/(c-d)*c)^(1/2)*(-d/(c+d)*sin(f*x+e)+d/(c+d))^(1/2)*(-d/(c-d)*sin(f*x+e) 
-d/(c-d))^(1/2)*EllipticE((d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2),((c-d)/(c+d 
))^(1/2))*c^2*d^2+20*(d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2)*(-d/(c+d)*sin(f* 
x+e)+d/(c+d))^(1/2)*(-d/(c-d)*sin(f*x+e)-d/(c-d))^(1/2)*EllipticE((d/(c-d) 
*sin(f*x+e)+1/(c-d)*c)^(1/2),((c-d)/(c+d))^(1/2))*c*d^3-9*(d/(c-d)*sin(f*x 
+e)+1/(c-d)*c)^(1/2)*(-d/(c+d)*sin(f*x+e)+d/(c+d))^(1/2)*(-d/(c-d)*sin(f*x 
+e)-d/(c-d))^(1/2)*EllipticE((d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2),((c-d)/( 
c+d))^(1/2))*d^4+12*(d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2)*(-d/(c+d)*sin(f*x 
+e)+d/(c+d))^(1/2)*(-d/(c-d)*sin(f*x+e)-d/(c-d))^(1/2)*EllipticF((d/(c-d)* 
sin(f*x+e)+1/(c-d)*c)^(1/2),((c-d)/(c+d))^(1/2))*c^3*d-4*(d/(c-d)*sin(f*x+ 
e)+1/(c-d)*c)^(1/2)*(-d/(c+d)*sin(f*x+e)+d/(c+d))^(1/2)*(-d/(c-d)*sin(f*x+ 
e)-d/(c-d))^(1/2)*EllipticF((d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2),((c-d)/(c 
+d))^(1/2))*c^2*d^2-12*(d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2)*(-d/(c+d)*sin( 
f*x+e)+d/(c+d))^(1/2)*(-d/(c-d)*sin(f*x+e)-d/(c-d))^(1/2)*EllipticF((d/...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 776, normalized size of antiderivative = 3.15 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{a+a \sin (e+f x)} \, dx=\text {Too large to display} \] Input:

integrate((c+d*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e)),x, algorithm="fricas")
 

Output:

1/9*((6*c^3 + 5*c^2*d - 18*c*d^2 + 15*d^3 + (6*c^3 + 5*c^2*d - 18*c*d^2 + 
15*d^3)*cos(f*x + e) + (6*c^3 + 5*c^2*d - 18*c*d^2 + 15*d^3)*sin(f*x + e)) 
*sqrt(1/2*I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^ 
3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d) 
 + (6*c^3 + 5*c^2*d - 18*c*d^2 + 15*d^3 + (6*c^3 + 5*c^2*d - 18*c*d^2 + 15 
*d^3)*cos(f*x + e) + (6*c^3 + 5*c^2*d - 18*c*d^2 + 15*d^3)*sin(f*x + e))*s 
qrt(-1/2*I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^ 
3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d) 
 - 3*(-3*I*c^2*d + 20*I*c*d^2 - 9*I*d^3 + (-3*I*c^2*d + 20*I*c*d^2 - 9*I*d 
^3)*cos(f*x + e) + (-3*I*c^2*d + 20*I*c*d^2 - 9*I*d^3)*sin(f*x + e))*sqrt( 
1/2*I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c* 
d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9 
*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d)) - 3 
*(3*I*c^2*d - 20*I*c*d^2 + 9*I*d^3 + (3*I*c^2*d - 20*I*c*d^2 + 9*I*d^3)*co 
s(f*x + e) + (3*I*c^2*d - 20*I*c*d^2 + 9*I*d^3)*sin(f*x + e))*sqrt(-1/2*I* 
d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/ 
d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c 
*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d)) - 3*(2* 
d^3*cos(f*x + e)^2 + 3*c^2*d - 6*c*d^2 + 3*d^3 + (3*c^2*d - 6*c*d^2 + 5*d^ 
3)*cos(f*x + e) + (2*d^3*cos(f*x + e) - 3*c^2*d + 6*c*d^2 - 3*d^3)*sin(...
 

Sympy [F]

\[ \int \frac {(c+d \sin (e+f x))^{5/2}}{a+a \sin (e+f x)} \, dx=\frac {\int \frac {c^{2} \sqrt {c + d \sin {\left (e + f x \right )}}}{\sin {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}}{\sin {\left (e + f x \right )} + 1}\, dx + \int \frac {2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}}{\sin {\left (e + f x \right )} + 1}\, dx}{a} \] Input:

integrate((c+d*sin(f*x+e))**(5/2)/(a+a*sin(f*x+e)),x)
 

Output:

(Integral(c**2*sqrt(c + d*sin(e + f*x))/(sin(e + f*x) + 1), x) + Integral( 
d**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**2/(sin(e + f*x) + 1), x) + Int 
egral(2*c*d*sqrt(c + d*sin(e + f*x))*sin(e + f*x)/(sin(e + f*x) + 1), x))/ 
a
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {(c+d \sin (e+f x))^{5/2}}{a+a \sin (e+f x)} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{a \sin \left (f x + e\right ) + a} \,d x } \] Input:

integrate((c+d*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e)),x, algorithm="maxima")
 

Output:

integrate((d*sin(f*x + e) + c)^(5/2)/(a*sin(f*x + e) + a), x)
 

Giac [F]

\[ \int \frac {(c+d \sin (e+f x))^{5/2}}{a+a \sin (e+f x)} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{a \sin \left (f x + e\right ) + a} \,d x } \] Input:

integrate((c+d*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e)),x, algorithm="giac")
 

Output:

integrate((d*sin(f*x + e) + c)^(5/2)/(a*sin(f*x + e) + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{a+a \sin (e+f x)} \, dx=\int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}}{a+a\,\sin \left (e+f\,x\right )} \,d x \] Input:

int((c + d*sin(e + f*x))^(5/2)/(a + a*sin(e + f*x)),x)
 

Output:

int((c + d*sin(e + f*x))^(5/2)/(a + a*sin(e + f*x)), x)
 

Reduce [F]

\[ \int \frac {(c+d \sin (e+f x))^{5/2}}{a+a \sin (e+f x)} \, dx=\frac {\left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}}{\sin \left (f x +e \right )+1}d x \right ) c^{2}+\left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )+1}d x \right ) d^{2}+2 \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )+1}d x \right ) c d}{a} \] Input:

int((c+d*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e)),x)
 

Output:

(int(sqrt(sin(e + f*x)*d + c)/(sin(e + f*x) + 1),x)*c**2 + int((sqrt(sin(e 
 + f*x)*d + c)*sin(e + f*x)**2)/(sin(e + f*x) + 1),x)*d**2 + 2*int((sqrt(s 
in(e + f*x)*d + c)*sin(e + f*x))/(sin(e + f*x) + 1),x)*c*d)/a