\(\int \frac {(c+d \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^2} \, dx\) [518]

Optimal result

Integrand size = 27, antiderivative size = 256 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^2} \, dx=-\frac {(c-d) (c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 a^2 f (1+\sin (e+f x))}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3 f (a+a \sin (e+f x))^2}-\frac {\left (c^2+5 c d-12 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 a^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(c+5 d) \left (c^2-d^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 a^2 f \sqrt {c+d \sin (e+f x)}} \] Output:

-1/3*(c-d)*(c+5*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/a^2/f/(1+sin(f*x+e))- 
1/3*(c-d)*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/f/(a+a*sin(f*x+e))^2+1/3*(c^2+ 
5*c*d-12*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^2/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+1/3*(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^2/f/(c+d*sin(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 2.34 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.21 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^2} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 \left (-\left (\left (c^2+5 c d-6 d^2\right ) (c+d \sin (e+f x))\right )+\frac {(c-d) \left (7 d \cos \left (\frac {1}{2} (e+f x)\right )-(c+6 d) \cos \left (\frac {3}{2} (e+f x)\right )+(3 c+11 d) \sin \left (\frac {1}{2} (e+f x)\right )\right ) (c+d \sin (e+f x))}{\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}+d^2 (-11 c+5 d) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+\left (c^2+5 c d-12 d^2\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right )}{3 a^2 f (1+\sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}} \] Input:

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

Output:

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(-((c^2 + 5*c*d - 6*d^2)*(c + d*S 
in[e + f*x])) + ((c - d)*(7*d*Cos[(e + f*x)/2] - (c + 6*d)*Cos[(3*(e + f*x 
))/2] + (3*c + 11*d)*Sin[(e + f*x)/2])*(c + d*Sin[e + f*x]))/(Cos[(e + f*x 
)/2] + Sin[(e + f*x)/2])^3 + d^2*(-11*c + 5*d)*EllipticF[(-2*e + Pi - 2*f* 
x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + (c^2 + 5*c*d - 1 
2*d^2)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*Ellipt 
icF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)])*Sqrt[(c + d*Sin[e + f*x])/(c + 
d)]))/(3*a^2*f*(1 + Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]])
 

Rubi [A] (verified)

Time = 1.42 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.02, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {3042, 3244, 27, 3042, 3456, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3244

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (a \left (2 c^2+7 d c-3 d^2\right )-a (c-7 d) d \sin (e+f x)\right )}{\sin (e+f x) a+a}dx}{6 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3 f (a \sin (e+f x)+a)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (a \left (2 c^2+7 d c-3 d^2\right )-a (c-7 d) d \sin (e+f x)\right )}{\sin (e+f x) a+a}dx}{6 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3 f (a \sin (e+f x)+a)^2}\)

\(\Big \downarrow \) 3456

\(\displaystyle \frac {\frac {\int \frac {a^2 (11 c-5 d) d^2-a^2 d \left (c^2+5 d c-12 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{a^2}-\frac {2 (c-d) (c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (\sin (e+f x)+1)}}{6 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3 f (a \sin (e+f x)+a)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {a^2 (11 c-5 d) d^2-a^2 d \left (c^2+5 d c-12 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{a^2}-\frac {2 (c-d) (c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (\sin (e+f x)+1)}}{6 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3 f (a \sin (e+f x)+a)^2}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {\frac {a^2 (c+5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx-a^2 \left (c^2+5 c d-12 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{a^2}-\frac {2 (c-d) (c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (\sin (e+f x)+1)}}{6 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3 f (a \sin (e+f x)+a)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {a^2 (c+5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx-a^2 \left (c^2+5 c d-12 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{a^2}-\frac {2 (c-d) (c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (\sin (e+f x)+1)}}{6 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3 f (a \sin (e+f x)+a)^2}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {\frac {a^2 (c+5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx-\frac {a^2 \left (c^2+5 c d-12 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{a^2}-\frac {2 (c-d) (c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (\sin (e+f x)+1)}}{6 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3 f (a \sin (e+f x)+a)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {a^2 (c+5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx-\frac {a^2 \left (c^2+5 c d-12 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{a^2}-\frac {2 (c-d) (c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (\sin (e+f x)+1)}}{6 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3 f (a \sin (e+f x)+a)^2}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {\frac {a^2 (c+5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx-\frac {2 a^2 \left (c^2+5 c d-12 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{a^2}-\frac {2 (c-d) (c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (\sin (e+f x)+1)}}{6 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3 f (a \sin (e+f x)+a)^2}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {\frac {\frac {a^2 (c+5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}-\frac {2 a^2 \left (c^2+5 c d-12 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{a^2}-\frac {2 (c-d) (c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (\sin (e+f x)+1)}}{6 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3 f (a \sin (e+f x)+a)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {a^2 (c+5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}-\frac {2 a^2 \left (c^2+5 c d-12 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{a^2}-\frac {2 (c-d) (c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (\sin (e+f x)+1)}}{6 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3 f (a \sin (e+f x)+a)^2}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {\frac {\frac {2 a^2 (c+5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}-\frac {2 a^2 \left (c^2+5 c d-12 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{a^2}-\frac {2 (c-d) (c+5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (\sin (e+f x)+1)}}{6 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3 f (a \sin (e+f x)+a)^2}\)

Input:

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

Output:

-1/3*((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(f*(a + a*Sin[e + f 
*x])^2) + ((-2*(c - d)*(c + 5*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f 
*(1 + Sin[e + f*x])) + ((-2*a^2*(c^2 + 5*c*d - 12*d^2)*EllipticE[(e - Pi/2 
 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(f*Sqrt[(c + d*Sin[e + 
 f*x])/(c + d)]) + (2*a^2*(c + 5*d)*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x) 
/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(f*Sqrt[c + d*Sin[e 
 + f*x]]))/a^2)/(6*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 3244

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

rule 3456

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + 
(f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim 
p[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/( 
a*f*(2*m + 1))), x] - Simp[1/(a*b*(2*m + 1))   Int[(a + b*Sin[e + f*x])^(m 
+ 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + 
 b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] & 
& NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (In 
tegerQ[2*n] || EqQ[c, 0])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1364\) vs. \(2(241)=482\).

Time = 22.64 (sec) , antiderivative size = 1365, normalized size of antiderivative = 5.33

method	result	size
default	\(\text {Expression too large to display}\)	\(1365\)

Input:

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

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 1066, normalized size of antiderivative = 4.16 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/9*((4*c^3 + 20*c^2*d + 18*c*d^2 - 30*d^3 - (2*c^3 + 10*c^2*d + 9*c*d^2 
- 15*d^3)*cos(f*x + e)^2 + (2*c^3 + 10*c^2*d + 9*c*d^2 - 15*d^3)*cos(f*x + 
 e) + (4*c^3 + 20*c^2*d + 18*c*d^2 - 30*d^3 + (2*c^3 + 10*c^2*d + 9*c*d^2 
- 15*d^3)*cos(f*x + e))*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) + (4*c^3 + 20*c^2*d + 18*c*d^2 - 30* 
d^3 - (2*c^3 + 10*c^2*d + 9*c*d^2 - 15*d^3)*cos(f*x + e)^2 + (2*c^3 + 10*c 
^2*d + 9*c*d^2 - 15*d^3)*cos(f*x + e) + (4*c^3 + 20*c^2*d + 18*c*d^2 - 30* 
d^3 + (2*c^3 + 10*c^2*d + 9*c*d^2 - 15*d^3)*cos(f*x + e))*sin(f*x + e))*sq 
rt(-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*(-2*I*c^2*d - 10*I*c*d^2 + 24*I*d^3 + (I*c^2*d + 5*I*c*d^2 - 12*I*d^3) 
*cos(f*x + e)^2 + (-I*c^2*d - 5*I*c*d^2 + 12*I*d^3)*cos(f*x + e) + (-2*I*c 
^2*d - 10*I*c*d^2 + 24*I*d^3 + (-I*c^2*d - 5*I*c*d^2 + 12*I*d^3)*cos(f*x + 
 e))*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*I*c^2*d + 10*I*c*d^2 - 24*I*d^3 + (-I*c^2*d - 
5*I*c*d^2 + 12*I*d^3)*cos(f*x + e)^2 + (I*c^2*d + 5*I*c*d^2 - 12*I*d^3)*co 
s(f*x + e) + (2*I*c^2*d + 10*I*c*d^2 - 24*I*d^3 + (I*c^2*d + 5*I*c*d^2 ...
 

Sympy [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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