\(\int \frac {(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{9/2}} \, dx\) [594]

Optimal result

Integrand size = 29, antiderivative size = 254 \[ \int \frac {(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{9/2}} \, dx=\frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{7 d (c+d) f (c+d \sin (e+f x))^{7/2}}+\frac {6 a^3 (c-d) (c+5 d) \cos (e+f x)}{35 d^2 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {2 a^3 \left (3 c^2+22 c d+115 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^3 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {4 a^3 \left (3 c^2+22 c d+115 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^4 f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \] Output:

2/7*a^2*(c-d)*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)/d/(c+d)/f/(c+d*sin(f*x+e)) 
^(7/2)+6/35*a^3*(c-d)*(c+5*d)*cos(f*x+e)/d^2/(c+d)^2/f/(a+a*sin(f*x+e))^(1 
/2)/(c+d*sin(f*x+e))^(5/2)-2/105*a^3*(3*c^2+22*c*d+115*d^2)*cos(f*x+e)/d^2 
/(c+d)^3/f/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(3/2)-4/105*a^3*(3*c^2+ 
22*c*d+115*d^2)*cos(f*x+e)/d^2/(c+d)^4/f/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f 
*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 6.48 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.85 \[ \int \frac {(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{9/2}} \, dx=\frac {a^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} \left (-623 c^3-495 c^2 d-977 c d^2-145 d^3+\left (21 c^3+157 c^2 d+827 c d^2+115 d^3\right ) \cos (2 (e+f x))-\left (196 c^3+1865 c^2 d+694 c d^2+465 d^3\right ) \sin (e+f x)+3 c^2 d \sin (3 (e+f x))+22 c d^2 \sin (3 (e+f x))+115 d^3 \sin (3 (e+f x))\right )}{105 (c+d)^4 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c+d \sin (e+f x))^{7/2}} \] Input:

Integrate[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(9/2),x]
 

Output:

(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(-62 
3*c^3 - 495*c^2*d - 977*c*d^2 - 145*d^3 + (21*c^3 + 157*c^2*d + 827*c*d^2 
+ 115*d^3)*Cos[2*(e + f*x)] - (196*c^3 + 1865*c^2*d + 694*c*d^2 + 465*d^3) 
*Sin[e + f*x] + 3*c^2*d*Sin[3*(e + f*x)] + 22*c*d^2*Sin[3*(e + f*x)] + 115 
*d^3*Sin[3*(e + f*x)]))/(105*(c + d)^4*f*(Cos[(e + f*x)/2] + Sin[(e + f*x) 
/2])*(c + d*Sin[e + f*x])^(7/2))
 

Rubi [A] (verified)

Time = 1.16 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {3042, 3241, 27, 3042, 3459, 3042, 3251, 3042, 3250}

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.

\(\displaystyle \int \frac {(a \sin (e+f x)+a)^{5/2}}{(c+d \sin (e+f x))^{9/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a \sin (e+f x)+a)^{5/2}}{(c+d \sin (e+f x))^{9/2}}dx\)

\(\Big \downarrow \) 3241

\(\displaystyle \frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{7 d f (c+d) (c+d \sin (e+f x))^{7/2}}-\frac {2 a \int \frac {\sqrt {\sin (e+f x) a+a} (a (c-15 d)-a (3 c+11 d) \sin (e+f x))}{2 (c+d \sin (e+f x))^{7/2}}dx}{7 d (c+d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{7 d f (c+d) (c+d \sin (e+f x))^{7/2}}-\frac {a \int \frac {\sqrt {\sin (e+f x) a+a} (a (c-15 d)-a (3 c+11 d) \sin (e+f x))}{(c+d \sin (e+f x))^{7/2}}dx}{7 d (c+d)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{7 d f (c+d) (c+d \sin (e+f x))^{7/2}}-\frac {a \int \frac {\sqrt {\sin (e+f x) a+a} (a (c-15 d)-a (3 c+11 d) \sin (e+f x))}{(c+d \sin (e+f x))^{7/2}}dx}{7 d (c+d)}\)

\(\Big \downarrow \) 3459

\(\displaystyle \frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{7 d f (c+d) (c+d \sin (e+f x))^{7/2}}-\frac {a \left (-\frac {a \left (3 c^2+22 c d+115 d^2\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{5/2}}dx}{5 d (c+d)}-\frac {6 a^2 (c-d) (c+5 d) \cos (e+f x)}{5 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}\right )}{7 d (c+d)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{7 d f (c+d) (c+d \sin (e+f x))^{7/2}}-\frac {a \left (-\frac {a \left (3 c^2+22 c d+115 d^2\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{5/2}}dx}{5 d (c+d)}-\frac {6 a^2 (c-d) (c+5 d) \cos (e+f x)}{5 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}\right )}{7 d (c+d)}\)

\(\Big \downarrow \) 3251

\(\displaystyle \frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{7 d f (c+d) (c+d \sin (e+f x))^{7/2}}-\frac {a \left (-\frac {a \left (3 c^2+22 c d+115 d^2\right ) \left (\frac {2 \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{3/2}}dx}{3 (c+d)}-\frac {2 a \cos (e+f x)}{3 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}\right )}{5 d (c+d)}-\frac {6 a^2 (c-d) (c+5 d) \cos (e+f x)}{5 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}\right )}{7 d (c+d)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{7 d f (c+d) (c+d \sin (e+f x))^{7/2}}-\frac {a \left (-\frac {a \left (3 c^2+22 c d+115 d^2\right ) \left (\frac {2 \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{3/2}}dx}{3 (c+d)}-\frac {2 a \cos (e+f x)}{3 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}\right )}{5 d (c+d)}-\frac {6 a^2 (c-d) (c+5 d) \cos (e+f x)}{5 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}\right )}{7 d (c+d)}\)

\(\Big \downarrow \) 3250

\(\displaystyle \frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{7 d f (c+d) (c+d \sin (e+f x))^{7/2}}-\frac {a \left (-\frac {6 a^2 (c-d) (c+5 d) \cos (e+f x)}{5 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}-\frac {a \left (3 c^2+22 c d+115 d^2\right ) \left (-\frac {4 a \cos (e+f x)}{3 f (c+d)^2 \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}-\frac {2 a \cos (e+f x)}{3 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}\right )}{5 d (c+d)}\right )}{7 d (c+d)}\)

Input:

Int[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(9/2),x]
 

Output:

(2*a^2*(c - d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(7*d*(c + d)*f*(c + 
d*Sin[e + f*x])^(7/2)) - (a*((-6*a^2*(c - d)*(c + 5*d)*Cos[e + f*x])/(5*d* 
(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2)) - (a*(3*c^2 
 + 22*c*d + 115*d^2)*((-2*a*Cos[e + f*x])/(3*(c + d)*f*Sqrt[a + a*Sin[e + 
f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (4*a*Cos[e + f*x])/(3*(c + d)^2*f*Sqrt 
[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])))/(5*d*(c + d))))/(7*d*(c + 
 d))
 

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 3241

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*(b*c - a*d)*Cos[e + f*x]*(a + b 
*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a* 
d))), x] + Simp[b^2/(d*(n + 1)*(b*c + a*d))   Int[(a + b*Sin[e + f*x])^(m - 
 2)*(c + d*Sin[e + f*x])^(n + 1)*Simp[a*c*(m - 2) - b*d*(m - 2*n - 4) - (b* 
c*(m - 1) - a*d*(m + 2*n + 1))*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[m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || 
 (IntegerQ[m] && EqQ[c, 0]))
 

rule 3250

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + ( 
f_.)*(x_)])^(3/2), x_Symbol] :> Simp[-2*b^2*(Cos[e + f*x]/(f*(b*c + a*d)*Sq 
rt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*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]
 

rule 3251

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + ( 
f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*((c + d*Sin[e 
+ f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]])), x] + Sim 
p[(2*n + 3)*((b*c - a*d)/(2*b*(n + 1)*(c^2 - d^2)))   Int[Sqrt[a + b*Sin[e 
+ f*x]]*(c + d*Sin[e + f*x])^(n + 1), 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[n, 
-1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]
 

rule 3459

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + ( 
f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp 
[(-b^2)*(B*c - A*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1) 
*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[(A*b*d*(2*n + 3) - B*(b* 
c - 2*a*d*(n + 1)))/(2*d*(n + 1)*(b*c + a*d))   Int[Sqrt[a + b*Sin[e + f*x] 
]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{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[n, 
-1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(716\) vs. \(2(230)=460\).

Time = 33.08 (sec) , antiderivative size = 717, normalized size of antiderivative = 2.82

method	result	size
default	\(\frac {2 \left (\left (\left (-84 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-112 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+301\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+84 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-280 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-105 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c^{4}+\left (\left (216 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-1236 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-1438 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+169\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+216 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}+588 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-3262 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+2289 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c^{3} d +\left (\left (96 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+1416 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-8532 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+6054 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+75\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-96 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}+1800 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}+3708 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-6378 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+891 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c^{2} d^{2}+\left (\left (704 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+7048 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-9468 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+1446 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+15\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-704 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}+9864 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}-15900 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+6470 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+255 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c \,d^{3}+\cos \left (\frac {f x}{2}+\frac {e}{2}\right ) \left (\left (3680 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-3680 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-4600 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+2760 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+680 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+240 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+30\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-30 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) d^{4}\right ) \sqrt {\left (2 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) a}\, a^{2}}{105 f \left (c^{4}+4 c^{3} d +6 c^{2} d^{2}+4 c \,d^{3}+d^{4}\right ) \left (\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+\sin \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \left (c +2 d \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{\frac {9}{2}}}\)	\(717\)

Input:

int((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(9/2),x,method=_RETURNVERBOSE)
 

Output:

2/105/f/(c^4+4*c^3*d+6*c^2*d^2+4*c*d^3+d^4)*(((-84*cos(1/2*f*x+1/2*e)^4-11 
2*cos(1/2*f*x+1/2*e)^2+301)*sin(1/2*f*x+1/2*e)+84*cos(1/2*f*x+1/2*e)^5-280 
*cos(1/2*f*x+1/2*e)^3-105*cos(1/2*f*x+1/2*e))*c^4+((216*cos(1/2*f*x+1/2*e) 
^6-1236*cos(1/2*f*x+1/2*e)^4-1438*cos(1/2*f*x+1/2*e)^2+169)*sin(1/2*f*x+1/ 
2*e)+216*cos(1/2*f*x+1/2*e)^7+588*cos(1/2*f*x+1/2*e)^5-3262*cos(1/2*f*x+1/ 
2*e)^3+2289*cos(1/2*f*x+1/2*e))*c^3*d+((96*cos(1/2*f*x+1/2*e)^8+1416*cos(1 
/2*f*x+1/2*e)^6-8532*cos(1/2*f*x+1/2*e)^4+6054*cos(1/2*f*x+1/2*e)^2+75)*si 
n(1/2*f*x+1/2*e)-96*cos(1/2*f*x+1/2*e)^9+1800*cos(1/2*f*x+1/2*e)^7+3708*co 
s(1/2*f*x+1/2*e)^5-6378*cos(1/2*f*x+1/2*e)^3+891*cos(1/2*f*x+1/2*e))*c^2*d 
^2+((704*cos(1/2*f*x+1/2*e)^8+7048*cos(1/2*f*x+1/2*e)^6-9468*cos(1/2*f*x+1 
/2*e)^4+1446*cos(1/2*f*x+1/2*e)^2+15)*sin(1/2*f*x+1/2*e)-704*cos(1/2*f*x+1 
/2*e)^9+9864*cos(1/2*f*x+1/2*e)^7-15900*cos(1/2*f*x+1/2*e)^5+6470*cos(1/2* 
f*x+1/2*e)^3+255*cos(1/2*f*x+1/2*e))*c*d^3+cos(1/2*f*x+1/2*e)*((3680*cos(1 
/2*f*x+1/2*e)^6-3680*cos(1/2*f*x+1/2*e)^5*sin(1/2*f*x+1/2*e)-4600*cos(1/2* 
f*x+1/2*e)^4+2760*sin(1/2*f*x+1/2*e)*cos(1/2*f*x+1/2*e)^3+680*cos(1/2*f*x+ 
1/2*e)^2+240*sin(1/2*f*x+1/2*e)*cos(1/2*f*x+1/2*e)+30)*sin(1/2*f*x+1/2*e)^ 
2-30*sin(1/2*f*x+1/2*e)*cos(1/2*f*x+1/2*e))*d^4)*((2*sin(1/2*f*x+1/2*e)*co 
s(1/2*f*x+1/2*e)+1)*a)^(1/2)*a^2/(cos(1/2*f*x+1/2*e)+sin(1/2*f*x+1/2*e))/( 
c+2*d*sin(1/2*f*x+1/2*e)*cos(1/2*f*x+1/2*e))^(9/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1033 vs. \(2 (230) = 460\).

Time = 0.18 (sec) , antiderivative size = 1033, normalized size of antiderivative = 4.07 \[ \int \frac {(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{9/2}} \, dx =\text {Too large to display} \] Input:

integrate((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(9/2),x, algorithm="fric 
as")
 

Output:

-2/105*(224*a^2*c^3 - 608*a^2*c^2*d + 544*a^2*c*d^2 - 160*a^2*d^3 - 2*(3*a 
^2*c^2*d + 22*a^2*c*d^2 + 115*a^2*d^3)*cos(f*x + e)^4 - (21*a^2*c^3 + 157* 
a^2*c^2*d + 827*a^2*c*d^2 + 115*a^2*d^3)*cos(f*x + e)^3 + (77*a^2*c^3 + 78 
3*a^2*c^2*d - 425*a^2*c*d^2 + 405*a^2*d^3)*cos(f*x + e)^2 + 2*(161*a^2*c^3 
 + 163*a^2*c^2*d + 451*a^2*c*d^2 + 65*a^2*d^3)*cos(f*x + e) - (224*a^2*c^3 
 - 608*a^2*c^2*d + 544*a^2*c*d^2 - 160*a^2*d^3 + 2*(3*a^2*c^2*d + 22*a^2*c 
*d^2 + 115*a^2*d^3)*cos(f*x + e)^3 - (21*a^2*c^3 + 151*a^2*c^2*d + 783*a^2 
*c*d^2 - 115*a^2*d^3)*cos(f*x + e)^2 - 2*(49*a^2*c^3 + 467*a^2*c^2*d + 179 
*a^2*c*d^2 + 145*a^2*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) 
+ a)*sqrt(d*sin(f*x + e) + c)/((c^4*d^4 + 4*c^3*d^5 + 6*c^2*d^6 + 4*c*d^7 
+ d^8)*f*cos(f*x + e)^5 + (4*c^5*d^3 + 17*c^4*d^4 + 28*c^3*d^5 + 22*c^2*d^ 
6 + 8*c*d^7 + d^8)*f*cos(f*x + e)^4 - 2*(3*c^6*d^2 + 12*c^5*d^3 + 19*c^4*d 
^4 + 16*c^3*d^5 + 9*c^2*d^6 + 4*c*d^7 + d^8)*f*cos(f*x + e)^3 - 2*(2*c^7*d 
 + 11*c^6*d^2 + 28*c^5*d^3 + 43*c^4*d^4 + 42*c^3*d^5 + 25*c^2*d^6 + 8*c*d^ 
7 + d^8)*f*cos(f*x + e)^2 + (c^8 + 4*c^7*d + 12*c^6*d^2 + 28*c^5*d^3 + 38* 
c^4*d^4 + 28*c^3*d^5 + 12*c^2*d^6 + 4*c*d^7 + d^8)*f*cos(f*x + e) + (c^8 + 
 8*c^7*d + 28*c^6*d^2 + 56*c^5*d^3 + 70*c^4*d^4 + 56*c^3*d^5 + 28*c^2*d^6 
+ 8*c*d^7 + d^8)*f + ((c^4*d^4 + 4*c^3*d^5 + 6*c^2*d^6 + 4*c*d^7 + d^8)*f* 
cos(f*x + e)^4 - 4*(c^5*d^3 + 4*c^4*d^4 + 6*c^3*d^5 + 4*c^2*d^6 + c*d^7)*f 
*cos(f*x + e)^3 - 2*(3*c^6*d^2 + 14*c^5*d^3 + 27*c^4*d^4 + 28*c^3*d^5 +...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \] Input:

integrate((a+a*sin(f*x+e))**(5/2)/(c+d*sin(f*x+e))**(9/2),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 703 vs. \(2 (230) = 460\).

Time = 0.23 (sec) , antiderivative size = 703, normalized size of antiderivative = 2.77 \[ \int \frac {(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{9/2}} \, dx =\text {Too large to display} \] Input:

integrate((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(9/2),x, algorithm="maxi 
ma")
                                                                                    
                                                                                    
 

Output:

-2/105*((301*c^4 + 169*c^3*d + 75*c^2*d^2 + 15*c*d^3)*a^(5/2) - 3*(35*c^4 
- 763*c^3*d - 297*c^2*d^2 - 85*c*d^3 - 10*d^4)*a^(5/2)*sin(f*x + e)/(cos(f 
*x + e) + 1) + 6*(182*c^4 - 127*c^3*d + 1059*c^2*d^2 + 251*c*d^3 + 35*d^4) 
*a^(5/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 14*(50*c^4 - 421*c^3*d + 20 
1*c^2*d^2 - 535*c*d^3 - 55*d^4)*a^(5/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^ 
3 + 126*(11*c^4 - 36*c^3*d + 80*c^2*d^2 - 40*c*d^3 + 25*d^4)*a^(5/2)*sin(f 
*x + e)^4/(cos(f*x + e) + 1)^4 - 126*(11*c^4 - 36*c^3*d + 80*c^2*d^2 - 40* 
c*d^3 + 25*d^4)*a^(5/2)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 14*(50*c^4 - 
 421*c^3*d + 201*c^2*d^2 - 535*c*d^3 - 55*d^4)*a^(5/2)*sin(f*x + e)^6/(cos 
(f*x + e) + 1)^6 - 6*(182*c^4 - 127*c^3*d + 1059*c^2*d^2 + 251*c*d^3 + 35* 
d^4)*a^(5/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 3*(35*c^4 - 763*c^3*d - 
 297*c^2*d^2 - 85*c*d^3 - 10*d^4)*a^(5/2)*sin(f*x + e)^8/(cos(f*x + e) + 1 
)^8 - (301*c^4 + 169*c^3*d + 75*c^2*d^2 + 15*c*d^3)*a^(5/2)*sin(f*x + e)^9 
/(cos(f*x + e) + 1)^9)*(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)^2/((c^4 + 
 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 + 2*(c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c* 
d^3 + d^4)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + (c^4 + 4*c^3*d + 6*c^2*d^ 
2 + 4*c*d^3 + d^4)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4)*(c + 2*d*sin(f*x + 
 e)/(cos(f*x + e) + 1) + c*sin(f*x + e)^2/(cos(f*x + e) + 1)^2)^(9/2)*f)
 

Giac [F(-1)]

Timed out. \[ \int \frac {(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \] Input:

integrate((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(9/2),x, algorithm="giac 
")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 28.33 (sec) , antiderivative size = 862, normalized size of antiderivative = 3.39 \[ \int \frac {(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{9/2}} \, dx=\text {Too large to display} \] Input:

int((a + a*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^(9/2),x)
 

Output:

((c + d*sin(e + f*x))^(1/2)*((8*a^2*exp(e*8i + f*x*8i)*(a + a*sin(e + f*x) 
)^(1/2)*(22*c*d + 3*c^2 + 115*d^2))/(105*d^3*f*(c + d)^4) + (8*a^2*exp(e*1 
i + f*x*1i)*(a + a*sin(e + f*x))^(1/2)*(c*d*22i + c^2*3i + d^2*115i))/(105 
*d^3*f*(c + d)^4) - (8*a^2*exp(e*4i + f*x*4i)*(a + a*sin(e + f*x))^(1/2)*( 
36*c*d^2 - 25*c^2*d + 30*c^3 - 5*d^3))/(3*d^4*f*(c + d)^4) - (8*a^2*exp(e* 
5i + f*x*5i)*(a + a*sin(e + f*x))^(1/2)*(c*d^2*36i - c^2*d*25i + c^3*30i - 
 d^3*5i))/(3*d^4*f*(c + d)^4) - (8*a^2*exp(e*6i + f*x*6i)*(a + a*sin(e + f 
*x))^(1/2)*(244*c^2*d - 19*c*d^2 + 25*c^3 + 50*d^3))/(15*d^4*f*(c + d)^4) 
- (8*a^2*exp(e*3i + f*x*3i)*(a + a*sin(e + f*x))^(1/2)*(c^2*d*244i - c*d^2 
*19i + c^3*25i + d^3*50i))/(15*d^4*f*(c + d)^4) + (8*a^2*c*exp(e*2i + f*x* 
2i)*(a + a*sin(e + f*x))^(1/2)*(22*c*d + 3*c^2 + 115*d^2))/(15*d^4*f*(c + 
d)^4) + (8*a^2*c*exp(e*7i + f*x*7i)*(a + a*sin(e + f*x))^(1/2)*(c*d*22i + 
c^2*3i + d^2*115i))/(15*d^4*f*(c + d)^4)))/(exp(e*9i + f*x*9i) + ((c*1i + 
d*1i)^4*1i)/(c + d)^4 - (4*exp(e*3i + f*x*3i)*(6*c*d^2 + 6*c^2*d + 8*c^3 + 
 d^3))/d^3 - (4*exp(e*7i + f*x*7i)*(2*c*d + 6*c^2 + d^2))/d^2 + (exp(e*1i 
+ f*x*1i)*(8*c + d))/d + (2*exp(e*5i + f*x*5i)*(12*c*d^3 + 16*c^3*d + 8*c^ 
4 + 3*d^4 + 24*c^2*d^2))/d^4 - (exp(e*6i + f*x*6i)*(c*1i + d*1i)^4*(6*c*d^ 
2 + 6*c^2*d + 8*c^3 + d^3)*4i)/(d^3*(c + d)^4) - (exp(e*2i + f*x*2i)*(c*1i 
 + d*1i)^4*(2*c*d + 6*c^2 + d^2)*4i)/(d^2*(c + d)^4) + (exp(e*8i + f*x*8i) 
*(8*c + d)*(c*1i + d*1i)^4*1i)/(d*(c + d)^4) + (exp(e*4i + f*x*4i)*(c*1...
 

Reduce [F]

\[ \int \frac {(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{9/2}} \, dx=\sqrt {a}\, a^{2} \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{5} d^{5}+5 \sin \left (f x +e \right )^{4} c \,d^{4}+10 \sin \left (f x +e \right )^{3} c^{2} d^{3}+10 \sin \left (f x +e \right )^{2} c^{3} d^{2}+5 \sin \left (f x +e \right ) c^{4} d +c^{5}}d x +2 \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{5} d^{5}+5 \sin \left (f x +e \right )^{4} c \,d^{4}+10 \sin \left (f x +e \right )^{3} c^{2} d^{3}+10 \sin \left (f x +e \right )^{2} c^{3} d^{2}+5 \sin \left (f x +e \right ) c^{4} d +c^{5}}d x \right )+\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sqrt {\sin \left (f x +e \right )+1}}{\sin \left (f x +e \right )^{5} d^{5}+5 \sin \left (f x +e \right )^{4} c \,d^{4}+10 \sin \left (f x +e \right )^{3} c^{2} d^{3}+10 \sin \left (f x +e \right )^{2} c^{3} d^{2}+5 \sin \left (f x +e \right ) c^{4} d +c^{5}}d x \right ) \] Input:

int((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(9/2),x)
 

Output:

sqrt(a)*a**2*(int((sqrt(sin(e + f*x)*d + c)*sqrt(sin(e + f*x) + 1)*sin(e + 
 f*x)**2)/(sin(e + f*x)**5*d**5 + 5*sin(e + f*x)**4*c*d**4 + 10*sin(e + f* 
x)**3*c**2*d**3 + 10*sin(e + f*x)**2*c**3*d**2 + 5*sin(e + f*x)*c**4*d + c 
**5),x) + 2*int((sqrt(sin(e + f*x)*d + c)*sqrt(sin(e + f*x) + 1)*sin(e + f 
*x))/(sin(e + f*x)**5*d**5 + 5*sin(e + f*x)**4*c*d**4 + 10*sin(e + f*x)**3 
*c**2*d**3 + 10*sin(e + f*x)**2*c**3*d**2 + 5*sin(e + f*x)*c**4*d + c**5), 
x) + int((sqrt(sin(e + f*x)*d + c)*sqrt(sin(e + f*x) + 1))/(sin(e + f*x)** 
5*d**5 + 5*sin(e + f*x)**4*c*d**4 + 10*sin(e + f*x)**3*c**2*d**3 + 10*sin( 
e + f*x)**2*c**3*d**2 + 5*sin(e + f*x)*c**4*d + c**5),x))