\(\int \frac {(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{9/2}} \, dx\) [586]

Optimal result

Integrand size = 29, antiderivative size = 229 \[ \int \frac {(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{9/2}} \, dx=\frac {2 a^2 (c-d) \cos (e+f x)}{7 d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}-\frac {2 a^2 (c+13 d) \cos (e+f x)}{35 d (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {8 a^2 (c+13 d) \cos (e+f x)}{105 d (c+d)^3 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {16 a^2 (c+13 d) \cos (e+f x)}{105 d (c+d)^4 f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \] Output:

2/7*a^2*(c-d)*cos(f*x+e)/d/(c+d)/f/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e)) 
^(7/2)-2/35*a^2*(c+13*d)*cos(f*x+e)/d/(c+d)^2/f/(a+a*sin(f*x+e))^(1/2)/(c+ 
d*sin(f*x+e))^(5/2)-8/105*a^2*(c+13*d)*cos(f*x+e)/d/(c+d)^3/f/(a+a*sin(f*x 
+e))^(1/2)/(c+d*sin(f*x+e))^(3/2)-16/105*a^2*(c+13*d)*cos(f*x+e)/d/(c+d)^4 
/f/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 2.71 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.84 \[ \int \frac {(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{9/2}} \, dx=-\frac {2 a \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} \left (175 c^3+147 c^2 d+253 c d^2+41 d^3-2 d \left (7 c^2+92 c d+13 d^2\right ) \cos (2 (e+f x))+\left (35 c^3+469 c^2 d+191 c d^2+117 d^3\right ) \sin (e+f x)-2 c d^2 \sin (3 (e+f x))-26 d^3 \sin (3 (e+f x))\right )}{105 (c+d)^4 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c+d \sin (e+f x))^{7/2}} \] Input:

Integrate[(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(9/2),x]
 

Output:

(-2*a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(17 
5*c^3 + 147*c^2*d + 253*c*d^2 + 41*d^3 - 2*d*(7*c^2 + 92*c*d + 13*d^2)*Cos 
[2*(e + f*x)] + (35*c^3 + 469*c^2*d + 191*c*d^2 + 117*d^3)*Sin[e + f*x] - 
2*c*d^2*Sin[3*(e + f*x)] - 26*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 = 1.02 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {3042, 3241, 27, 2011, 3042, 3251, 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.

\(\displaystyle \int \frac {(a \sin (e+f x)+a)^{3/2}}{(c+d \sin (e+f x))^{9/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a \sin (e+f x)+a)^{3/2}}{(c+d \sin (e+f x))^{9/2}}dx\)

\(\Big \downarrow \) 3241

\(\displaystyle \frac {2 a^2 (c-d) \cos (e+f x)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}-\frac {2 a \int -\frac {a (c+13 d)+a \sin (e+f x) (c+13 d)}{2 \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^{7/2}}dx}{7 d (c+d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \int \frac {a (c+13 d)+a \sin (e+f x) (c+13 d)}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^{7/2}}dx}{7 d (c+d)}+\frac {2 a^2 (c-d) \cos (e+f x)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}\)

\(\Big \downarrow \) 2011

\(\displaystyle \frac {a (c+13 d) \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{7/2}}dx}{7 d (c+d)}+\frac {2 a^2 (c-d) \cos (e+f x)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a (c+13 d) \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{7/2}}dx}{7 d (c+d)}+\frac {2 a^2 (c-d) \cos (e+f x)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}\)

\(\Big \downarrow \) 3251

\(\displaystyle \frac {a (c+13 d) \left (\frac {4 \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{5/2}}dx}{5 (c+d)}-\frac {2 a \cos (e+f x)}{5 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}\right )}{7 d (c+d)}+\frac {2 a^2 (c-d) \cos (e+f x)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a (c+13 d) \left (\frac {4 \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{5/2}}dx}{5 (c+d)}-\frac {2 a \cos (e+f x)}{5 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}\right )}{7 d (c+d)}+\frac {2 a^2 (c-d) \cos (e+f x)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}\)

\(\Big \downarrow \) 3251

\(\displaystyle \frac {a (c+13 d) \left (\frac {4 \left (\frac {2 \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{3/2}}dx}{3 (c+d)}-\frac {2 a \cos (e+f x)}{3 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}\right )}{5 (c+d)}-\frac {2 a \cos (e+f x)}{5 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}\right )}{7 d (c+d)}+\frac {2 a^2 (c-d) \cos (e+f x)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a (c+13 d) \left (\frac {4 \left (\frac {2 \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{3/2}}dx}{3 (c+d)}-\frac {2 a \cos (e+f x)}{3 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}\right )}{5 (c+d)}-\frac {2 a \cos (e+f x)}{5 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}\right )}{7 d (c+d)}+\frac {2 a^2 (c-d) \cos (e+f x)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}\)

\(\Big \downarrow \) 3250

\(\displaystyle \frac {2 a^2 (c-d) \cos (e+f x)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}+\frac {a (c+13 d) \left (\frac {4 \left (-\frac {4 a \cos (e+f x)}{3 f (c+d)^2 \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}-\frac {2 a \cos (e+f x)}{3 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}\right )}{5 (c+d)}-\frac {2 a \cos (e+f x)}{5 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}\right )}{7 d (c+d)}\)

Input:

Int[(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(9/2),x]
 

Output:

(2*a^2*(c - d)*Cos[e + f*x])/(7*d*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + 
d*Sin[e + f*x])^(7/2)) + (a*(c + 13*d)*((-2*a*Cos[e + f*x])/(5*(c + d)*f*S 
qrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2)) + (4*((-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*(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 2011

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> 
 Simp[(b/d)^m   Int[u*(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c + d*x 
, a + 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(636\) vs. \(2(205)=410\).

Time = 32.25 (sec) , antiderivative size = 637, normalized size of antiderivative = 2.78

method	result	size
default	\(\frac {2 \left (\left (\left (-70 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+175\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-70 \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 (-252 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-1036 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+133\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+252 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-1540 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+1155 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c^{3} d +\left (\left (288 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-3636 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+2712 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+69\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+288 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}+2772 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-3696 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+567 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) d^{2} c^{2}+\left (\left (128 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+3520 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-4596 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+732 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+15\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-128 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}+4032 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}-6732 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+2612 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+201 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) d^{3} c +\cos \left (\frac {f x}{2}+\frac {e}{2}\right ) \left (\left (1664 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-1664 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-2080 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+1248 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+260 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+156 \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}{105 f \left (c^{4}+4 c^{3} d +6 d^{2} c^{2}+4 d^{3} c +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}}}\)	\(637\)

Input:

int((a+a*sin(f*x+e))^(3/2)/(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)*(((-70*cos(1/2*f*x+1/2*e)^2+17 
5)*sin(1/2*f*x+1/2*e)-70*cos(1/2*f*x+1/2*e)^3-105*cos(1/2*f*x+1/2*e))*c^4+ 
((-252*cos(1/2*f*x+1/2*e)^4-1036*cos(1/2*f*x+1/2*e)^2+133)*sin(1/2*f*x+1/2 
*e)+252*cos(1/2*f*x+1/2*e)^5-1540*cos(1/2*f*x+1/2*e)^3+1155*cos(1/2*f*x+1/ 
2*e))*c^3*d+((288*cos(1/2*f*x+1/2*e)^6-3636*cos(1/2*f*x+1/2*e)^4+2712*cos( 
1/2*f*x+1/2*e)^2+69)*sin(1/2*f*x+1/2*e)+288*cos(1/2*f*x+1/2*e)^7+2772*cos( 
1/2*f*x+1/2*e)^5-3696*cos(1/2*f*x+1/2*e)^3+567*cos(1/2*f*x+1/2*e))*d^2*c^2 
+((128*cos(1/2*f*x+1/2*e)^8+3520*cos(1/2*f*x+1/2*e)^6-4596*cos(1/2*f*x+1/2 
*e)^4+732*cos(1/2*f*x+1/2*e)^2+15)*sin(1/2*f*x+1/2*e)-128*cos(1/2*f*x+1/2* 
e)^9+4032*cos(1/2*f*x+1/2*e)^7-6732*cos(1/2*f*x+1/2*e)^5+2612*cos(1/2*f*x+ 
1/2*e)^3+201*cos(1/2*f*x+1/2*e))*d^3*c+cos(1/2*f*x+1/2*e)*((1664*cos(1/2*f 
*x+1/2*e)^6-1664*cos(1/2*f*x+1/2*e)^5*sin(1/2*f*x+1/2*e)-2080*cos(1/2*f*x+ 
1/2*e)^4+1248*sin(1/2*f*x+1/2*e)*cos(1/2*f*x+1/2*e)^3+260*cos(1/2*f*x+1/2* 
e)^2+156*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)*cos(1/ 
2*f*x+1/2*e)+1)*a)^(1/2)*a/(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. 937 vs. \(2 (205) = 410\).

Time = 0.16 (sec) , antiderivative size = 937, normalized size of antiderivative = 4.09 \[ \int \frac {(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{9/2}} \, dx =\text {Too large to display} \] Input:

integrate((a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(9/2),x, algorithm="fric 
as")
 

Output:

2/105*(8*(a*c*d^2 + 13*a*d^3)*cos(f*x + e)^4 - 140*a*c^3 + 308*a*c^2*d - 2 
44*a*c*d^2 + 76*a*d^3 + 4*(7*a*c^2*d + 92*a*c*d^2 + 13*a*d^3)*cos(f*x + e) 
^3 - (35*a*c^3 + 441*a*c^2*d - 167*a*c*d^2 + 195*a*d^3)*cos(f*x + e)^2 - ( 
175*a*c^3 + 161*a*c^2*d + 437*a*c*d^2 + 67*a*d^3)*cos(f*x + e) + (140*a*c^ 
3 - 308*a*c^2*d + 244*a*c*d^2 - 76*a*d^3 + 8*(a*c*d^2 + 13*a*d^3)*cos(f*x 
+ e)^3 - 4*(7*a*c^2*d + 90*a*c*d^2 - 13*a*d^3)*cos(f*x + e)^2 - (35*a*c^3 
+ 469*a*c^2*d + 193*a*c*d^2 + 143*a*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 + 2 
8*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 + 17*c^2*d^6 + 6*c*d^7 + d^8)*f*cos(f*x + e)^2 + 4*(c^7*d + 4*c 
^6*d^2 + 7*c^5*d^3 + 8*c^4*d^4 + 7*c^3*d^5 + 4*c^2*d^6 + c*d^7)*f*cos(f...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \] Input:

integrate((a+a*sin(f*x+e))**(3/2)/(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. 750 vs. \(2 (205) = 410\).

Time = 0.24 (sec) , antiderivative size = 750, normalized size of antiderivative = 3.28 \[ \int \frac {(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{9/2}} \, dx =\text {Too large to display} \] Input:

integrate((a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(9/2),x, algorithm="maxi 
ma")
                                                                                    
                                                                                    
 

Output:

-2/105*((175*c^4 + 133*c^3*d + 69*c^2*d^2 + 15*c*d^3)*a^(3/2) - 3*(35*c^4 
- 385*c^3*d - 189*c^2*d^2 - 67*c*d^3 - 10*d^4)*a^(3/2)*sin(f*x + e)/(cos(f 
*x + e) + 1) + 18*(35*c^4 - 28*c^3*d + 166*c^2*d^2 + 44*c*d^3 + 7*d^4)*a^( 
3/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 14*(35*c^4 - 220*c^3*d + 102*c^ 
2*d^2 - 244*c*d^3 - 25*d^4)*a^(3/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 
42*(20*c^4 - 61*c^3*d + 117*c^2*d^2 - 55*c*d^3 + 35*d^4)*a^(3/2)*sin(f*x + 
 e)^4/(cos(f*x + e) + 1)^4 - 42*(20*c^4 - 61*c^3*d + 117*c^2*d^2 - 55*c*d^ 
3 + 35*d^4)*a^(3/2)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 14*(35*c^4 - 220 
*c^3*d + 102*c^2*d^2 - 244*c*d^3 - 25*d^4)*a^(3/2)*sin(f*x + e)^6/(cos(f*x 
 + e) + 1)^6 - 18*(35*c^4 - 28*c^3*d + 166*c^2*d^2 + 44*c*d^3 + 7*d^4)*a^( 
3/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 3*(35*c^4 - 385*c^3*d - 189*c^2 
*d^2 - 67*c*d^3 - 10*d^4)*a^(3/2)*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - (1 
75*c^4 + 133*c^3*d + 69*c^2*d^2 + 15*c*d^3)*a^(3/2)*sin(f*x + e)^9/(cos(f* 
x + e) + 1)^9)*(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)^3/((c^4 + 4*c^3*d 
 + 6*c^2*d^2 + 4*c*d^3 + d^4 + 3*(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 + 3*(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^4 + 4*c^3*d + 6*c^2* 
d^2 + 4*c*d^3 + d^4)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6)*(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. \[ \int \frac {(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \] Input:

integrate((a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(9/2),x, algorithm="giac 
")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 29.55 (sec) , antiderivative size = 807, normalized size of antiderivative = 3.52 \[ \int \frac {(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{9/2}} \, dx =\text {Too large to display} \] Input:

int((a + a*sin(e + f*x))^(3/2)/(c + d*sin(e + f*x))^(9/2),x)
 

Output:

((c + d*sin(e + f*x))^(1/2)*((32*a*exp(e*8i + f*x*8i)*(c + 13*d)*(a + a*si 
n(e + f*x))^(1/2))/(105*d^2*f*(c + d)^4) - (16*a*exp(e*4i + f*x*4i)*(a + a 
*sin(e + f*x))^(1/2)*(9*c*d^2 - 5*c^2*d + 9*c^3 - d^3))/(3*d^4*f*(c + d)^4 
) - (16*a*exp(e*5i + f*x*5i)*(a + a*sin(e + f*x))^(1/2)*(c*d^2*9i - c^2*d* 
5i + c^3*9i - d^3*1i))/(3*d^4*f*(c + d)^4) - (16*a*exp(e*6i + f*x*6i)*(a + 
 a*sin(e + f*x))^(1/2)*(c*d^2 + 65*c^2*d + 5*c^3 + 13*d^3))/(15*d^4*f*(c + 
 d)^4) - (16*a*exp(e*3i + f*x*3i)*(a + a*sin(e + f*x))^(1/2)*(c*d^2*1i + c 
^2*d*65i + c^3*5i + d^3*13i))/(15*d^4*f*(c + d)^4) + (32*a*exp(e*1i + f*x* 
1i)*(c*1i + d*13i)*(a + a*sin(e + f*x))^(1/2))/(105*d^2*f*(c + d)^4) + (32 
*a*c*exp(e*7i + f*x*7i)*(c*1i + d*13i)*(a + a*sin(e + f*x))^(1/2))/(15*d^3 
*f*(c + d)^4) + (32*a*c*exp(e*2i + f*x*2i)*(c + 13*d)*(a + a*sin(e + f*x)) 
^(1/2))/(15*d^3*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*1i + d*1i)^4*(12 
*c*d^3 + 16*c^3*d + 8*c^4 + 3*d^4 + 24*c^2*d^2)*2i)/(d^4*(c + d)^4))
 

Reduce [F]

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

int((a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(9/2),x)
 

Output:

sqrt(a)*a*(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))