\(\int \frac {(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{11/2}} \, dx\) [595]

Optimal result

Integrand size = 29, antiderivative size = 317 \[ \int \frac {(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{11/2}} \, dx=\frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{9 d (c+d) f (c+d \sin (e+f x))^{9/2}}+\frac {2 a^3 (c-d) (3 c+19 d) \cos (e+f x)}{63 d^2 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}-\frac {2 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^3 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {8 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{315 d^2 (c+d)^4 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {16 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{315 d^2 (c+d)^5 f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \] Output:

2/9*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)) 
^(9/2)+2/63*a^3*(c-d)*(3*c+19*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))^(7/2)-2/105*a^3*(c^2+10*c*d+73*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))^(5/2)-8/315*a^3*(c^2+10 
*c*d+73*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))^(3/2)-16/315*a^3*(c^2+10*c*d+73*d^2)*cos(f*x+e)/d^2/(c+d)^5/f/(a+a*sin 
(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 9.44 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.96 \[ \int \frac {(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{11/2}} \, dx=-\frac {a^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} \left (1869 c^4+2088 c^3 d+5776 c^2 d^2+1804 c d^3+727 d^4-\left (63 c^4+648 c^3 d+4790 c^2 d^2+1424 c d^3+803 d^4\right ) \cos (2 (e+f x))+2 d^2 \left (c^2+10 c d+73 d^2\right ) \cos (4 (e+f x))+588 c^4 \sin (e+f x)+7326 c^3 d \sin (e+f x)+4370 c^2 d^2 \sin (e+f x)+5498 c d^3 \sin (e+f x)+698 d^4 \sin (e+f x)-18 c^3 d \sin (3 (e+f x))-182 c^2 d^2 \sin (3 (e+f x))-1334 c d^3 \sin (3 (e+f x))-146 d^4 \sin (3 (e+f x))\right )}{315 (c+d)^5 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c+d \sin (e+f x))^{9/2}} \] Input:

Integrate[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(11/2),x]
 

Output:

-1/315*(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x] 
)]*(1869*c^4 + 2088*c^3*d + 5776*c^2*d^2 + 1804*c*d^3 + 727*d^4 - (63*c^4 
+ 648*c^3*d + 4790*c^2*d^2 + 1424*c*d^3 + 803*d^4)*Cos[2*(e + f*x)] + 2*d^ 
2*(c^2 + 10*c*d + 73*d^2)*Cos[4*(e + f*x)] + 588*c^4*Sin[e + f*x] + 7326*c 
^3*d*Sin[e + f*x] + 4370*c^2*d^2*Sin[e + f*x] + 5498*c*d^3*Sin[e + f*x] + 
698*d^4*Sin[e + f*x] - 18*c^3*d*Sin[3*(e + f*x)] - 182*c^2*d^2*Sin[3*(e + 
f*x)] - 1334*c*d^3*Sin[3*(e + f*x)] - 146*d^4*Sin[3*(e + f*x)]))/((c + d)^ 
5*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c + d*Sin[e + f*x])^(9/2))
 

Rubi [A] (verified)

Time = 1.45 (sec) , antiderivative size = 314, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {3042, 3241, 27, 3042, 3459, 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.

\(\displaystyle \int \frac {(a \sin (e+f x)+a)^{5/2}}{(c+d \sin (e+f x))^{11/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a \sin (e+f x)+a)^{5/2}}{(c+d \sin (e+f x))^{11/2}}dx\)

\(\Big \downarrow \) 3241

\(\displaystyle \frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{9 d f (c+d) (c+d \sin (e+f x))^{9/2}}-\frac {2 a \int \frac {\sqrt {\sin (e+f x) a+a} (a (c-19 d)-3 a (c+5 d) \sin (e+f x))}{2 (c+d \sin (e+f x))^{9/2}}dx}{9 d (c+d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{9 d f (c+d) (c+d \sin (e+f x))^{9/2}}-\frac {a \int \frac {\sqrt {\sin (e+f x) a+a} (a (c-19 d)-3 a (c+5 d) \sin (e+f x))}{(c+d \sin (e+f x))^{9/2}}dx}{9 d (c+d)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{9 d f (c+d) (c+d \sin (e+f x))^{9/2}}-\frac {a \int \frac {\sqrt {\sin (e+f x) a+a} (a (c-19 d)-3 a (c+5 d) \sin (e+f x))}{(c+d \sin (e+f x))^{9/2}}dx}{9 d (c+d)}\)

\(\Big \downarrow \) 3459

\(\displaystyle \frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{9 d f (c+d) (c+d \sin (e+f x))^{9/2}}-\frac {a \left (-\frac {3 a \left (c^2+10 c d+73 d^2\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{7/2}}dx}{7 d (c+d)}-\frac {2 a^2 (c-d) (3 c+19 d) \cos (e+f x)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}\right )}{9 d (c+d)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{9 d f (c+d) (c+d \sin (e+f x))^{9/2}}-\frac {a \left (-\frac {3 a \left (c^2+10 c d+73 d^2\right ) \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{7/2}}dx}{7 d (c+d)}-\frac {2 a^2 (c-d) (3 c+19 d) \cos (e+f x)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}\right )}{9 d (c+d)}\)

\(\Big \downarrow \) 3251

\(\displaystyle \frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{9 d f (c+d) (c+d \sin (e+f x))^{9/2}}-\frac {a \left (-\frac {3 a \left (c^2+10 c d+73 d^2\right ) \left (\frac {4 \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{5/2}}dx}{5 (c+d)}-\frac {2 a \cos (e+f x)}{5 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}\right )}{7 d (c+d)}-\frac {2 a^2 (c-d) (3 c+19 d) \cos (e+f x)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}\right )}{9 d (c+d)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{9 d f (c+d) (c+d \sin (e+f x))^{9/2}}-\frac {a \left (-\frac {3 a \left (c^2+10 c d+73 d^2\right ) \left (\frac {4 \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{5/2}}dx}{5 (c+d)}-\frac {2 a \cos (e+f x)}{5 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}\right )}{7 d (c+d)}-\frac {2 a^2 (c-d) (3 c+19 d) \cos (e+f x)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}\right )}{9 d (c+d)}\)

\(\Big \downarrow \) 3251

\(\displaystyle \frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{9 d f (c+d) (c+d \sin (e+f x))^{9/2}}-\frac {a \left (-\frac {3 a \left (c^2+10 c d+73 d^2\right ) \left (\frac {4 \left (\frac {2 \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{3/2}}dx}{3 (c+d)}-\frac {2 a \cos (e+f x)}{3 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}\right )}{5 (c+d)}-\frac {2 a \cos (e+f x)}{5 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}\right )}{7 d (c+d)}-\frac {2 a^2 (c-d) (3 c+19 d) \cos (e+f x)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}\right )}{9 d (c+d)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{9 d f (c+d) (c+d \sin (e+f x))^{9/2}}-\frac {a \left (-\frac {3 a \left (c^2+10 c d+73 d^2\right ) \left (\frac {4 \left (\frac {2 \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{3/2}}dx}{3 (c+d)}-\frac {2 a \cos (e+f x)}{3 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}\right )}{5 (c+d)}-\frac {2 a \cos (e+f x)}{5 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}\right )}{7 d (c+d)}-\frac {2 a^2 (c-d) (3 c+19 d) \cos (e+f x)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}\right )}{9 d (c+d)}\)

\(\Big \downarrow \) 3250

\(\displaystyle \frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{9 d f (c+d) (c+d \sin (e+f x))^{9/2}}-\frac {a \left (-\frac {2 a^2 (c-d) (3 c+19 d) \cos (e+f x)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}-\frac {3 a \left (c^2+10 c d+73 d^2\right ) \left (\frac {4 \left (-\frac {4 a \cos (e+f x)}{3 f (c+d)^2 \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}-\frac {2 a \cos (e+f x)}{3 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}\right )}{5 (c+d)}-\frac {2 a \cos (e+f x)}{5 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}\right )}{7 d (c+d)}\right )}{9 d (c+d)}\)

Input:

Int[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(11/2),x]
 

Output:

(2*a^2*(c - d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(9*d*(c + d)*f*(c + 
d*Sin[e + f*x])^(9/2)) - (a*((-2*a^2*(c - d)*(3*c + 19*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)) - (3*a*( 
c^2 + 10*c*d + 73*d^2)*((-2*a*Cos[e + f*x])/(5*(c + d)*f*Sqrt[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))))/(9*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. \(946\) vs. \(2(287)=574\).

Time = 42.21 (sec) , antiderivative size = 947, normalized size of antiderivative = 2.99

method	result	size
default	\(\text {Expression too large to display}\)	\(947\)

Input:

int((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(11/2),x,method=_RETURNVERBOSE 
)
 

Output:

-2/315/f/(c^5+5*c^4*d+10*c^3*d^2+10*c^2*d^3+5*c*d^4+d^5)*(((252*cos(1/2*f* 
x+1/2*e)^4+336*cos(1/2*f*x+1/2*e)^2-903)*sin(1/2*f*x+1/2*e)-252*cos(1/2*f* 
x+1/2*e)^5+840*cos(1/2*f*x+1/2*e)^3+315*cos(1/2*f*x+1/2*e))*c^5+(-792*cos( 
1/2*f*x+1/2*e)^7-2184*cos(1/2*f*x+1/2*e)^5+12054*cos(1/2*f*x+1/2*e)^3-8358 
*cos(1/2*f*x+1/2*e)+(-792*cos(1/2*f*x+1/2*e)^6+4560*cos(1/2*f*x+1/2*e)^4+5 
310*cos(1/2*f*x+1/2*e)^2-720)*sin(1/2*f*x+1/2*e))*c^4*d+(704*cos(1/2*f*x+1 
/2*e)^9-9504*cos(1/2*f*x+1/2*e)^7-16776*cos(1/2*f*x+1/2*e)^5+30840*cos(1/2 
*f*x+1/2*e)^3-4770*cos(1/2*f*x+1/2*e)+(-704*cos(1/2*f*x+1/2*e)^8-6688*cos( 
1/2*f*x+1/2*e)^6+41064*cos(1/2*f*x+1/2*e)^4-28408*cos(1/2*f*x+1/2*e)^2-494 
)*sin(1/2*f*x+1/2*e))*c^3*d^2+(6336*cos(1/2*f*x+1/2*e)^9-73040*cos(1/2*f*x 
+1/2*e)^7+113024*cos(1/2*f*x+1/2*e)^5-44092*cos(1/2*f*x+1/2*e)^3-2284*cos( 
1/2*f*x+1/2*e)+256*cos(1/2*f*x+1/2*e)^11+(256*cos(1/2*f*x+1/2*e)^10-7616*c 
os(1/2*f*x+1/2*e)^8-45136*cos(1/2*f*x+1/2*e)^6+65520*cos(1/2*f*x+1/2*e)^4- 
10540*cos(1/2*f*x+1/2*e)^2-200)*sin(1/2*f*x+1/2*e))*c^2*d^3+(44352*cos(1/2 
*f*x+1/2*e)^9-109472*cos(1/2*f*x+1/2*e)^7+71780*cos(1/2*f*x+1/2*e)^5-8560* 
cos(1/2*f*x+1/2*e)^3-625*cos(1/2*f*x+1/2*e)+2560*cos(1/2*f*x+1/2*e)^11+(25 
60*cos(1/2*f*x+1/2*e)^10-57152*cos(1/2*f*x+1/2*e)^8+93536*cos(1/2*f*x+1/2* 
e)^6-35076*cos(1/2*f*x+1/2*e)^4-3208*cos(1/2*f*x+1/2*e)^2-35)*sin(1/2*f*x+ 
1/2*e))*c*d^4+cos(1/2*f*x+1/2*e)*((-18688*cos(1/2*f*x+1/2*e)^8-18688*cos(1 
/2*f*x+1/2*e)^7*sin(1/2*f*x+1/2*e)+32704*cos(1/2*f*x+1/2*e)^6+23360*cos...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1492 vs. \(2 (287) = 574\).

Time = 0.22 (sec) , antiderivative size = 1492, normalized size of antiderivative = 4.71 \[ \int \frac {(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{11/2}} \, dx=\text {Too large to display} \] Input:

integrate((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(11/2),x, algorithm="fri 
cas")
 

Output:

2/315*(672*a^2*c^4 - 2304*a^2*c^3*d + 3008*a^2*c^2*d^2 - 1792*a^2*c*d^3 + 
416*a^2*d^4 + 8*(a^2*c^2*d^2 + 10*a^2*c*d^3 + 73*a^2*d^4)*cos(f*x + e)^5 - 
 4*(9*a^2*c^3*d + 89*a^2*c^2*d^2 + 647*a^2*c*d^3 - 73*a^2*d^4)*cos(f*x + e 
)^4 - (63*a^2*c^4 + 648*a^2*c^3*d + 4798*a^2*c^2*d^2 + 1504*a^2*c*d^3 + 13 
87*a^2*d^4)*cos(f*x + e)^3 + (231*a^2*c^4 + 3060*a^2*c^3*d - 2158*a^2*c^2* 
d^2 + 4580*a^2*c*d^3 - 673*a^2*d^4)*cos(f*x + e)^2 + 2*(483*a^2*c^4 + 684* 
a^2*c^3*d + 2642*a^2*c^2*d^2 + 812*a^2*c*d^3 + 419*a^2*d^4)*cos(f*x + e) - 
 (672*a^2*c^4 - 2304*a^2*c^3*d + 3008*a^2*c^2*d^2 - 1792*a^2*c*d^3 + 416*a 
^2*d^4 + 8*(a^2*c^2*d^2 + 10*a^2*c*d^3 + 73*a^2*d^4)*cos(f*x + e)^4 + 4*(9 
*a^2*c^3*d + 91*a^2*c^2*d^2 + 667*a^2*c*d^3 + 73*a^2*d^4)*cos(f*x + e)^3 - 
 3*(21*a^2*c^4 + 204*a^2*c^3*d + 1478*a^2*c^2*d^2 - 388*a^2*c*d^3 + 365*a^ 
2*d^4)*cos(f*x + e)^2 - 2*(147*a^2*c^4 + 1836*a^2*c^3*d + 1138*a^2*c^2*d^2 
 + 1708*a^2*c*d^3 + 211*a^2*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f* 
x + e) + a)*sqrt(d*sin(f*x + e) + c)/((c^5*d^5 + 5*c^4*d^6 + 10*c^3*d^7 + 
10*c^2*d^8 + 5*c*d^9 + d^10)*f*cos(f*x + e)^6 - 5*(c^6*d^4 + 5*c^5*d^5 + 1 
0*c^4*d^6 + 10*c^3*d^7 + 5*c^2*d^8 + c*d^9)*f*cos(f*x + e)^5 - (10*c^7*d^3 
 + 55*c^6*d^4 + 128*c^5*d^5 + 165*c^4*d^6 + 130*c^3*d^7 + 65*c^2*d^8 + 20* 
c*d^9 + 3*d^10)*f*cos(f*x + e)^4 + 10*(c^8*d^2 + 5*c^7*d^3 + 11*c^6*d^4 + 
15*c^5*d^5 + 15*c^4*d^6 + 11*c^3*d^7 + 5*c^2*d^8 + c*d^9)*f*cos(f*x + e)^3 
 + (5*c^9*d + 35*c^8*d^2 + 120*c^7*d^3 + 260*c^6*d^4 + 378*c^5*d^5 + 37...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{11/2}} \, dx=\text {Timed out} \] Input:

integrate((a+a*sin(f*x+e))**(5/2)/(c+d*sin(f*x+e))**(11/2),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 984 vs. \(2 (287) = 574\).

Time = 0.33 (sec) , antiderivative size = 984, normalized size of antiderivative = 3.10 \[ \int \frac {(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{11/2}} \, dx=\text {Too large to display} \] Input:

integrate((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(11/2),x, algorithm="max 
ima")
                                                                                    
                                                                                    
 

Output:

-2/315*((903*c^5 + 720*c^4*d + 494*c^3*d^2 + 200*c^2*d^3 + 35*c*d^4)*a^(5/ 
2) - (315*c^5 - 8358*c^4*d - 4770*c^3*d^2 - 2284*c^2*d^3 - 625*c*d^4 - 70* 
d^5)*a^(5/2)*sin(f*x + e)/(cos(f*x + e) + 1) + (4179*c^5 - 1710*c^4*d + 30 
878*c^3*d^2 + 11540*c^2*d^3 + 3383*c*d^4 + 450*d^5)*a^(5/2)*sin(f*x + e)^2 
/(cos(f*x + e) + 1)^2 - 3*(805*c^5 - 9912*c^4*d + 2330*c^3*d^2 - 18504*c^2 
*d^3 - 3895*c*d^4 - 504*d^5)*a^(5/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 
 6*(1239*c^5 - 3100*c^4*d + 12918*c^3*d^2 - 3560*c^2*d^3 + 8043*c*d^4 + 70 
0*d^5)*a^(5/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 42*(149*c^5 - 894*c^4 
*d + 1402*c^3*d^2 - 2052*c^2*d^3 + 745*c*d^4 - 390*d^5)*a^(5/2)*sin(f*x + 
e)^5/(cos(f*x + e) + 1)^5 + 42*(149*c^5 - 894*c^4*d + 1402*c^3*d^2 - 2052* 
c^2*d^3 + 745*c*d^4 - 390*d^5)*a^(5/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 
 - 6*(1239*c^5 - 3100*c^4*d + 12918*c^3*d^2 - 3560*c^2*d^3 + 8043*c*d^4 + 
700*d^5)*a^(5/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 3*(805*c^5 - 9912*c 
^4*d + 2330*c^3*d^2 - 18504*c^2*d^3 - 3895*c*d^4 - 504*d^5)*a^(5/2)*sin(f* 
x + e)^8/(cos(f*x + e) + 1)^8 - (4179*c^5 - 1710*c^4*d + 30878*c^3*d^2 + 1 
1540*c^2*d^3 + 3383*c*d^4 + 450*d^5)*a^(5/2)*sin(f*x + e)^9/(cos(f*x + e) 
+ 1)^9 + (315*c^5 - 8358*c^4*d - 4770*c^3*d^2 - 2284*c^2*d^3 - 625*c*d^4 - 
 70*d^5)*a^(5/2)*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - (903*c^5 + 720*c^ 
4*d + 494*c^3*d^2 + 200*c^2*d^3 + 35*c*d^4)*a^(5/2)*sin(f*x + e)^11/(cos(f 
*x + e) + 1)^11)*(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)^3/((c^5 + 5*...
 

Giac [F(-1)]

Timed out. \[ \int \frac {(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{11/2}} \, dx=\text {Timed out} \] Input:

integrate((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(11/2),x, algorithm="gia 
c")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 29.66 (sec) , antiderivative size = 1155, normalized size of antiderivative = 3.64 \[ \int \frac {(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{11/2}} \, dx=\text {Too large to display} \] Input:

int((a + a*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^(11/2),x)
 

Output:

-((c + d*sin(e + f*x))^(1/2)*((32*a^2*exp(e*1i + f*x*1i)*(a + a*sin(e + f* 
x))^(1/2)*(c*d*10i + c^2*1i + d^2*73i))/(315*d^3*f*(c + d)^5) - (32*a^2*ex 
p(e*10i + f*x*10i)*(a + a*sin(e + f*x))^(1/2)*(10*c*d + c^2 + 73*d^2))/(31 
5*d^3*f*(c + d)^5) - (32*a^2*exp(e*6i + f*x*6i)*(a + a*sin(e + f*x))^(1/2) 
*(25*c^4 - 25*c^3*d - 15*c*d^3 + 6*d^4 + 57*c^2*d^2))/(5*d^5*f*(c + d)^5) 
+ (32*a^2*exp(e*5i + f*x*5i)*(a + a*sin(e + f*x))^(1/2)*(c^4*25i - c^3*d*2 
5i - c*d^3*15i + d^4*6i + c^2*d^2*57i))/(5*d^5*f*(c + d)^5) - (16*a^2*exp( 
e*4i + f*x*4i)*(a + a*sin(e + f*x))^(1/2)*(194*c*d^3 + 318*c^3*d + 25*c^4 
- 5*d^4 - 20*c^2*d^2))/(15*d^5*f*(c + d)^5) + (16*a^2*exp(e*7i + f*x*7i)*( 
a + a*sin(e + f*x))^(1/2)*(c*d^3*194i + c^3*d*318i + c^4*25i - d^4*5i - c^ 
2*d^2*20i))/(15*d^5*f*(c + d)^5) + (16*a^2*exp(e*8i + f*x*8i)*(a + a*sin(e 
 + f*x))^(1/2)*(10*c*d^3 + 70*c^3*d + 7*c^4 + 73*d^4 + 512*c^2*d^2))/(35*d 
^5*f*(c + d)^5) - (16*a^2*exp(e*3i + f*x*3i)*(a + a*sin(e + f*x))^(1/2)*(c 
*d^3*10i + c^3*d*70i + c^4*7i + d^4*73i + c^2*d^2*512i))/(35*d^5*f*(c + d) 
^5) + (32*a^2*c*exp(e*2i + f*x*2i)*(a + a*sin(e + f*x))^(1/2)*(10*c*d + c^ 
2 + 73*d^2))/(35*d^4*f*(c + d)^5) - (32*a^2*c*exp(e*9i + f*x*9i)*(a + a*si 
n(e + f*x))^(1/2)*(c*d*10i + c^2*1i + d^2*73i))/(35*d^4*f*(c + d)^5)))/(ex 
p(e*11i + f*x*11i) - (c*1i + d*1i)^5/(c + d)^5 + (10*exp(e*7i + f*x*7i)*(4 
*c*d^3 + 8*c^3*d + 8*c^4 + d^4 + 12*c^2*d^2))/d^4 + (5*exp(e*3i + f*x*3i)* 
(8*c*d^2 + 8*c^2*d + 16*c^3 + d^3))/d^3 - (5*exp(e*9i + f*x*9i)*(2*c*d ...
 

Reduce [F]

\[ \int \frac {(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{11/2}} \, dx=\sqrt {a}\, a^{2} \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{6} d^{6}+6 \sin \left (f x +e \right )^{5} c \,d^{5}+15 \sin \left (f x +e \right )^{4} c^{2} d^{4}+20 \sin \left (f x +e \right )^{3} c^{3} d^{3}+15 \sin \left (f x +e \right )^{2} c^{4} d^{2}+6 \sin \left (f x +e \right ) c^{5} d +c^{6}}d x +2 \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{6} d^{6}+6 \sin \left (f x +e \right )^{5} c \,d^{5}+15 \sin \left (f x +e \right )^{4} c^{2} d^{4}+20 \sin \left (f x +e \right )^{3} c^{3} d^{3}+15 \sin \left (f x +e \right )^{2} c^{4} d^{2}+6 \sin \left (f x +e \right ) c^{5} d +c^{6}}d x \right )+\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sqrt {\sin \left (f x +e \right )+1}}{\sin \left (f x +e \right )^{6} d^{6}+6 \sin \left (f x +e \right )^{5} c \,d^{5}+15 \sin \left (f x +e \right )^{4} c^{2} d^{4}+20 \sin \left (f x +e \right )^{3} c^{3} d^{3}+15 \sin \left (f x +e \right )^{2} c^{4} d^{2}+6 \sin \left (f x +e \right ) c^{5} d +c^{6}}d x \right ) \] Input:

int((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(11/2),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)**6*d**6 + 6*sin(e + f*x)**5*c*d**5 + 15*sin(e + f* 
x)**4*c**2*d**4 + 20*sin(e + f*x)**3*c**3*d**3 + 15*sin(e + f*x)**2*c**4*d 
**2 + 6*sin(e + f*x)*c**5*d + c**6),x) + 2*int((sqrt(sin(e + f*x)*d + c)*s 
qrt(sin(e + f*x) + 1)*sin(e + f*x))/(sin(e + f*x)**6*d**6 + 6*sin(e + f*x) 
**5*c*d**5 + 15*sin(e + f*x)**4*c**2*d**4 + 20*sin(e + f*x)**3*c**3*d**3 + 
 15*sin(e + f*x)**2*c**4*d**2 + 6*sin(e + f*x)*c**5*d + c**6),x) + int((sq 
rt(sin(e + f*x)*d + c)*sqrt(sin(e + f*x) + 1))/(sin(e + f*x)**6*d**6 + 6*s 
in(e + f*x)**5*c*d**5 + 15*sin(e + f*x)**4*c**2*d**4 + 20*sin(e + f*x)**3* 
c**3*d**3 + 15*sin(e + f*x)**2*c**4*d**2 + 6*sin(e + f*x)*c**5*d + c**6),x 
))