\(\int \frac {(c+d \sin (e+f x))^{5/2}}{(3+3 \sin (e+f x))^3} \, dx\) [516]

Optimal result

Integrand size = 27, antiderivative size = 309 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+3 \sin (e+f x))^3} \, dx=-\frac {2 (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{45 f (3+3 \sin (e+f x))^2}-\frac {\left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{30 f (27+27 \sin (e+f x))}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (3+3 \sin (e+f x))^3}-\frac {\left (4 c^2+15 c d+27 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{810 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(c+d) \left (4 c^2+11 c d+15 d^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{810 f \sqrt {c+d \sin (e+f x)}} \]

[Out]

-1/5*(c-d)*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/f/(a+a*sin(f*x+e))^3-2/15*(c-d)*(c+3*d)*cos(f*x+e)*(c+d*sin(f*x+e
))^(1/2)/a/f/(a+a*sin(f*x+e))^2-1/30*(4*c^2+15*c*d+27*d^2)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/f/(a^3+a^3*sin(f*
x+e))+1/30*(4*c^2+15*c*d+27*d^2)*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*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^3/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-1/30
*(c+d)*(4*c^2+11*c*d+15*d^2)*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticF(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))/(c+d))^(1/2)/a^3/f/(c+d*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2844, 3056, 3057, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+3 \sin (e+f x))^3} \, dx=-\frac {\left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{30 f \left (a^3 \sin (e+f x)+a^3\right )}+\frac {(c+d) \left (4 c^2+11 c d+15 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{30 a^3 f \sqrt {c+d \sin (e+f x)}}-\frac {\left (4 c^2+15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{30 a^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a \sin (e+f x)+a)^3}-\frac {2 (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 a f (a \sin (e+f x)+a)^2} \]

[In]

Int[(c + d*Sin[e + f*x])^(5/2)/(a + a*Sin[e + f*x])^3,x]

[Out]

(-2*(c - d)*(c + 3*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(15*a*f*(a + a*Sin[e + f*x])^2) - ((4*c^2 + 15*c*
d + 27*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(30*f*(a^3 + a^3*Sin[e + f*x])) - ((c - d)*Cos[e + f*x]*(c
+ d*Sin[e + f*x])^(3/2))/(5*f*(a + a*Sin[e + f*x])^3) - ((4*c^2 + 15*c*d + 27*d^2)*EllipticE[(e - Pi/2 + f*x)/
2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(30*a^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((c + d)*(4*c^2 +
11*c*d + 15*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(30*a^3*f*Sq
rt[c + d*Sin[e + f*x]])

Rule 2732

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 2734

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[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 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/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 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[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 2831

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[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 2844

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(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] + Dist[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 3056

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(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] - Dist[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] /; 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
[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])

Rule 3057

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

Rubi steps \begin{align*} \text {integral}& = -\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a+a \sin (e+f x))^3}-\frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (-\frac {1}{2} a \left (4 c^2+9 c d-3 d^2\right )-\frac {1}{2} a d (c+9 d) \sin (e+f x)\right )}{(a+a \sin (e+f x))^2} \, dx}{5 a^2} \\ & = -\frac {2 (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 a f (a+a \sin (e+f x))^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a+a \sin (e+f x))^3}-\frac {\int \frac {-\frac {1}{2} a^2 \left (4 c^3+13 c^2 d+19 c d^2-6 d^3\right )-\frac {1}{2} a^2 d \left (2 c^2+7 c d+21 d^2\right ) \sin (e+f x)}{(a+a \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{15 a^4} \\ & = -\frac {2 (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 a f (a+a \sin (e+f x))^2}-\frac {\left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{30 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a+a \sin (e+f x))^3}+\frac {\int \frac {-\frac {1}{4} a^3 (c-15 d) (c-d) d^2-\frac {1}{4} a^3 (c-d) d \left (4 c^2+15 c d+27 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{15 a^6 (c-d)} \\ & = -\frac {2 (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 a f (a+a \sin (e+f x))^2}-\frac {\left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{30 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a+a \sin (e+f x))^3}+\frac {\left ((c+d) \left (4 c^2+11 c d+15 d^2\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{60 a^3}-\frac {\left (4 c^2+15 c d+27 d^2\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{60 a^3} \\ & = -\frac {2 (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 a f (a+a \sin (e+f x))^2}-\frac {\left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{30 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a+a \sin (e+f x))^3}-\frac {\left (\left (4 c^2+15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{60 a^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left ((c+d) \left (4 c^2+11 c d+15 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{60 a^3 \sqrt {c+d \sin (e+f x)}} \\ & = -\frac {2 (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 a f (a+a \sin (e+f x))^2}-\frac {\left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{30 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a+a \sin (e+f x))^3}-\frac {\left (4 c^2+15 c d+27 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)}}{30 a^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(c+d) \left (4 c^2+11 c d+15 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}}}{30 a^3 f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.12 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.24 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+3 \sin (e+f x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 \left (-\left (\left (4 c^2+15 c d+27 d^2\right ) (c+d \sin (e+f x))\right )-\frac {\left (-2 d (35 c+57 d) \cos \left (\frac {1}{2} (e+f x)\right )+\left (20 c^2+74 c d+90 d^2\right ) \cos \left (\frac {3}{2} (e+f x)\right )+2 \left (-3 \left (6 c^2+11 c d+29 d^2\right )+2 \left (2 c^2+7 c d-9 d^2\right ) \cos (e+f x)+\left (4 c^2+15 c d+27 d^2\right ) \cos (2 (e+f x))\right ) \sin \left (\frac {1}{2} (e+f x)\right )\right ) (c+d \sin (e+f x))}{2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}+(c-15 d) d^2 \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+\left (4 c^2+15 c d+27 d^2\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right )}{810 f (1+\sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}} \]

[In]

Integrate[(c + d*Sin[e + f*x])^(5/2)/(3 + 3*Sin[e + f*x])^3,x]

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(-((4*c^2 + 15*c*d + 27*d^2)*(c + d*Sin[e + f*x])) - ((-2*d*(35*c + 5
7*d)*Cos[(e + f*x)/2] + (20*c^2 + 74*c*d + 90*d^2)*Cos[(3*(e + f*x))/2] + 2*(-3*(6*c^2 + 11*c*d + 29*d^2) + 2*
(2*c^2 + 7*c*d - 9*d^2)*Cos[e + f*x] + (4*c^2 + 15*c*d + 27*d^2)*Cos[2*(e + f*x)])*Sin[(e + f*x)/2])*(c + d*Si
n[e + f*x]))/(2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) + (c - 15*d)*d^2*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*
d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + (4*c^2 + 15*c*d + 27*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)])
)/(810*f*(1 + Sin[e + f*x])^3*Sqrt[c + d*Sin[e + f*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1614\) vs. \(2(364)=728\).

Time = 175.87 (sec) , antiderivative size = 1615, normalized size of antiderivative = 5.23

method	result	size
default	\(\text {Expression too large to display}\)	\(1615\)

[In]

int((c+d*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/a^3*(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)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e
))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+3*d^2*(c-d)*(-(-d*sin(f*x+e)^2-c*sin(f*x+e)+d*sin(f*x+e)+c)/(c-d)/((sin(f
*x+e)+1)*(sin(f*x+e)-1)*(-d*sin(f*x+e)-c))^(1/2)-2*d/(2*c-2*d)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-si
n(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c
+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-d/(c-d)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+
e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*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))))+3*d*(c^2-2*c*d+d^2)*(-1/3/(c-d)*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+1)^2-1/3*(-d*sin(f*
x+e)^2-c*sin(f*x+e)+d*sin(f*x+e)+c)/(c-d)^2*(c-3*d)/((sin(f*x+e)+1)*(sin(f*x+e)-1)*(-d*sin(f*x+e)-c))^(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)*(1/(c-d)*(-sin(f
*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*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*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)
*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*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)*(-1/5/(c-d)*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+1)^3-2/15*(c-3*d)/(c-d)^2*(-(-d*sin(f*
x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+1)^2-1/30*(-d*sin(f*x+e)^2-c*sin(f*x+e)+d*sin(f*x+e)+c)/(c-d)^3*(4*c^2
-15*c*d+27*d^2)/((sin(f*x+e)+1)*(sin(f*x+e)-1)*(-d*sin(f*x+e)-c))^(1/2)+2*(-c*d^2-15*d^3)/(60*c^3-180*c^2*d+18
0*c*d^2-60*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)*(1/(c-d)*(-sin(f*x+e)-1)
*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)
)-1/30*d*(4*c^2-15*c*d+27*d^2)/(c-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)*(
1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*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)))))/cos(f*x+e)
/(c+d*sin(f*x+e))^(1/2)/f

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.15 (sec) , antiderivative size = 1513, normalized size of antiderivative = 4.90 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+3 \sin (e+f x))^3} \, dx=\text {Too large to display} \]

[In]

integrate((c+d*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

1/180*((sqrt(2)*(8*c^3 + 30*c^2*d + 51*c*d^2 + 45*d^3)*cos(f*x + e)^3 + 3*sqrt(2)*(8*c^3 + 30*c^2*d + 51*c*d^2
+ 45*d^3)*cos(f*x + e)^2 - 2*sqrt(2)*(8*c^3 + 30*c^2*d + 51*c*d^2 + 45*d^3)*cos(f*x + e) + (sqrt(2)*(8*c^3 +
30*c^2*d + 51*c*d^2 + 45*d^3)*cos(f*x + e)^2 - 2*sqrt(2)*(8*c^3 + 30*c^2*d + 51*c*d^2 + 45*d^3)*cos(f*x + e) -
4*sqrt(2)*(8*c^3 + 30*c^2*d + 51*c*d^2 + 45*d^3))*sin(f*x + e) - 4*sqrt(2)*(8*c^3 + 30*c^2*d + 51*c*d^2 + 45*
d^3))*sqrt(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) + (sqrt(2)*(8*c^3 + 30*c^2*d + 51*c*d^2 + 45*d^3)*cos(f*x + e)^3 + 3*s
qrt(2)*(8*c^3 + 30*c^2*d + 51*c*d^2 + 45*d^3)*cos(f*x + e)^2 - 2*sqrt(2)*(8*c^3 + 30*c^2*d + 51*c*d^2 + 45*d^3
)*cos(f*x + e) + (sqrt(2)*(8*c^3 + 30*c^2*d + 51*c*d^2 + 45*d^3)*cos(f*x + e)^2 - 2*sqrt(2)*(8*c^3 + 30*c^2*d
+ 51*c*d^2 + 45*d^3)*cos(f*x + e) - 4*sqrt(2)*(8*c^3 + 30*c^2*d + 51*c*d^2 + 45*d^3))*sin(f*x + e) - 4*sqrt(2)
*(8*c^3 + 30*c^2*d + 51*c*d^2 + 45*d^3))*sqrt(-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*(sqrt(2)*(-4*I*c^2*d - 15*I*c
*d^2 - 27*I*d^3)*cos(f*x + e)^3 + 3*sqrt(2)*(-4*I*c^2*d - 15*I*c*d^2 - 27*I*d^3)*cos(f*x + e)^2 + 2*sqrt(2)*(4
*I*c^2*d + 15*I*c*d^2 + 27*I*d^3)*cos(f*x + e) + (sqrt(2)*(-4*I*c^2*d - 15*I*c*d^2 - 27*I*d^3)*cos(f*x + e)^2
+ 2*sqrt(2)*(4*I*c^2*d + 15*I*c*d^2 + 27*I*d^3)*cos(f*x + e) + 4*sqrt(2)*(4*I*c^2*d + 15*I*c*d^2 + 27*I*d^3))*
sin(f*x + e) + 4*sqrt(2)*(4*I*c^2*d + 15*I*c*d^2 + 27*I*d^3))*sqrt(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*(sqrt(2)*(4*I*c^2*d + 15*I*c*d^2 + 27*I*d^3)
*cos(f*x + e)^3 + 3*sqrt(2)*(4*I*c^2*d + 15*I*c*d^2 + 27*I*d^3)*cos(f*x + e)^2 + 2*sqrt(2)*(-4*I*c^2*d - 15*I*
c*d^2 - 27*I*d^3)*cos(f*x + e) + (sqrt(2)*(4*I*c^2*d + 15*I*c*d^2 + 27*I*d^3)*cos(f*x + e)^2 + 2*sqrt(2)*(-4*I
*c^2*d - 15*I*c*d^2 - 27*I*d^3)*cos(f*x + e) + 4*sqrt(2)*(-4*I*c^2*d - 15*I*c*d^2 - 27*I*d^3))*sin(f*x + e) +
4*sqrt(2)*(-4*I*c^2*d - 15*I*c*d^2 - 27*I*d^3))*sqrt(-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)) - 6*((4*c^2*d + 15*c*d^2 + 27*d^3)*cos(f*x + e)^3 - 6*c^2*d
+ 12*c*d^2 - 6*d^3 - (8*c^2*d + 29*c*d^2 + 9*d^3)*cos(f*x + e)^2 - 2*(9*c^2*d + 16*c*d^2 + 21*d^3)*cos(f*x +
e) + (6*c^2*d - 12*c*d^2 + 6*d^3 - (4*c^2*d + 15*c*d^2 + 27*d^3)*cos(f*x + e)^2 - 4*(3*c^2*d + 11*c*d^2 + 9*d^
3)*cos(f*x + e))*sin(f*x + e))*sqrt(d*sin(f*x + e) + c))/(a^3*d*f*cos(f*x + e)^3 + 3*a^3*d*f*cos(f*x + e)^2 -
2*a^3*d*f*cos(f*x + e) - 4*a^3*d*f + (a^3*d*f*cos(f*x + e)^2 - 2*a^3*d*f*cos(f*x + e) - 4*a^3*d*f)*sin(f*x + e
))

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+3 \sin (e+f x))^3} \, dx=\text {Timed out} \]

[In]

integrate((c+d*sin(f*x+e))**(5/2)/(a+a*sin(f*x+e))**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+3 \sin (e+f x))^3} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate((c+d*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^3,x, algorithm="maxima")

[Out]

integrate((d*sin(f*x + e) + c)^(5/2)/(a*sin(f*x + e) + a)^3, x)

Giac [F]

\[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+3 \sin (e+f x))^3} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate((c+d*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^3,x, algorithm="giac")

[Out]

integrate((d*sin(f*x + e) + c)^(5/2)/(a*sin(f*x + e) + a)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+3 \sin (e+f x))^3} \, dx=\int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3} \,d x \]

[In]

int((c + d*sin(e + f*x))^(5/2)/(a + a*sin(e + f*x))^3,x)

[Out]

int((c + d*sin(e + f*x))^(5/2)/(a + a*sin(e + f*x))^3, x)