\(\int \frac {(c+d \sin (e+f x))^{5/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx\) [600]

Optimal result

Integrand size = 29, antiderivative size = 251 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=-\frac {2 d^{5/2} \arctan \left (\frac {\sqrt {3} \sqrt {d} \cos (e+f x)}{\sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{9 \sqrt {3} f}-\frac {\sqrt {c-d} \left (3 c^2+14 c d+43 d^2\right ) \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {c-d} \cos (e+f x)}{\sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{144 \sqrt {6} f}-\frac {(c-d) (3 c+11 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{48 f (3+3 \sin (e+f x))^{3/2}}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 f (3+3 \sin (e+f x))^{5/2}} \]

[Out]

-2*d^(5/2)*arctan(cos(f*x+e)*a^(1/2)*d^(1/2)/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2))/a^(5/2)/f-1/4*(c-d
)*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/f/(a+a*sin(f*x+e))^(5/2)-1/32*(3*c^2+14*c*d+43*d^2)*arctanh(1/2*cos(f*x+e)
*a^(1/2)*(c-d)^(1/2)*2^(1/2)/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2))*(c-d)^(1/2)/a^(5/2)/f*2^(1/2)-1/16
*(c-d)*(3*c+11*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/a/f/(a+a*sin(f*x+e))^(3/2)

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2844, 3056, 3061, 2861, 214, 2854, 211} \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=-\frac {2 d^{5/2} \arctan \left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{a^{5/2} f}-\frac {\sqrt {c-d} \left (3 c^2+14 c d+43 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} f}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 f (a \sin (e+f x)+a)^{5/2}}-\frac {(c-d) (3 c+11 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{16 a f (a \sin (e+f x)+a)^{3/2}} \]

[In]

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

[Out]

(-2*d^(5/2)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(a^(5/
2)*f) - (Sqrt[c - d]*(3*c^2 + 14*c*d + 43*d^2)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*
Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*f) - ((c - d)*(3*c + 11*d)*Cos[e + f*x]*Sqrt[c +
d*Sin[e + f*x]])/(16*a*f*(a + a*Sin[e + f*x])^(3/2)) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(4*f
*(a + a*Sin[e + f*x])^(5/2))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

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 2854

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[
-2*(b/f), Subst[Int[1/(b + d*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[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 2861

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> D
ist[-2*(a/f), Subst[Int[1/(2*b^2 - (a*c - b*d)*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[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 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 3061

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

Rubi steps \begin{align*} \text {integral}& = -\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (-\frac {1}{2} a (3 c-d) (c+3 d)-4 a d^2 \sin (e+f x)\right )}{(a+a \sin (e+f x))^{3/2}} \, dx}{4 a^2} \\ & = -\frac {(c-d) (3 c+11 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{16 a f (a+a \sin (e+f x))^{3/2}}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {\int \frac {-\frac {1}{4} a^2 (3 c-d) \left (c^2+4 c d+11 d^2\right )-8 a^2 d^3 \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{8 a^4} \\ & = -\frac {(c-d) (3 c+11 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{16 a f (a+a \sin (e+f x))^{3/2}}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 f (a+a \sin (e+f x))^{5/2}}+\frac {d^3 \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{a^3}+\frac {\left ((c-d) \left (3 c^2+14 c d+43 d^2\right )\right ) \int \frac {1}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{32 a^2} \\ & = -\frac {(c-d) (3 c+11 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{16 a f (a+a \sin (e+f x))^{3/2}}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {\left (2 d^3\right ) \text {Subst}\left (\int \frac {1}{a+d x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{a^2 f}-\frac {\left ((c-d) \left (3 c^2+14 c d+43 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{2 a^2-(a c-a d) x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{16 a f} \\ & = -\frac {2 d^{5/2} \arctan \left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{a^{5/2} f}-\frac {\sqrt {c-d} \left (3 c^2+14 c d+43 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} f}-\frac {(c-d) (3 c+11 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{16 a f (a+a \sin (e+f x))^{3/2}}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 f (a+a \sin (e+f x))^{5/2}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 15.40 (sec) , antiderivative size = 1845, normalized size of antiderivative = 7.35 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (-\frac {(c-d)^2}{4 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}-\frac {3 (c-d) (c+5 d)}{16 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}+\frac {3 \left (c^2 \sin \left (\frac {1}{2} (e+f x)\right )+4 c d \sin \left (\frac {1}{2} (e+f x)\right )-5 d^2 \sin \left (\frac {1}{2} (e+f x)\right )\right )}{8 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {c^2 \sin \left (\frac {1}{2} (e+f x)\right )-2 c d \sin \left (\frac {1}{2} (e+f x)\right )+d^2 \sin \left (\frac {1}{2} (e+f x)\right )}{2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}\right ) \sqrt {c+d \sin (e+f x)}}{f (3+3 \sin (e+f x))^{5/2}}+\frac {\left (\sqrt {2} \left (3 c^3+11 c^2 d+29 c d^2-43 d^3\right ) \log \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )-\sqrt {2} \left (3 c^3+11 c^2 d+29 c d^2-43 d^3\right ) \log \left (c-d+2 \sqrt {c-d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )-32 i \sqrt {c-d} d^{5/2} \left (\log \left (\frac {c-i \left (d+(1+i) \sqrt {2} \sqrt {d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}\right )+(-i c+d) \tan \left (\frac {1}{2} (e+f x)\right )}{16 d^{7/2} \left (i+\tan \left (\frac {1}{2} (e+f x)\right )\right )}\right )-\log \left (\frac {c+i d+(1+i) \sqrt {2} \sqrt {d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(i c+d) \tan \left (\frac {1}{2} (e+f x)\right )}{16 d^{7/2} \left (-i+\tan \left (\frac {1}{2} (e+f x)\right )\right )}\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\frac {3 c^3}{32 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c+d \sin (e+f x)}}+\frac {11 c^2 d}{32 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c+d \sin (e+f x)}}+\frac {29 c d^2}{32 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c+d \sin (e+f x)}}-\frac {11 d^3}{32 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c+d \sin (e+f x)}}+\frac {d^3 \sin (e+f x)}{\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c+d \sin (e+f x)}}\right )}{f (3+3 \sin (e+f x))^{5/2} \left (\frac {\left (3 c^3+11 c^2 d+29 c d^2-43 d^3\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right )}{\sqrt {2} \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )}-\frac {\sqrt {2} \left (3 c^3+11 c^2 d+29 c d^2-43 d^3\right ) \left (\frac {1}{2} (-c+d) \sec ^2\left (\frac {1}{2} (e+f x)\right )+\frac {\sqrt {c-d} d \cos (e+f x) \sqrt {\frac {1}{1+\cos (e+f x)}}}{\sqrt {c+d \sin (e+f x)}}+\sqrt {c-d} \left (\frac {1}{1+\cos (e+f x)}\right )^{3/2} \sin (e+f x) \sqrt {c+d \sin (e+f x)}\right )}{c-d+2 \sqrt {c-d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )}-32 i \sqrt {c-d} d^{5/2} \left (\frac {16 d^{7/2} \left (i+\tan \left (\frac {1}{2} (e+f x)\right )\right ) \left (\frac {\frac {1}{2} (-i c+d) \sec ^2\left (\frac {1}{2} (e+f x)\right )-i \left (\frac {(1+i) d^{3/2} \cos (e+f x) \sqrt {\frac {1}{1+\cos (e+f x)}}}{\sqrt {2} \sqrt {c+d \sin (e+f x)}}+\frac {(1+i) \sqrt {d} \left (\frac {1}{1+\cos (e+f x)}\right )^{3/2} \sin (e+f x) \sqrt {c+d \sin (e+f x)}}{\sqrt {2}}\right )}{16 d^{7/2} \left (i+\tan \left (\frac {1}{2} (e+f x)\right )\right )}-\frac {\sec ^2\left (\frac {1}{2} (e+f x)\right ) \left (c-i \left (d+(1+i) \sqrt {2} \sqrt {d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}\right )+(-i c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )}{32 d^{7/2} \left (i+\tan \left (\frac {1}{2} (e+f x)\right )\right )^2}\right )}{c-i \left (d+(1+i) \sqrt {2} \sqrt {d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}\right )+(-i c+d) \tan \left (\frac {1}{2} (e+f x)\right )}-\frac {16 d^{7/2} \left (-i+\tan \left (\frac {1}{2} (e+f x)\right )\right ) \left (\frac {\frac {1}{2} (i c+d) \sec ^2\left (\frac {1}{2} (e+f x)\right )+\frac {(1+i) d^{3/2} \cos (e+f x) \sqrt {\frac {1}{1+\cos (e+f x)}}}{\sqrt {2} \sqrt {c+d \sin (e+f x)}}+\frac {(1+i) \sqrt {d} \left (\frac {1}{1+\cos (e+f x)}\right )^{3/2} \sin (e+f x) \sqrt {c+d \sin (e+f x)}}{\sqrt {2}}}{16 d^{7/2} \left (-i+\tan \left (\frac {1}{2} (e+f x)\right )\right )}-\frac {\sec ^2\left (\frac {1}{2} (e+f x)\right ) \left (c+i d+(1+i) \sqrt {2} \sqrt {d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(i c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )}{32 d^{7/2} \left (-i+\tan \left (\frac {1}{2} (e+f x)\right )\right )^2}\right )}{c+i d+(1+i) \sqrt {2} \sqrt {d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(i c+d) \tan \left (\frac {1}{2} (e+f x)\right )}\right )\right )} \]

[In]

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

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(-1/4*(c - d)^2/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - (3*(c - d)*
(c + 5*d))/(16*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])) + (3*(c^2*Sin[(e + f*x)/2] + 4*c*d*Sin[(e + f*x)/2] - 5*
d^2*Sin[(e + f*x)/2]))/(8*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2) + (c^2*Sin[(e + f*x)/2] - 2*c*d*Sin[(e + f*
x)/2] + d^2*Sin[(e + f*x)/2])/(2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4))*Sqrt[c + d*Sin[e + f*x]])/(f*(3 + 3
*Sin[e + f*x])^(5/2)) + ((Sqrt[2]*(3*c^3 + 11*c^2*d + 29*c*d^2 - 43*d^3)*Log[1 + Tan[(e + f*x)/2]] - Sqrt[2]*(
3*c^3 + 11*c^2*d + 29*c*d^2 - 43*d^3)*Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e
+ f*x]] + (-c + d)*Tan[(e + f*x)/2]] - (32*I)*Sqrt[c - d]*d^(5/2)*(Log[(c - I*(d + (1 + I)*Sqrt[2]*Sqrt[d]*Sq
rt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]]) + ((-I)*c + d)*Tan[(e + f*x)/2])/(16*d^(7/2)*(I + Tan[(e
+ f*x)/2]))] - Log[(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]]
+ (I*c + d)*Tan[(e + f*x)/2])/(16*d^(7/2)*(-I + Tan[(e + f*x)/2]))]))*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*
((3*c^3)/(32*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) + (11*c^2*d)/(32*(Cos[(e + f*x)/2
] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) + (29*c*d^2)/(32*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c
+ d*Sin[e + f*x]]) - (11*d^3)/(32*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) + (d^3*Sin[e
+ f*x])/((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]])))/(f*(3 + 3*Sin[e + f*x])^(5/2)*(((3
*c^3 + 11*c^2*d + 29*c*d^2 - 43*d^3)*Sec[(e + f*x)/2]^2)/(Sqrt[2]*(1 + Tan[(e + f*x)/2])) - (Sqrt[2]*(3*c^3 +
11*c^2*d + 29*c*d^2 - 43*d^3)*(((-c + d)*Sec[(e + f*x)/2]^2)/2 + (Sqrt[c - d]*d*Cos[e + f*x]*Sqrt[(1 + Cos[e +
f*x])^(-1)])/Sqrt[c + d*Sin[e + f*x]] + Sqrt[c - d]*((1 + Cos[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*S
in[e + f*x]]))/(c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e
+ f*x)/2]) - (32*I)*Sqrt[c - d]*d^(5/2)*((16*d^(7/2)*(I + Tan[(e + f*x)/2])*(((((-I)*c + d)*Sec[(e + f*x)/2]^
2)/2 - I*(((1 + I)*d^(3/2)*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/(Sqrt[2]*Sqrt[c + d*Sin[e + f*x]]) + ((
1 + I)*Sqrt[d]*((1 + Cos[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/Sqrt[2]))/(16*d^(7/2)*(I
+ Tan[(e + f*x)/2])) - (Sec[(e + f*x)/2]^2*(c - I*(d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*
Sqrt[c + d*Sin[e + f*x]]) + ((-I)*c + d)*Tan[(e + f*x)/2]))/(32*d^(7/2)*(I + Tan[(e + f*x)/2])^2)))/(c - I*(d
+ (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]]) + ((-I)*c + d)*Tan[(e + f*x)
/2]) - (16*d^(7/2)*(-I + Tan[(e + f*x)/2])*((((I*c + d)*Sec[(e + f*x)/2]^2)/2 + ((1 + I)*d^(3/2)*Cos[e + f*x]*
Sqrt[(1 + Cos[e + f*x])^(-1)])/(Sqrt[2]*Sqrt[c + d*Sin[e + f*x]]) + ((1 + I)*Sqrt[d]*((1 + Cos[e + f*x])^(-1))
^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/Sqrt[2])/(16*d^(7/2)*(-I + Tan[(e + f*x)/2])) - (Sec[(e + f*x)/2
]^2*(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (I*c + d)*Tan[
(e + f*x)/2]))/(32*d^(7/2)*(-I + Tan[(e + f*x)/2])^2)))/(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f
*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (I*c + d)*Tan[(e + f*x)/2]))))

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(13852\) vs. \(2(219)=438\).

Time = 1.82 (sec) , antiderivative size = 13853, normalized size of antiderivative = 55.19

\[\text {output too large to display}\]

[In]

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

[Out]

result too large to display

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 712 vs. \(2 (219) = 438\).

Time = 0.85 (sec) , antiderivative size = 3855, normalized size of antiderivative = 15.36 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/32*(sqrt(1/2)*((3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e)^3 - 12*a*c^2 - 56*a*c*d - 172*a*d^2 + 3*(3*a*c^
2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e)^2 - 2*(3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e) - (12*a*c^2 + 56*a*c*
d + 172*a*d^2 - (3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e)^2 + 2*(3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e
))*sin(f*x + e))*sqrt((c - d)/a)*log((4*sqrt(1/2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt((c -
d)/a)*(cos(f*x + e) - sin(f*x + e) + 1) - (c - 3*d)*cos(f*x + e)^2 - (3*c - d)*cos(f*x + e) + ((c - 3*d)*cos(f
*x + e) - 2*c - 2*d)*sin(f*x + e) - 2*c - 2*d)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e
) - 2)) + 8*(a*d^2*cos(f*x + e)^3 + 3*a*d^2*cos(f*x + e)^2 - 2*a*d^2*cos(f*x + e) - 4*a*d^2 + (a*d^2*cos(f*x +
e)^2 - 2*a*d^2*cos(f*x + e) - 4*a*d^2)*sin(f*x + e))*sqrt(-d/a)*log((128*d^4*cos(f*x + e)^5 + 128*(2*c*d^3 -
d^4)*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 32*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x +
e)^3 - 32*(c^3*d - 2*c^2*d^2 + 9*c*d^3 - 4*d^4)*cos(f*x + e)^2 - 8*(16*d^3*cos(f*x + e)^4 + 24*(c*d^2 - d^3)*
cos(f*x + e)^3 - c^3 + 17*c^2*d - 59*c*d^2 + 51*d^3 - 2*(5*c^2*d - 26*c*d^2 + 33*d^3)*cos(f*x + e)^2 - (c^3 -
7*c^2*d + 31*c*d^2 - 25*d^3)*cos(f*x + e) + (16*d^3*cos(f*x + e)^3 + c^3 - 17*c^2*d + 59*c*d^2 - 51*d^3 - 8*(3
*c*d^2 - 5*d^3)*cos(f*x + e)^2 - 2*(5*c^2*d - 14*c*d^2 + 13*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x +
e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-d/a) + (c^4 - 28*c^3*d + 230*c^2*d^2 - 476*c*d^3 + 289*d^4)*cos(f*x + e
) + (128*d^4*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 256*(c*d^3 - d^4)*cos(f*x + e)^3 - 3
2*(5*c^2*d^2 - 6*c*d^3 + 5*d^4)*cos(f*x + e)^2 + 32*(c^3*d - 7*c^2*d^2 + 15*c*d^3 - 9*d^4)*cos(f*x + e))*sin(f
*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) + 2*(3*(c^2 + 4*c*d - 5*d^2)*cos(f*x + e)^2 + 4*c^2 - 8*c*d + 4*d^
2 + (7*c^2 + 4*c*d - 11*d^2)*cos(f*x + e) - (4*c^2 - 8*c*d + 4*d^2 - 3*(c^2 + 4*c*d - 5*d^2)*cos(f*x + e))*sin
(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 -
2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e)), 1/32*
(sqrt(1/2)*((3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e)^3 - 12*a*c^2 - 56*a*c*d - 172*a*d^2 + 3*(3*a*c^2 + 14
*a*c*d + 43*a*d^2)*cos(f*x + e)^2 - 2*(3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e) - (12*a*c^2 + 56*a*c*d + 17
2*a*d^2 - (3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e)^2 + 2*(3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e))*sin
(f*x + e))*sqrt((c - d)/a)*log((4*sqrt(1/2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt((c - d)/a)*
(cos(f*x + e) - sin(f*x + e) + 1) - (c - 3*d)*cos(f*x + e)^2 - (3*c - d)*cos(f*x + e) + ((c - 3*d)*cos(f*x + e
) - 2*c - 2*d)*sin(f*x + e) - 2*c - 2*d)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)
) + 16*(a*d^2*cos(f*x + e)^3 + 3*a*d^2*cos(f*x + e)^2 - 2*a*d^2*cos(f*x + e) - 4*a*d^2 + (a*d^2*cos(f*x + e)^2
- 2*a*d^2*cos(f*x + e) - 4*a*d^2)*sin(f*x + e))*sqrt(d/a)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*
d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(d/a)/(2*d^3*cos(f*x +
e)^3 - (3*c*d^2 - d^3)*cos(f*x + e)*sin(f*x + e) - (c^2*d - c*d^2 + 2*d^3)*cos(f*x + e))) + 2*(3*(c^2 + 4*c*d
- 5*d^2)*cos(f*x + e)^2 + 4*c^2 - 8*c*d + 4*d^2 + (7*c^2 + 4*c*d - 11*d^2)*cos(f*x + e) - (4*c^2 - 8*c*d + 4*
d^2 - 3*(c^2 + 4*c*d - 5*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(
a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3
*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e)), -1/16*(sqrt(1/2)*((3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e)^3 - 1
2*a*c^2 - 56*a*c*d - 172*a*d^2 + 3*(3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e)^2 - 2*(3*a*c^2 + 14*a*c*d + 43
*a*d^2)*cos(f*x + e) - (12*a*c^2 + 56*a*c*d + 172*a*d^2 - (3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e)^2 + 2*(
3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(-(c - d)/a)*arctan(-2*sqrt(1/2)*sqrt(a*sin(f*x
+ e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-(c - d)/a)/((c - d)*cos(f*x + e))) - 4*(a*d^2*cos(f*x + e)^3 + 3*a*d
^2*cos(f*x + e)^2 - 2*a*d^2*cos(f*x + e) - 4*a*d^2 + (a*d^2*cos(f*x + e)^2 - 2*a*d^2*cos(f*x + e) - 4*a*d^2)*s
in(f*x + e))*sqrt(-d/a)*log((128*d^4*cos(f*x + e)^5 + 128*(2*c*d^3 - d^4)*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c
^2*d^2 + 4*c*d^3 + d^4 - 32*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e)^3 - 32*(c^3*d - 2*c^2*d^2 + 9*c*d^3 -
4*d^4)*cos(f*x + e)^2 - 8*(16*d^3*cos(f*x + e)^4 + 24*(c*d^2 - d^3)*cos(f*x + e)^3 - c^3 + 17*c^2*d - 59*c*d^
2 + 51*d^3 - 2*(5*c^2*d - 26*c*d^2 + 33*d^3)*cos(f*x + e)^2 - (c^3 - 7*c^2*d + 31*c*d^2 - 25*d^3)*cos(f*x + e)
+ (16*d^3*cos(f*x + e)^3 + c^3 - 17*c^2*d + 59*c*d^2 - 51*d^3 - 8*(3*c*d^2 - 5*d^3)*cos(f*x + e)^2 - 2*(5*c^2
*d - 14*c*d^2 + 13*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-d/
a) + (c^4 - 28*c^3*d + 230*c^2*d^2 - 476*c*d^3 + 289*d^4)*cos(f*x + e) + (128*d^4*cos(f*x + e)^4 + c^4 + 4*c^3
*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 256*(c*d^3 - d^4)*cos(f*x + e)^3 - 32*(5*c^2*d^2 - 6*c*d^3 + 5*d^4)*cos(f*x +
e)^2 + 32*(c^3*d - 7*c^2*d^2 + 15*c*d^3 - 9*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1
)) - (3*(c^2 + 4*c*d - 5*d^2)*cos(f*x + e)^2 + 4*c^2 - 8*c*d + 4*d^2 + (7*c^2 + 4*c*d - 11*d^2)*cos(f*x + e) -
(4*c^2 - 8*c*d + 4*d^2 - 3*(c^2 + 4*c*d - 5*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*
sin(f*x + e) + c))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*co
s(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e)), -1/16*(sqrt(1/2)*((3*a*c^2 + 14*a*c*d + 43*a*d^2
)*cos(f*x + e)^3 - 12*a*c^2 - 56*a*c*d - 172*a*d^2 + 3*(3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e)^2 - 2*(3*a
*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e) - (12*a*c^2 + 56*a*c*d + 172*a*d^2 - (3*a*c^2 + 14*a*c*d + 43*a*d^2)*
cos(f*x + e)^2 + 2*(3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(-(c - d)/a)*arctan(-2*sqrt
(1/2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-(c - d)/a)/((c - d)*cos(f*x + e))) - 8*(a*d^2*co
s(f*x + e)^3 + 3*a*d^2*cos(f*x + e)^2 - 2*a*d^2*cos(f*x + e) - 4*a*d^2 + (a*d^2*cos(f*x + e)^2 - 2*a*d^2*cos(f
*x + e) - 4*a*d^2)*sin(f*x + e))*sqrt(d/a)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d
^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(d/a)/(2*d^3*cos(f*x + e)^3 - (3*c*d^2
- d^3)*cos(f*x + e)*sin(f*x + e) - (c^2*d - c*d^2 + 2*d^3)*cos(f*x + e))) - (3*(c^2 + 4*c*d - 5*d^2)*cos(f*x
+ e)^2 + 4*c^2 - 8*c*d + 4*d^2 + (7*c^2 + 4*c*d - 11*d^2)*cos(f*x + e) - (4*c^2 - 8*c*d + 4*d^2 - 3*(c^2 + 4*c
*d - 5*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(a^3*f*cos(f*x + e)
^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) -
4*a^3*f)*sin(f*x + e))]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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