\(\int \frac {1}{(3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx\) [563]

Optimal result

Integrand size = 27, antiderivative size = 383 \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=-\frac {\left (c^2-10 c d+73 d^2\right ) \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} \cos (e+f x)}{\sqrt {3+3 \sin (e+f x)}}\right )}{48 \sqrt {6} (c-d)^5 f}+\frac {d^{5/2} \left (21 c^2+30 c d+13 d^2\right ) \text {arctanh}\left (\frac {\sqrt {3} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {3+3 \sin (e+f x)}}\right )}{12 \sqrt {3} (c-d)^5 (c+d)^{5/2} f}-\frac {\cos (e+f x)}{4 (c-d) f (3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac {(3 c-19 d) \cos (e+f x)}{48 (c-d)^2 f (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}-\frac {d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{144 (c-d)^3 (c+d) f \sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac {d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{48 (c-d)^4 (c+d)^2 f \sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))} \]

[Out]

3/4*d^(5/2)*(21*c^2+30*c*d+13*d^2)*arctanh(cos(f*x+e)*a^(1/2)*d^(1/2)/(c+d)^(1/2)/(a+a*sin(f*x+e))^(1/2))/a^(5
/2)/(c-d)^5/(c+d)^(5/2)/f-1/4*cos(f*x+e)/(c-d)/f/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^2-1/16*(3*c-19*d)*cos
(f*x+e)/a/(c-d)^2/f/(a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^2-3/32*(c^2-10*c*d+73*d^2)*arctanh(1/2*cos(f*x+e)*
a^(1/2)*2^(1/2)/(a+a*sin(f*x+e))^(1/2))/a^(5/2)/(c-d)^5/f*2^(1/2)-1/16*d*(3*c^2-20*c*d-31*d^2)*cos(f*x+e)/a^2/
(c-d)^3/(c+d)/f/(c+d*sin(f*x+e))^2/(a+a*sin(f*x+e))^(1/2)-3/16*d*(c+3*d)*(c^2-10*c*d-7*d^2)*cos(f*x+e)/a^2/(c-
d)^4/(c+d)^2/f/(c+d*sin(f*x+e))/(a+a*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 1.05 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2845, 3057, 3063, 3064, 2728, 212, 2852, 214} \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=-\frac {3 \left (c^2-10 c d+73 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{16 \sqrt {2} a^{5/2} f (c-d)^5}+\frac {3 d^{5/2} \left (21 c^2+30 c d+13 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{4 a^{5/2} f (c-d)^5 (c+d)^{5/2}}-\frac {3 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{16 a^2 f (c-d)^4 (c+d)^2 \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac {d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{16 a^2 f (c-d)^3 (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2}-\frac {(3 c-19 d) \cos (e+f x)}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}-\frac {\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2} \]

[In]

Int[1/((a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^3),x]

[Out]

(-3*(c^2 - 10*c*d + 73*d^2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(16*Sqrt[2]*a^
(5/2)*(c - d)^5*f) + (3*d^(5/2)*(21*c^2 + 30*c*d + 13*d^2)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]
*Sqrt[a + a*Sin[e + f*x]])])/(4*a^(5/2)*(c - d)^5*(c + d)^(5/2)*f) - Cos[e + f*x]/(4*(c - d)*f*(a + a*Sin[e +
f*x])^(5/2)*(c + d*Sin[e + f*x])^2) - ((3*c - 19*d)*Cos[e + f*x])/(16*a*(c - d)^2*f*(a + a*Sin[e + f*x])^(3/2)
*(c + d*Sin[e + f*x])^2) - (d*(3*c^2 - 20*c*d - 31*d^2)*Cos[e + f*x])/(16*a^2*(c - d)^3*(c + d)*f*Sqrt[a + a*S
in[e + f*x]]*(c + d*Sin[e + f*x])^2) - (3*d*(c + 3*d)*(c^2 - 10*c*d - 7*d^2)*Cos[e + f*x])/(16*a^2*(c - d)^4*(
c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 2728

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, b*(C
os[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2845

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[b^2*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] + Dis
t[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*c*(m + 1) - a*d*
(2*m + n + 2) + b*d*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d,
0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] &&  !GtQ[n, 0] && (IntegersQ[2*m, 2*n] || (IntegerQ
[m] && EqQ[c, 0]))

Rule 2852

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

Rule 3063

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*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x]
)^(n + 1)/(f*(n + 1)*(c^2 - d^2))), x] + Dist[1/(b*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin
[e + f*x])^(n + 1)*Simp[A*(a*d*m + b*c*(n + 1)) - B*(a*c*m + b*d*(n + 1)) + b*(B*c - A*d)*(m + n + 2)*Sin[e +
f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2
- d^2, 0] && LtQ[n, -1] && (IntegerQ[n] || EqQ[m + 1/2, 0])

Rule 3064

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[(
B*c - A*d)/(b*c - a*d), Int[Sqrt[a + b*Sin[e + f*x]]/(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 {\cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac {\int \frac {-\frac {3}{2} a (c-4 d)-\frac {7}{2} a d \sin (e+f x)}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3} \, dx}{4 a^2 (c-d)} \\ & = -\frac {\cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac {(3 c-19 d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}+\frac {\int \frac {\frac {1}{4} a^2 \left (3 c^2-15 c d+124 d^2\right )+\frac {5}{4} a^2 (3 c-19 d) d \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3} \, dx}{8 a^4 (c-d)^2} \\ & = -\frac {\cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac {(3 c-19 d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}-\frac {d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{16 a^2 (c-d)^3 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac {\int \frac {-\frac {3}{2} a^3 \left (c^3-6 c^2 d+43 c d^2+42 d^3\right )-\frac {3}{2} a^3 d \left (3 c^2-20 c d-31 d^2\right ) \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2} \, dx}{16 a^5 (c-d)^3 (c+d)} \\ & = -\frac {\cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac {(3 c-19 d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}-\frac {d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{16 a^2 (c-d)^3 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac {3 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{16 a^2 (c-d)^4 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))}+\frac {\int \frac {\frac {3}{2} a^4 \left (c^4-7 c^3 d+47 c^2 d^2+99 c d^3+52 d^4\right )+\frac {3}{2} a^4 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))} \, dx}{16 a^6 (c-d)^4 (c+d)^2} \\ & = -\frac {\cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac {(3 c-19 d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}-\frac {d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{16 a^2 (c-d)^3 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac {3 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{16 a^2 (c-d)^4 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))}-\frac {\left (3 d^3 \left (21 c^2+30 c d+13 d^2\right )\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx}{8 a^3 (c-d)^5 (c+d)^2}+\frac {\left (3 \left (c^2-10 c d+73 d^2\right )\right ) \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx}{32 a^2 (c-d)^5} \\ & = -\frac {\cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac {(3 c-19 d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}-\frac {d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{16 a^2 (c-d)^3 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac {3 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{16 a^2 (c-d)^4 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))}+\frac {\left (3 d^3 \left (21 c^2+30 c d+13 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{a c+a d-d x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{4 a^2 (c-d)^5 (c+d)^2 f}-\frac {\left (3 \left (c^2-10 c d+73 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{16 a^2 (c-d)^5 f} \\ & = -\frac {3 \left (c^2-10 c d+73 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} (c-d)^5 f}+\frac {3 d^{5/2} \left (21 c^2+30 c d+13 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{4 a^{5/2} (c-d)^5 (c+d)^{5/2} f}-\frac {\cos (e+f x)}{4 (c-d) f (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2}-\frac {(3 c-19 d) \cos (e+f x)}{16 a (c-d)^2 f (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2}-\frac {d \left (3 c^2-20 c d-31 d^2\right ) \cos (e+f x)}{16 a^2 (c-d)^3 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac {3 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{16 a^2 (c-d)^4 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 14.00 (sec) , antiderivative size = 1048, normalized size of antiderivative = 2.74 \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} (c+d \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 ) \left (\frac {8 \sin \left (\frac {1}{2} (e+f x)\right )}{(c-d)^3}-\frac {4 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}{(c-d)^3}+\frac {6 (c-9 d) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}{(c-d)^4}-\frac {3 (c-9 d) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}{(c-d)^4}+\frac {(3+3 i) (-1)^{3/4} \left (c^2-10 c d+73 d^2\right ) \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}{(c-d)^5}+\frac {3 d^{5/2} \left (21 c^2+30 c d+13 d^2\right ) \left (e+f x-2 \log \left (\sec ^2\left (\frac {1}{4} (e+f x)\right )\right )+\text {RootSum}\left [c+4 d \text {$\#$1}+2 c \text {$\#$1}^2-4 d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {-d \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right )+\sqrt {d} \sqrt {c+d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right )-c \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}+2 \sqrt {d} \sqrt {c+d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}+3 d \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}^2-\sqrt {d} \sqrt {c+d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}^2-c \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}^3}{-d-c \text {$\#$1}+3 d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}{(c-d)^5 (c+d)^{5/2}}+\frac {3 d^{5/2} \left (21 c^2+30 c d+13 d^2\right ) \left (e+f x-2 \log \left (\sec ^2\left (\frac {1}{4} (e+f x)\right )\right )+\text {RootSum}\left [c+4 d \text {$\#$1}+2 c \text {$\#$1}^2-4 d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {-d \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right )-\sqrt {d} \sqrt {c+d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right )-c \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}-2 \sqrt {d} \sqrt {c+d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}+3 d \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}^2+\sqrt {d} \sqrt {c+d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}^2-c \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}^3}{-d-c \text {$\#$1}+3 d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}{(-c+d)^5 (c+d)^{5/2}}+\frac {8 d^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}{(c-d)^3 (c+d) (c+d \sin (e+f x))^2}+\frac {12 d^3 (5 c+3 d) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}{(c-d)^4 (c+d)^2 (c+d \sin (e+f x))}\right )}{144 \sqrt {3} f (1+\sin (e+f x))^{5/2}} \]

[In]

Integrate[1/((3 + 3*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^3),x]

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*((8*Sin[(e + f*x)/2])/(c - d)^3 - (4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/
2]))/(c - d)^3 + (6*(c - 9*d)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)/(c - d)^4 - (3*(c - 9*
d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)/(c - d)^4 + ((3 + 3*I)*(-1)^(3/4)*(c^2 - 10*c*d + 73*d^2)*ArcTanh[
(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)/(c - d)^5 + (3*d^(5/2
)*(21*c^2 + 30*c*d + 13*d^2)*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + RootSum[c + 4*d*#1 + 2*c*#1^2 - 4*d*#1^3 +
c*#1^4 & , (-(d*Log[-#1 + Tan[(e + f*x)/4]]) + Sqrt[d]*Sqrt[c + d]*Log[-#1 + Tan[(e + f*x)/4]] - c*Log[-#1 +
Tan[(e + f*x)/4]]*#1 + 2*Sqrt[d]*Sqrt[c + d]*Log[-#1 + Tan[(e + f*x)/4]]*#1 + 3*d*Log[-#1 + Tan[(e + f*x)/4]]*
#1^2 - Sqrt[d]*Sqrt[c + d]*Log[-#1 + Tan[(e + f*x)/4]]*#1^2 - c*Log[-#1 + Tan[(e + f*x)/4]]*#1^3)/(-d - c*#1 +
3*d*#1^2 - c*#1^3) & ])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)/((c - d)^5*(c + d)^(5/2)) + (3*d^(5/2)*(21*c
^2 + 30*c*d + 13*d^2)*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + RootSum[c + 4*d*#1 + 2*c*#1^2 - 4*d*#1^3 + c*#1^4
& , (-(d*Log[-#1 + Tan[(e + f*x)/4]]) - Sqrt[d]*Sqrt[c + d]*Log[-#1 + Tan[(e + f*x)/4]] - c*Log[-#1 + Tan[(e
+ f*x)/4]]*#1 - 2*Sqrt[d]*Sqrt[c + d]*Log[-#1 + Tan[(e + f*x)/4]]*#1 + 3*d*Log[-#1 + Tan[(e + f*x)/4]]*#1^2 +
Sqrt[d]*Sqrt[c + d]*Log[-#1 + Tan[(e + f*x)/4]]*#1^2 - c*Log[-#1 + Tan[(e + f*x)/4]]*#1^3)/(-d - c*#1 + 3*d*#1
^2 - c*#1^3) & ])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)/((-c + d)^5*(c + d)^(5/2)) + (8*d^3*(Cos[(e + f*x)/
2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)/((c - d)^3*(c + d)*(c + d*Sin[e + f*x])^2) + (
12*d^3*(5*c + 3*d)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)/((c - d)^4*(
c + d)^2*(c + d*Sin[e + f*x]))))/(144*Sqrt[3]*f*(1 + Sin[e + f*x])^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3534\) vs. \(2(357)=714\).

Time = 2.88 (sec) , antiderivative size = 3535, normalized size of antiderivative = 9.23

method	result	size
default	\(\text {Expression too large to display}\)	\(3535\)

[In]

int(1/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

-1/32*(-a*(sin(f*x+e)-1))^(1/2)*(-24*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d
)^(1/2)*sin(f*x+e)^4*a^2*c^3*d^3-42*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)
^(1/2)*sin(f*x+e)*a^2*c^5*d+276*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/
2)*sin(f*x+e)*a^2*c^4*d^2+1140*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2
)*sin(f*x+e)*a^2*c^3*d^3+1254*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)
*sin(f*x+e)*a^2*c^2*d^4+438*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*s
in(f*x+e)*a^2*c*d^5+162*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f
*x+e)^4*a^2*c^2*d^4+408*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f
*x+e)^4*a^2*c*d^5-720*a^(5/2)*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*c^3*d^4-312*a^(5/2)*arcta
nh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*c^2*d^5+20*(-a*(sin(f*x+e)-1))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/
2)*c^6+56*(-a*(sin(f*x+e)-1))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*d^6-312*a^(5/2)*arctanh((-a*(sin(f*x+e)-1))^(1/2
)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^4*d^7-624*a^(5/2)*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin
(f*x+e)^3*d^7-312*a^(5/2)*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^2*d^7-6*(-a*(sin(f
*x+e)-1))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*c^6-72*(-a*(sin(f*x+e)-1))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*d^6-504*a
^(5/2)*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*c^4*d^3+6*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1)
)^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)^3*a^2*c^5*d-42*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(
1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)^3*a^2*c^4*d^2+276*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(
1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)^3*a^2*c^3*d^3+1140*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^
(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)^3*a^2*c^2*d^4+1254*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))
^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)^3*a^2*c*d^5-12*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1
/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)^2*a^2*c^5*d+69*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)
*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)^2*a^2*c^4*d^2+1032*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2
)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)^2*a^2*c^3*d^3+2013*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/
2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)^2*a^2*c^2*d^4+1284*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1
/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)^2*a^2*c*d^5+3*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*
2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)^4*a^2*c^4*d^2-2064*a^(5/2)*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(
a*(c+d)*d)^(1/2))*sin(f*x+e)^3*c*d^6-504*a^(5/2)*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*
x+e)^2*c^4*d^3-2736*a^(5/2)*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^2*c^3*d^4-3696*a
^(5/2)*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^2*c^2*d^5-1968*a^(5/2)*arctanh((-a*(s
in(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^2*c*d^6-192*(-a*(sin(f*x+e)-1))^(1/2)*a^(3/2)*(a*(c+d)*d)^
(1/2)*sin(f*x+e)^2*c^2*d^4-6*(-a*(sin(f*x+e)-1))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)^2*c^4*d^2+48*(-a*(
sin(f*x+e)-1))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)^2*c^3*d^3+180*(-a*(sin(f*x+e)-1))^(3/2)*a^(1/2)*(a*(
c+d)*d)^(1/2)*sin(f*x+e)^2*c^2*d^4-96*(-a*(sin(f*x+e)-1))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)^2*c*d^5+3
*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)^2*a^2*c^6+219*2^(
1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)^2*a^2*d^6-12*(-a*(sin
(f*x+e)-1))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)*c^5*d-204*(-a*(sin(f*x+e)-1))^(3/2)*a^(1/2)*(a*(c+d)*d)
^(1/2)*sin(f*x+e)*c*d^5+6*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin
(f*x+e)*a^2*c^6-24*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*a^2*c^5*d+
408*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*a^2*c^3*d^3+219*2^(1/2)*a
rctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*a^2*c^2*d^4+219*2^(1/2)*arctanh(1/2*(-
a*(sin(f*x+e)-1))^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)^4*a^2*d^6-232*(-a*(sin(f*x+e)-1))^(1/2)*
a^(3/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)^2*c^3*d^3+20*(-a*(sin(f*x+e)-1))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*sin(f*x+
e)^2*c^4*d^2+232*(-a*(sin(f*x+e)-1))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)^2*c*d^5+40*(-a*(sin(f*x+e)-1))
^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)*c^5*d-192*(-a*(sin(f*x+e)-1))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*sin(
f*x+e)*c^4*d^2-544*(-a*(sin(f*x+e)-1))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)*c^3*d^3+80*(-a*(sin(f*x+e)-1
))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)*c^2*d^4+504*(-a*(sin(f*x+e)-1))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*
sin(f*x+e)*c*d^5+172*(-a*(sin(f*x+e)-1))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)^2*d^6-1008*a^(5/2)*arctanh
((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)*c^4*d^3-2448*a^(5/2)*arctanh((-a*(sin(f*x+e)-1))^(1
/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)*c^3*d^4-2064*a^(5/2)*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2)
)*sin(f*x+e)*c^2*d^5-624*a^(5/2)*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)*c*d^6+112*(
-a*(sin(f*x+e)-1))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)*d^6-96*(-a*(sin(f*x+e)-1))^(1/2)*a^(3/2)*(a*(c+d
)*d)^(1/2)*c^5*d-136*(-a*(sin(f*x+e)-1))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*c^4*d^2-40*(-a*(sin(f*x+e)-1))^(1/2)*
a^(3/2)*(a*(c+d)*d)^(1/2)*c^3*d^3+60*(-a*(sin(f*x+e)-1))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*c^2*d^4+136*(-a*(sin(
f*x+e)-1))^(1/2)*a^(3/2)*(a*(c+d)*d)^(1/2)*c*d^5-126*(-a*(sin(f*x+e)-1))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*sin(f
*x+e)^2*d^6-144*(-a*(sin(f*x+e)-1))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)*d^6+48*(-a*(sin(f*x+e)-1))^(3/2
)*a^(1/2)*(a*(c+d)*d)^(1/2)*c^5*d+60*(-a*(sin(f*x+e)-1))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*c^4*d^2-48*(-a*(sin(f
*x+e)-1))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*c^3*d^3+66*(-a*(sin(f*x+e)-1))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*c^2*d
^4-48*(-a*(sin(f*x+e)-1))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*c*d^5+3*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2
)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*a^2*c^6-504*a^(5/2)*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2)
)*sin(f*x+e)^4*c^2*d^5-720*a^(5/2)*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^4*c*d^6-1
008*a^(5/2)*arctanh((-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^3*c^3*d^4-2448*a^(5/2)*arctanh((
-a*(sin(f*x+e)-1))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^3*c^2*d^5+162*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))
^(1/2)*2^(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*a^2*c^4*d^2+96*(-a*(sin(f*x+e)-1))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*s
in(f*x+e)*c^4*d^2+120*(-a*(sin(f*x+e)-1))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)*c^3*d^3+144*(-a*(sin(f*x+
e)-1))^(3/2)*a^(1/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)*c^2*d^4+438*2^(1/2)*arctanh(1/2*(-a*(sin(f*x+e)-1))^(1/2)*2^
(1/2)/a^(1/2))*(a*(c+d)*d)^(1/2)*sin(f*x+e)^3*a^2*d^6)/a^(9/2)/(a*(c+d)*d)^(1/2)/(c+d*sin(f*x+e))^2/(c+d)^2/(s
in(f*x+e)+1)/(c-d)^5/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2857 vs. \(2 (357) = 714\).

Time = 2.08 (sec) , antiderivative size = 5999, normalized size of antiderivative = 15.66 \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \]

[In]

integrate(1/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\text {Timed out} \]

[In]

integrate(1/(a+a*sin(f*x+e))**(5/2)/(c+d*sin(f*x+e))**3,x)

[Out]

Timed out

Maxima [F(-1)]

Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\text {Timed out} \]

[In]

integrate(1/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^3,x, algorithm="maxima")

[Out]

Timed out

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1399 vs. \(2 (357) = 714\).

Time = 0.78 (sec) , antiderivative size = 1399, normalized size of antiderivative = 3.65 \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \]

[In]

integrate(1/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^3,x, algorithm="giac")

[Out]

1/32*(24*sqrt(2)*(21*sqrt(a)*c^2*d^3 + 30*sqrt(a)*c*d^4 + 13*sqrt(a)*d^5)*arctan(sqrt(2)*d*sin(-1/4*pi + 1/2*f
*x + 1/2*e)/sqrt(-c*d - d^2))/((sqrt(2)*a^3*c^7*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 3*sqrt(2)*a^3*c^6*d*sgn(
cos(-1/4*pi + 1/2*f*x + 1/2*e)) + sqrt(2)*a^3*c^5*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 5*sqrt(2)*a^3*c^4*
d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 5*sqrt(2)*a^3*c^3*d^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - sqrt(2)*
a^3*c^2*d^5*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 3*sqrt(2)*a^3*c*d^6*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - sq
rt(2)*a^3*d^7*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sqrt(-c*d - d^2)) + 3*(sqrt(a)*c^2 - 10*sqrt(a)*c*d + 73*sq
rt(a)*d^2)*log(sin(-1/4*pi + 1/2*f*x + 1/2*e) + 1)/(sqrt(2)*a^3*c^5*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 5*sq
rt(2)*a^3*c^4*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 10*sqrt(2)*a^3*c^3*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e
)) - 10*sqrt(2)*a^3*c^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 5*sqrt(2)*a^3*c*d^4*sgn(cos(-1/4*pi + 1/2*f*
x + 1/2*e)) - sqrt(2)*a^3*d^5*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))) - 3*(sqrt(a)*c^2 - 10*sqrt(a)*c*d + 73*sqrt
(a)*d^2)*log(-sin(-1/4*pi + 1/2*f*x + 1/2*e) + 1)/(sqrt(2)*a^3*c^5*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 5*sqr
t(2)*a^3*c^4*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 10*sqrt(2)*a^3*c^3*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)
) - 10*sqrt(2)*a^3*c^2*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 5*sqrt(2)*a^3*c*d^4*sgn(cos(-1/4*pi + 1/2*f*x
+ 1/2*e)) - sqrt(2)*a^3*d^5*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))) - 2*(12*sqrt(a)*c^3*d^2*sin(-1/4*pi + 1/2*f*
x + 1/2*e)^7 - 84*sqrt(a)*c^2*d^3*sin(-1/4*pi + 1/2*f*x + 1/2*e)^7 - 444*sqrt(a)*c*d^4*sin(-1/4*pi + 1/2*f*x +
1/2*e)^7 - 252*sqrt(a)*d^5*sin(-1/4*pi + 1/2*f*x + 1/2*e)^7 - 12*sqrt(a)*c^4*d*sin(-1/4*pi + 1/2*f*x + 1/2*e)
^5 + 52*sqrt(a)*c^3*d^2*sin(-1/4*pi + 1/2*f*x + 1/2*e)^5 + 500*sqrt(a)*c^2*d^3*sin(-1/4*pi + 1/2*f*x + 1/2*e)^
5 + 1196*sqrt(a)*c*d^4*sin(-1/4*pi + 1/2*f*x + 1/2*e)^5 + 568*sqrt(a)*d^5*sin(-1/4*pi + 1/2*f*x + 1/2*e)^5 + 3
*sqrt(a)*c^5*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 + 5*sqrt(a)*c^4*d*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 - 146*sqrt(a)
*c^3*d^2*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 - 710*sqrt(a)*c^2*d^3*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 - 1057*sqrt(a
)*c*d^4*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 - 399*sqrt(a)*d^5*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 - 5*sqrt(a)*c^5*si
n(-1/4*pi + 1/2*f*x + 1/2*e) + 9*sqrt(a)*c^4*d*sin(-1/4*pi + 1/2*f*x + 1/2*e) + 86*sqrt(a)*c^3*d^2*sin(-1/4*pi
+ 1/2*f*x + 1/2*e) + 290*sqrt(a)*c^2*d^3*sin(-1/4*pi + 1/2*f*x + 1/2*e) + 303*sqrt(a)*c*d^4*sin(-1/4*pi + 1/2
*f*x + 1/2*e) + 85*sqrt(a)*d^5*sin(-1/4*pi + 1/2*f*x + 1/2*e))/((sqrt(2)*a^3*c^6*sgn(cos(-1/4*pi + 1/2*f*x + 1
/2*e)) - 2*sqrt(2)*a^3*c^5*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - sqrt(2)*a^3*c^4*d^2*sgn(cos(-1/4*pi + 1/2*f
*x + 1/2*e)) + 4*sqrt(2)*a^3*c^3*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - sqrt(2)*a^3*c^2*d^4*sgn(cos(-1/4*pi
+ 1/2*f*x + 1/2*e)) - 2*sqrt(2)*a^3*c*d^5*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + sqrt(2)*a^3*d^6*sgn(cos(-1/4*
pi + 1/2*f*x + 1/2*e)))*(2*d*sin(-1/4*pi + 1/2*f*x + 1/2*e)^4 - c*sin(-1/4*pi + 1/2*f*x + 1/2*e)^2 - 3*d*sin(-
1/4*pi + 1/2*f*x + 1/2*e)^2 + c + d)^2))/f

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx=\int \frac {1}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^3} \,d x \]

[In]

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

[Out]

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