Integrand size = 28, antiderivative size = 176 \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}} \, dx=\frac {7 \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{432 \sqrt {2} c^{3/2} f}+\frac {7 \cos (e+f x)}{432 f (c-c \sin (e+f x))^{3/2}}-\frac {7 \sec (e+f x)}{324 c f \sqrt {c-c \sin (e+f x)}}-\frac {7 \sec ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{810 c^2 f}-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{135 c^3 f} \]
7/16*cos(f*x+e)/a^3/f/(c-c*sin(f*x+e))^(3/2)-1/5*sec(f*x+e)^5*(c-c*sin(f*x +e))^(3/2)/a^3/c^3/f+7/32*arctanh(1/2*cos(f*x+e)*c^(1/2)*2^(1/2)/(c-c*sin( f*x+e))^(1/2))/a^3/c^(3/2)/f*2^(1/2)-7/12*sec(f*x+e)/a^3/c/f/(c-c*sin(f*x+ e))^(1/2)-7/30*sec(f*x+e)^3*(c-c*sin(f*x+e))^(1/2)/a^3/c^2/f
Result contains complex when optimal does not.
Time = 2.51 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.97 \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}} \, dx=\frac {\left (\frac {1}{51840}+\frac {i}{51840}\right ) \cos (e+f x) \left (840 \sqrt [4]{-1} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5+(1-i) (206+350 \cos (2 (e+f x))-231 \sin (e+f x)+105 \sin (3 (e+f x)))\right )}{c f (-1+\sin (e+f x)) (1+\sin (e+f x))^3 \sqrt {c-c \sin (e+f x)}} \]
((1/51840 + I/51840)*Cos[e + f*x]*(840*(-1)^(1/4)*ArcTan[(1/2 + I/2)*(-1)^ (1/4)*(1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*(Cos [(e + f*x)/2] + Sin[(e + f*x)/2])^5 + (1 - I)*(206 + 350*Cos[2*(e + f*x)] - 231*Sin[e + f*x] + 105*Sin[3*(e + f*x)])))/(c*f*(-1 + Sin[e + f*x])*(1 + Sin[e + f*x])^3*Sqrt[c - c*Sin[e + f*x]])
Time = 0.83 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.08, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {3042, 3215, 3042, 3154, 3042, 3154, 3042, 3166, 3042, 3129, 3042, 3128, 219}
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.
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle \frac {\int \sec ^6(e+f x) (c-c \sin (e+f x))^{3/2}dx}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(c-c \sin (e+f x))^{3/2}}{\cos (e+f x)^6}dx}{a^3 c^3}\) |
| \(\Big \downarrow \) 3154 |
| \(\displaystyle \frac {\frac {7}{10} c \int \sec ^4(e+f x) \sqrt {c-c \sin (e+f x)}dx-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {7}{10} c \int \frac {\sqrt {c-c \sin (e+f x)}}{\cos (e+f x)^4}dx-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3154 |
| \(\displaystyle \frac {\frac {7}{10} c \left (\frac {5}{6} c \int \frac {\sec ^2(e+f x)}{\sqrt {c-c \sin (e+f x)}}dx-\frac {\sec ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{3 f}\right )-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {7}{10} c \left (\frac {5}{6} c \int \frac {1}{\cos (e+f x)^2 \sqrt {c-c \sin (e+f x)}}dx-\frac {\sec ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{3 f}\right )-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3166 |
| \(\displaystyle \frac {\frac {7}{10} c \left (\frac {5}{6} c \left (\frac {3}{2} c \int \frac {1}{(c-c \sin (e+f x))^{3/2}}dx-\frac {\sec (e+f x)}{f \sqrt {c-c \sin (e+f x)}}\right )-\frac {\sec ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{3 f}\right )-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {7}{10} c \left (\frac {5}{6} c \left (\frac {3}{2} c \int \frac {1}{(c-c \sin (e+f x))^{3/2}}dx-\frac {\sec (e+f x)}{f \sqrt {c-c \sin (e+f x)}}\right )-\frac {\sec ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{3 f}\right )-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3129 |
| \(\displaystyle \frac {\frac {7}{10} c \left (\frac {5}{6} c \left (\frac {3}{2} c \left (\frac {\int \frac {1}{\sqrt {c-c \sin (e+f x)}}dx}{4 c}+\frac {\cos (e+f x)}{2 f (c-c \sin (e+f x))^{3/2}}\right )-\frac {\sec (e+f x)}{f \sqrt {c-c \sin (e+f x)}}\right )-\frac {\sec ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{3 f}\right )-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {7}{10} c \left (\frac {5}{6} c \left (\frac {3}{2} c \left (\frac {\int \frac {1}{\sqrt {c-c \sin (e+f x)}}dx}{4 c}+\frac {\cos (e+f x)}{2 f (c-c \sin (e+f x))^{3/2}}\right )-\frac {\sec (e+f x)}{f \sqrt {c-c \sin (e+f x)}}\right )-\frac {\sec ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{3 f}\right )-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle \frac {\frac {7}{10} c \left (\frac {5}{6} c \left (\frac {3}{2} c \left (\frac {\cos (e+f x)}{2 f (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {1}{2 c-\frac {c^2 \cos ^2(e+f x)}{c-c \sin (e+f x)}}d\left (-\frac {c \cos (e+f x)}{\sqrt {c-c \sin (e+f x)}}\right )}{2 c f}\right )-\frac {\sec (e+f x)}{f \sqrt {c-c \sin (e+f x)}}\right )-\frac {\sec ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{3 f}\right )-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {7}{10} c \left (\frac {5}{6} c \left (\frac {3}{2} c \left (\frac {\text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{2 \sqrt {2} c^{3/2} f}+\frac {\cos (e+f x)}{2 f (c-c \sin (e+f x))^{3/2}}\right )-\frac {\sec (e+f x)}{f \sqrt {c-c \sin (e+f x)}}\right )-\frac {\sec ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{3 f}\right )-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 f}}{a^3 c^3}\) |
(-1/5*(Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(3/2))/f + (7*c*(-1/3*(Sec[e + f*x]^3*Sqrt[c - c*Sin[e + f*x]])/f + (5*c*(-(Sec[e + f*x]/(f*Sqrt[c - c*Si n[e + f*x]])) + (3*c*(ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*S in[e + f*x]])]/(2*Sqrt[2]*c^(3/2)*f) + Cos[e + f*x]/(2*f*(c - c*Sin[e + f* x])^(3/2))))/2))/6))/10)/(a^3*c^3)
3.4.38.3.1 Defintions of rubi rules used
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2/d Subst[Int[1/(2*a - x^2), x], x, b*(Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[c + d*x]*((a + b*Sin[c + d*x])^n/(a*d*(2*n + 1))), x] + Simp[(n + 1)/(a*(2*n + 1)) Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*(p + 1))), x] + Simp[a*((m + p + 1)/(g^2*(p + 1))) Int[(g* Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[p, -2*m] && IntegersQ [m + 1/2, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_. )*(x_)]], x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(a*f*g*(p + 1)*S qrt[a + b*Sin[e + f*x]])), x] + Simp[a*((2*p + 1)/(2*g^2*(p + 1))) Int[(g *Cos[e + f*x])^(p + 2)/(a + b*Sin[e + f*x])^(3/2), x], x] /; FreeQ[{a, b, e , f, g}, x] && EqQ[a^2 - b^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[a^m*c^m Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[ b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && ((Lt Q[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Time = 1.14 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.97
| method | result | size |
| default | \(-\frac {105 \left (c \left (\sin \left (f x +e \right )+1\right )\right )^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sin \left (f x +e \right ) c -105 \left (c \left (\sin \left (f x +e \right )+1\right )\right )^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c +42 c^{\frac {7}{2}} \sin \left (f x +e \right )-350 \left (\sin ^{2}\left (f x +e \right )\right ) c^{\frac {7}{2}}-210 \left (\sin ^{3}\left (f x +e \right )\right ) c^{\frac {7}{2}}+278 c^{\frac {7}{2}}}{480 c^{\frac {9}{2}} a^{3} \left (\sin \left (f x +e \right )+1\right )^{2} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(170\) |
-1/480/c^(9/2)/a^3*(105*(c*(sin(f*x+e)+1))^(5/2)*2^(1/2)*arctanh(1/2*(c*(s in(f*x+e)+1))^(1/2)*2^(1/2)/c^(1/2))*sin(f*x+e)*c-105*(c*(sin(f*x+e)+1))^( 5/2)*2^(1/2)*arctanh(1/2*(c*(sin(f*x+e)+1))^(1/2)*2^(1/2)/c^(1/2))*c+42*c^ (7/2)*sin(f*x+e)-350*sin(f*x+e)^2*c^(7/2)-210*sin(f*x+e)^3*c^(7/2)+278*c^( 7/2))/(sin(f*x+e)+1)^2/cos(f*x+e)/(c-c*sin(f*x+e))^(1/2)/f
Time = 0.28 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.35 \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}} \, dx=\frac {105 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) + \cos \left (f x + e\right )^{3}\right )} \sqrt {c} \log \left (-\frac {c \cos \left (f x + e\right )^{2} + 2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} \sqrt {c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )} + 3 \, c \cos \left (f x + e\right ) + {\left (c \cos \left (f x + e\right ) - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) - 4 \, {\left (175 \, \cos \left (f x + e\right )^{2} + 21 \, {\left (5 \, \cos \left (f x + e\right )^{2} - 4\right )} \sin \left (f x + e\right ) - 36\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{960 \, {\left (a^{3} c^{2} f \cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) + a^{3} c^{2} f \cos \left (f x + e\right )^{3}\right )}} \]
1/960*(105*sqrt(2)*(cos(f*x + e)^3*sin(f*x + e) + cos(f*x + e)^3)*sqrt(c)* log(-(c*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*sqrt(c)*(cos( f*x + e) + sin(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f*x + e) - 2*c)*s in(f*x + e) + 2*c)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos (f*x + e) - 2)) - 4*(175*cos(f*x + e)^2 + 21*(5*cos(f*x + e)^2 - 4)*sin(f* x + e) - 36)*sqrt(-c*sin(f*x + e) + c))/(a^3*c^2*f*cos(f*x + e)^3*sin(f*x + e) + a^3*c^2*f*cos(f*x + e)^3)
Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 424 vs. \(2 (164) = 328\).
Time = 0.39 (sec) , antiderivative size = 424, normalized size of antiderivative = 2.41 \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}} \, dx=-\frac {\frac {15 \, \sqrt {2} {\left (\frac {14 \, {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} - 1\right )} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}}{a^{3} c^{\frac {3}{2}} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {210 \, \sqrt {2} \log \left (-\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1}\right )}{a^{3} c^{\frac {3}{2}} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {15 \, \sqrt {2} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}}{a^{3} c^{\frac {3}{2}} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {32 \, \sqrt {2} {\left (29 \, \sqrt {c} + \frac {100 \, \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + \frac {170 \, \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}} + \frac {120 \, \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{3}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} + \frac {45 \, \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{4}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{4}}\right )}}{a^{3} c^{2} {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + 1\right )}^{5} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{1920 \, f} \]
-1/1920*(15*sqrt(2)*(14*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) - 1)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)/(a^3*c^( 3/2)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2* e))) - 210*sqrt(2)*log(-(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1))/(a^3*c^(3/2)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))) + 15*sqrt(2)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(a^3*c^(3/2)*(cos(-1/4* pi + 1/2*f*x + 1/2*e) + 1)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))) - 32*sqrt( 2)*(29*sqrt(c) + 100*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/ 4*pi + 1/2*f*x + 1/2*e) + 1) + 170*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^2/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^2 + 120*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^3/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^3 + 45*sqrt (c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^4/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^4)/(a^3*c^2*((cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1/2 *f*x + 1/2*e) + 1) + 1)^5*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))))/f
Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]