Integrand size = 28, antiderivative size = 137 \[ \int \frac {(3+3 \sin (e+f x))^3}{(c-c \sin (e+f x))^{3/2}} \, dx=-\frac {270 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{c^{3/2} f}+\frac {27 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^{7/2}}+\frac {45 \cos ^3(e+f x)}{f (c-c \sin (e+f x))^{3/2}}+\frac {270 \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}} \]
a^3*c^2*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^(7/2)+5/3*a^3*cos(f*x+e)^3/f/(c-c* sin(f*x+e))^(3/2)-10*a^3*arctanh(1/2*cos(f*x+e)*c^(1/2)*2^(1/2)/(c-c*sin(f *x+e))^(1/2))*2^(1/2)/c^(3/2)/f+10*a^3*cos(f*x+e)/c/f/(c-c*sin(f*x+e))^(1/ 2)
Result contains complex when optimal does not.
Time = 0.82 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.24 \[ \int \frac {(3+3 \sin (e+f x))^3}{(c-c \sin (e+f x))^{3/2}} \, dx=-\frac {9 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (50 \cos \left (\frac {1}{2} (e+f x)\right )+25 \cos \left (\frac {3}{2} (e+f x)\right )+\cos \left (\frac {5}{2} (e+f x)\right )+50 \sin \left (\frac {1}{2} (e+f x)\right )-(120+120 i) \sqrt [4]{-1} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) (-1+\sin (e+f x))-25 \sin \left (\frac {3}{2} (e+f x)\right )+\sin \left (\frac {5}{2} (e+f x)\right )\right )}{2 c f (-1+\sin (e+f x)) \sqrt {c-c \sin (e+f x)}} \]
(-9*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(50*Cos[(e + f*x)/2] + 25*Cos[(3 *(e + f*x))/2] + Cos[(5*(e + f*x))/2] + 50*Sin[(e + f*x)/2] - (120 + 120*I )*(-1)^(1/4)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(-1 + S in[e + f*x]) - 25*Sin[(3*(e + f*x))/2] + Sin[(5*(e + f*x))/2]))/(2*c*f*(-1 + Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])
Time = 0.77 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.18, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {3042, 3215, 3042, 3159, 3042, 3158, 3042, 3158, 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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^3}{(c-c \sin (e+f x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^3}{(c-c \sin (e+f x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle a^3 c^3 \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{9/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \int \frac {\cos (e+f x)^6}{(c-c \sin (e+f x))^{9/2}}dx\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^3 c^3 \left (\frac {\cos ^5(e+f x)}{c f (c-c \sin (e+f x))^{7/2}}-\frac {5 \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{5/2}}dx}{2 c^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {\cos ^5(e+f x)}{c f (c-c \sin (e+f x))^{7/2}}-\frac {5 \int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^{5/2}}dx}{2 c^2}\right )\) |
| \(\Big \downarrow \) 3158 |
| \(\displaystyle a^3 c^3 \left (\frac {\cos ^5(e+f x)}{c f (c-c \sin (e+f x))^{7/2}}-\frac {5 \left (\frac {2 \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^{3/2}}dx}{c}-\frac {2 \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{3/2}}\right )}{2 c^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {\cos ^5(e+f x)}{c f (c-c \sin (e+f x))^{7/2}}-\frac {5 \left (\frac {2 \int \frac {\cos (e+f x)^2}{(c-c \sin (e+f x))^{3/2}}dx}{c}-\frac {2 \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{3/2}}\right )}{2 c^2}\right )\) |
| \(\Big \downarrow \) 3158 |
| \(\displaystyle a^3 c^3 \left (\frac {\cos ^5(e+f x)}{c f (c-c \sin (e+f x))^{7/2}}-\frac {5 \left (\frac {2 \left (\frac {2 \int \frac {1}{\sqrt {c-c \sin (e+f x)}}dx}{c}-\frac {2 \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}}\right )}{c}-\frac {2 \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{3/2}}\right )}{2 c^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {\cos ^5(e+f x)}{c f (c-c \sin (e+f x))^{7/2}}-\frac {5 \left (\frac {2 \left (\frac {2 \int \frac {1}{\sqrt {c-c \sin (e+f x)}}dx}{c}-\frac {2 \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}}\right )}{c}-\frac {2 \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{3/2}}\right )}{2 c^2}\right )\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle a^3 c^3 \left (\frac {\cos ^5(e+f x)}{c f (c-c \sin (e+f x))^{7/2}}-\frac {5 \left (\frac {2 \left (-\frac {4 \int \frac {1}{2 c-\frac {c^2 \cos ^2(e+f x)}{c-c \sin (e+f x)}}d\left (-\frac {c \cos (e+f x)}{\sqrt {c-c \sin (e+f x)}}\right )}{c f}-\frac {2 \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}}\right )}{c}-\frac {2 \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{3/2}}\right )}{2 c^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle a^3 c^3 \left (\frac {\cos ^5(e+f x)}{c f (c-c \sin (e+f x))^{7/2}}-\frac {5 \left (\frac {2 \left (\frac {2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{c^{3/2} f}-\frac {2 \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}}\right )}{c}-\frac {2 \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{3/2}}\right )}{2 c^2}\right )\) |
a^3*c^3*(Cos[e + f*x]^5/(c*f*(c - c*Sin[e + f*x])^(7/2)) - (5*((-2*Cos[e + f*x]^3)/(3*c*f*(c - c*Sin[e + f*x])^(3/2)) + (2*((2*Sqrt[2]*ArcTanh[(Sqrt [c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(c^(3/2)*f) - (2*Co s[e + f*x])/(c*f*Sqrt[c - c*Sin[e + f*x]])))/c))/(2*c^2))
3.4.12.3.1 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[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x ])^(m + 1)/(b*f*(m + p))), x] + Simp[g^2*((p - 1)/(a*(m + p))) 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] && LtQ[m, -1] && GtQ[p, 1] && (GtQ[m, -2] || EqQ[2*m + p + 1, 0] || (EqQ[m, -2] && IntegerQ[p])) && NeQ[m + p, 0] && In tegersQ[2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[2*g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f *x])^(m + 1)/(b*f*(2*m + p + 1))), x] + Simp[g^2*((p - 1)/(b^2*(2*m + p + 1 ))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] & & NeQ[2*m + p + 1, 0] && !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 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 = 2.90 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.38
| method | result | size |
| default | \(\frac {2 a^{3} \left (15 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} \sin \left (f x +e \right )-12 \sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, c^{\frac {3}{2}} \sin \left (f x +e \right )-\left (c \left (\sin \left (f x +e \right )+1\right )\right )^{\frac {3}{2}} \sqrt {c}\, \sin \left (f x +e \right )-15 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2}+18 \sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, c^{\frac {3}{2}}+\left (c \left (\sin \left (f x +e \right )+1\right )\right )^{\frac {3}{2}} \sqrt {c}\right ) \sqrt {c \left (\sin \left (f x +e \right )+1\right )}}{3 c^{\frac {7}{2}} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(189\) |
| parts | \(\frac {a^{3} \left (-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} \sin \left (f x +e \right )+2 \sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, c^{\frac {3}{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^{2}\right ) \sqrt {c \left (\sin \left (f x +e \right )+1\right )}}{4 c^{\frac {7}{2}} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {a^{3} \left (8 \left (c \left (\sin \left (f x +e \right )+1\right )\right )^{\frac {3}{2}} \sqrt {c}\, \sin \left (f x +e \right )-33 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} \sin \left (f x +e \right )-8 \left (c \left (\sin \left (f x +e \right )+1\right )\right )^{\frac {3}{2}} \sqrt {c}+24 \sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, c^{\frac {3}{2}} \sin \left (f x +e \right )+33 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2}-30 \sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, c^{\frac {3}{2}}\right ) \sqrt {c \left (\sin \left (f x +e \right )+1\right )}}{12 c^{\frac {7}{2}} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {3 a^{3} \left (3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c \sin \left (f x +e \right )-3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c +2 \sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {c}\right ) \sqrt {c \left (\sin \left (f x +e \right )+1\right )}}{4 c^{\frac {5}{2}} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {3 a^{3} \left (7 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c \sin \left (f x +e \right )-7 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c -8 \sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {c}\, \sin \left (f x +e \right )+10 \sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {c}\right ) \sqrt {c \left (\sin \left (f x +e \right )+1\right )}}{4 c^{\frac {5}{2}} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(583\) |
2/3*a^3*(15*2^(1/2)*arctanh(1/2*(c*(sin(f*x+e)+1))^(1/2)*2^(1/2)/c^(1/2))* c^2*sin(f*x+e)-12*(c*(sin(f*x+e)+1))^(1/2)*c^(3/2)*sin(f*x+e)-(c*(sin(f*x+ e)+1))^(3/2)*c^(1/2)*sin(f*x+e)-15*2^(1/2)*arctanh(1/2*(c*(sin(f*x+e)+1))^ (1/2)*2^(1/2)/c^(1/2))*c^2+18*(c*(sin(f*x+e)+1))^(1/2)*c^(3/2)+(c*(sin(f*x +e)+1))^(3/2)*c^(1/2))*(c*(sin(f*x+e)+1))^(1/2)/c^(7/2)/cos(f*x+e)/(c-c*si n(f*x+e))^(1/2)/f
Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (133) = 266\).
Time = 0.29 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.38 \[ \int \frac {(3+3 \sin (e+f x))^3}{(c-c \sin (e+f x))^{3/2}} \, dx=\frac {\frac {15 \, \sqrt {2} {\left (a^{3} c \cos \left (f x + e\right )^{2} - a^{3} c \cos \left (f x + e\right ) - 2 \, a^{3} c + {\left (a^{3} c \cos \left (f x + e\right ) + 2 \, a^{3} c\right )} \sin \left (f x + e\right )\right )} \log \left (-\frac {\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) - 2\right )} \sin \left (f x + e\right ) - \frac {2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )}}{\sqrt {c}} + 3 \, \cos \left (f x + e\right ) + 2}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right )}{\sqrt {c}} - 2 \, {\left (a^{3} \cos \left (f x + e\right )^{3} + 13 \, a^{3} \cos \left (f x + e\right )^{2} + 18 \, a^{3} \cos \left (f x + e\right ) + 6 \, a^{3} + {\left (a^{3} \cos \left (f x + e\right )^{2} - 12 \, a^{3} \cos \left (f x + e\right ) + 6 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{3 \, {\left (c^{2} f \cos \left (f x + e\right )^{2} - c^{2} f \cos \left (f x + e\right ) - 2 \, c^{2} f + {\left (c^{2} f \cos \left (f x + e\right ) + 2 \, c^{2} f\right )} \sin \left (f x + e\right )\right )}} \]
1/3*(15*sqrt(2)*(a^3*c*cos(f*x + e)^2 - a^3*c*cos(f*x + e) - 2*a^3*c + (a^ 3*c*cos(f*x + e) + 2*a^3*c)*sin(f*x + e))*log(-(cos(f*x + e)^2 + (cos(f*x + e) - 2)*sin(f*x + e) - 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*(cos(f*x + e) + sin(f*x + e) + 1)/sqrt(c) + 3*cos(f*x + e) + 2)/(cos(f*x + e)^2 + (cos( f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2))/sqrt(c) - 2*(a^3*cos(f*x + e)^3 + 13*a^3*cos(f*x + e)^2 + 18*a^3*cos(f*x + e) + 6*a^3 + (a^3*cos(f*x + e)^2 - 12*a^3*cos(f*x + e) + 6*a^3)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c))/(c^2*f*cos(f*x + e)^2 - c^2*f*cos(f*x + e) - 2*c^2*f + (c^2*f*cos(f* x + e) + 2*c^2*f)*sin(f*x + e))
\[ \int \frac {(3+3 \sin (e+f x))^3}{(c-c \sin (e+f x))^{3/2}} \, dx=a^{3} \left (\int \frac {3 \sin {\left (e + f x \right )}}{- c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )} + c \sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {3 \sin ^{2}{\left (e + f x \right )}}{- c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )} + c \sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {\sin ^{3}{\left (e + f x \right )}}{- c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )} + c \sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {1}{- c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )} + c \sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx\right ) \]
a**3*(Integral(3*sin(e + f*x)/(-c*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x) + c*sqrt(-c*sin(e + f*x) + c)), x) + Integral(3*sin(e + f*x)**2/(-c*sqrt(-c *sin(e + f*x) + c)*sin(e + f*x) + c*sqrt(-c*sin(e + f*x) + c)), x) + Integ ral(sin(e + f*x)**3/(-c*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x) + c*sqrt(-c *sin(e + f*x) + c)), x) + Integral(1/(-c*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x) + c*sqrt(-c*sin(e + f*x) + c)), x))
\[ \int \frac {(3+3 \sin (e+f x))^3}{(c-c \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 358 vs. \(2 (133) = 266\).
Time = 0.37 (sec) , antiderivative size = 358, normalized size of antiderivative = 2.61 \[ \int \frac {(3+3 \sin (e+f x))^3}{(c-c \sin (e+f x))^{3/2}} \, dx=-\frac {\frac {30 \, \sqrt {2} a^{3} \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 )}{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 {3 \, \sqrt {2} a^{3} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}}{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 {3 \, \sqrt {2} {\left (a^{3} + \frac {10 \, a^{3} {\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}\right )} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}}{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 {16 \, \sqrt {2} {\left (7 \, a^{3} \sqrt {c} - \frac {12 \, a^{3} \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 {9 \, a^{3} \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}}\right )}}{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 )}^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{6 \, f} \]
-1/6*(30*sqrt(2)*a^3*log(-(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*p i + 1/2*f*x + 1/2*e) + 1))/(c^(3/2)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))) + 3*sqrt(2)*a^3*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(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))) - 3*sqrt(2)*( a^3 + 10*a^3*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1))*(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)/(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))) - 16*sqrt(2)*( 7*a^3*sqrt(c) - 12*a^3*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) + 9*a^3*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)/(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)^3*sgn(sin(- 1/4*pi + 1/2*f*x + 1/2*e))))/f
Timed out. \[ \int \frac {(3+3 \sin (e+f x))^3}{(c-c \sin (e+f x))^{3/2}} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]