Integrand size = 28, antiderivative size = 142 \[ \int \frac {(3+3 \sin (e+f x))^2}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {9 \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{16 \sqrt {2} c^{7/2} f}+\frac {3 c \cos ^3(e+f x)}{f (c-c \sin (e+f x))^{9/2}}-\frac {9 \cos (e+f x)}{4 c f (c-c \sin (e+f x))^{5/2}}+\frac {9 \cos (e+f x)}{16 c^2 f (c-c \sin (e+f x))^{3/2}} \]
1/3*a^2*c*cos(f*x+e)^3/f/(c-c*sin(f*x+e))^(9/2)-1/4*a^2*cos(f*x+e)/c/f/(c- c*sin(f*x+e))^(5/2)+1/16*a^2*cos(f*x+e)/c^2/f/(c-c*sin(f*x+e))^(3/2)+1/32* a^2*arctanh(1/2*cos(f*x+e)*c^(1/2)*2^(1/2)/(c-c*sin(f*x+e))^(1/2))/c^(7/2) /f*2^(1/2)
Result contains complex when optimal does not.
Time = 1.10 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.91 \[ \int \frac {(3+3 \sin (e+f x))^2}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (32 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-28 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5-(3+3 i) \sqrt [4]{-1} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6+64 \sin \left (\frac {1}{2} (e+f x)\right )-56 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \sin \left (\frac {1}{2} (e+f x)\right )+6 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 \sin \left (\frac {1}{2} (e+f x)\right )\right )}{16 f (c-c \sin (e+f x))^{7/2}} \]
(3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(32*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) - 28*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 + 3*(Cos[(e + f*x)/2 ] - Sin[(e + f*x)/2])^5 - (3 + 3*I)*(-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])^6 + 64*Si n[(e + f*x)/2] - 56*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(e + f*x)/ 2] + 6*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*Sin[(e + f*x)/2]))/(16*f*(c - c*Sin[e + f*x])^(7/2))
Time = 0.69 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.17, 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, 3159, 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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^2}{(c-c \sin (e+f x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^2}{(c-c \sin (e+f x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle a^2 c^2 \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{11/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^{11/2}}dx\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^2 c^2 \left (\frac {\cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{9/2}}-\frac {\int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^{7/2}}dx}{2 c^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \left (\frac {\cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{9/2}}-\frac {\int \frac {\cos (e+f x)^2}{(c-c \sin (e+f x))^{7/2}}dx}{2 c^2}\right )\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^2 c^2 \left (\frac {\cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{9/2}}-\frac {\frac {\cos (e+f x)}{2 c f (c-c \sin (e+f x))^{5/2}}-\frac {\int \frac {1}{(c-c \sin (e+f x))^{3/2}}dx}{4 c^2}}{2 c^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \left (\frac {\cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{9/2}}-\frac {\frac {\cos (e+f x)}{2 c f (c-c \sin (e+f x))^{5/2}}-\frac {\int \frac {1}{(c-c \sin (e+f x))^{3/2}}dx}{4 c^2}}{2 c^2}\right )\) |
| \(\Big \downarrow \) 3129 |
| \(\displaystyle a^2 c^2 \left (\frac {\cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{9/2}}-\frac {\frac {\cos (e+f x)}{2 c f (c-c \sin (e+f x))^{5/2}}-\frac {\frac {\int \frac {1}{\sqrt {c-c \sin (e+f x)}}dx}{4 c}+\frac {\cos (e+f x)}{2 f (c-c \sin (e+f x))^{3/2}}}{4 c^2}}{2 c^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \left (\frac {\cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{9/2}}-\frac {\frac {\cos (e+f x)}{2 c f (c-c \sin (e+f x))^{5/2}}-\frac {\frac {\int \frac {1}{\sqrt {c-c \sin (e+f x)}}dx}{4 c}+\frac {\cos (e+f x)}{2 f (c-c \sin (e+f x))^{3/2}}}{4 c^2}}{2 c^2}\right )\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle a^2 c^2 \left (\frac {\cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{9/2}}-\frac {\frac {\cos (e+f x)}{2 c f (c-c \sin (e+f x))^{5/2}}-\frac {\frac {\cos (e+f x)}{2 f (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {1}{2 c-\frac {c^2 \cos ^2(e+f x)}{c-c \sin (e+f x)}}d\left (-\frac {c \cos (e+f x)}{\sqrt {c-c \sin (e+f x)}}\right )}{2 c f}}{4 c^2}}{2 c^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle a^2 c^2 \left (\frac {\cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{9/2}}-\frac {\frac {\cos (e+f x)}{2 c f (c-c \sin (e+f x))^{5/2}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{2 \sqrt {2} c^{3/2} f}+\frac {\cos (e+f x)}{2 f (c-c \sin (e+f x))^{3/2}}}{4 c^2}}{2 c^2}\right )\) |
a^2*c^2*(Cos[e + f*x]^3/(3*c*f*(c - c*Sin[e + f*x])^(9/2)) - (Cos[e + f*x] /(2*c*f*(c - c*Sin[e + f*x])^(5/2)) - (ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqr t[2]*Sqrt[c - c*Sin[e + f*x]])]/(2*Sqrt[2]*c^(3/2)*f) + Cos[e + f*x]/(2*f* (c - c*Sin[e + f*x])^(3/2)))/(4*c^2))/(2*c^2))
3.4.5.3.1 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[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 = 3.25 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.73
| method | result | size |
| default | \(-\frac {a^{2} \left (3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \left (\sin ^{3}\left (f x +e \right )\right ) c^{4}+24 \sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, c^{\frac {7}{2}}-32 \left (c \left (\sin \left (f x +e \right )+1\right )\right )^{\frac {3}{2}} c^{\frac {5}{2}}-6 \left (c \left (\sin \left (f x +e \right )+1\right )\right )^{\frac {5}{2}} c^{\frac {3}{2}}-9 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \left (\sin ^{2}\left (f x +e \right )\right ) c^{4}+9 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sin \left (f x +e \right ) c^{4}-3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4}\right ) \sqrt {c \left (\sin \left (f x +e \right )+1\right )}}{96 c^{\frac {15}{2}} \left (\sin \left (f x +e \right )-1\right )^{2} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(245\) |
| parts | \(\text {Expression too large to display}\) | \(748\) |
-1/96*a^2*(3*2^(1/2)*arctanh(1/2*(c*(sin(f*x+e)+1))^(1/2)*2^(1/2)/c^(1/2)) *sin(f*x+e)^3*c^4+24*(c*(sin(f*x+e)+1))^(1/2)*c^(7/2)-32*(c*(sin(f*x+e)+1) )^(3/2)*c^(5/2)-6*(c*(sin(f*x+e)+1))^(5/2)*c^(3/2)-9*2^(1/2)*arctanh(1/2*( c*(sin(f*x+e)+1))^(1/2)*2^(1/2)/c^(1/2))*sin(f*x+e)^2*c^4+9*2^(1/2)*arctan h(1/2*(c*(sin(f*x+e)+1))^(1/2)*2^(1/2)/c^(1/2))*sin(f*x+e)*c^4-3*2^(1/2)*a rctanh(1/2*(c*(sin(f*x+e)+1))^(1/2)*2^(1/2)/c^(1/2))*c^4)*(c*(sin(f*x+e)+1 ))^(1/2)/c^(15/2)/(sin(f*x+e)-1)^2/cos(f*x+e)/(c-c*sin(f*x+e))^(1/2)/f
Leaf count of result is larger than twice the leaf count of optimal. 440 vs. \(2 (133) = 266\).
Time = 0.28 (sec) , antiderivative size = 440, normalized size of antiderivative = 3.10 \[ \int \frac {(3+3 \sin (e+f x))^2}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {3 \, \sqrt {2} {\left (a^{2} \cos \left (f x + e\right )^{4} - 3 \, a^{2} \cos \left (f x + e\right )^{3} - 8 \, a^{2} \cos \left (f x + e\right )^{2} + 4 \, a^{2} \cos \left (f x + e\right ) + 8 \, a^{2} + {\left (a^{2} \cos \left (f x + e\right )^{3} + 4 \, a^{2} \cos \left (f x + e\right )^{2} - 4 \, a^{2} \cos \left (f x + e\right ) - 8 \, a^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {c} \log \left (-\frac {c \cos \left (f x + e\right )^{2} + 2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} \sqrt {c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )} + 3 \, c \cos \left (f x + e\right ) + {\left (c \cos \left (f x + e\right ) - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) - 4 \, {\left (3 \, a^{2} \cos \left (f x + e\right )^{3} + 25 \, a^{2} \cos \left (f x + e\right )^{2} - 10 \, a^{2} \cos \left (f x + e\right ) - 32 \, a^{2} + {\left (3 \, a^{2} \cos \left (f x + e\right )^{2} - 22 \, a^{2} \cos \left (f x + e\right ) - 32 \, a^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{192 \, {\left (c^{4} f \cos \left (f x + e\right )^{4} - 3 \, c^{4} f \cos \left (f x + e\right )^{3} - 8 \, c^{4} f \cos \left (f x + e\right )^{2} + 4 \, c^{4} f \cos \left (f x + e\right ) + 8 \, c^{4} f + {\left (c^{4} f \cos \left (f x + e\right )^{3} + 4 \, c^{4} f \cos \left (f x + e\right )^{2} - 4 \, c^{4} f \cos \left (f x + e\right ) - 8 \, c^{4} f\right )} \sin \left (f x + e\right )\right )}} \]
1/192*(3*sqrt(2)*(a^2*cos(f*x + e)^4 - 3*a^2*cos(f*x + e)^3 - 8*a^2*cos(f* x + e)^2 + 4*a^2*cos(f*x + e) + 8*a^2 + (a^2*cos(f*x + e)^3 + 4*a^2*cos(f* x + e)^2 - 4*a^2*cos(f*x + e) - 8*a^2)*sin(f*x + e))*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) + si n(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f*x + e) - 2*c)*sin(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*(3*a^2*cos(f*x + e)^3 + 25*a^2*cos(f*x + e)^2 - 10*a^2*cos(f*x + e) - 32*a^2 + (3*a^2*cos(f*x + e)^2 - 22*a^2*cos(f*x + e) - 32*a^2)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c))/(c^4*f*cos(f*x + e)^4 - 3*c^4*f*cos(f*x + e)^3 - 8*c^4*f*cos(f*x + e)^2 + 4*c^4*f*cos(f*x + e) + 8*c^4*f + (c^4*f*c os(f*x + e)^3 + 4*c^4*f*cos(f*x + e)^2 - 4*c^4*f*cos(f*x + e) - 8*c^4*f)*s in(f*x + e))
Timed out. \[ \int \frac {(3+3 \sin (e+f x))^2}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(3+3 \sin (e+f x))^2}{(c-c \sin (e+f x))^{7/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (133) = 266\).
Time = 0.41 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.72 \[ \int \frac {(3+3 \sin (e+f x))^2}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {\frac {12 \, \sqrt {2} a^{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 )}{c^{\frac {7}{2}} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {\sqrt {2} {\left (a^{2} \sqrt {c} + \frac {3 \, a^{2} \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 {3 \, a^{2} \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 {22 \, a^{2} \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}}\right )} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}}{c^{4} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {\sqrt {2} {\left (\frac {3 \, a^{2} c^{\frac {17}{2}} {\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 {3 \, a^{2} c^{\frac {17}{2}} {\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 {a^{2} c^{\frac {17}{2}} {\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}}\right )}}{c^{12} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{768 \, f} \]
1/768*(12*sqrt(2)*a^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))/(c^(7/2)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))) + sqrt(2)*(a^2*sqrt(c) + 3*a^2*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) - 3*a^2*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 - 22*a^2*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)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^3/(c^4*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))) + sqrt(2)*(3*a^2*c^(1 7/2)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) - 3*a^2*c^(17/2)*(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 - a^2*c^(17/2)*(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)/(c^12*sgn(sin(-1/4*pi + 1/2* f*x + 1/2*e))))/f
Timed out. \[ \int \frac {(3+3 \sin (e+f x))^2}{(c-c \sin (e+f x))^{7/2}} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2}} \,d x \]