Integrand size = 26, antiderivative size = 145 \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^{7/2}} \, dx=-\frac {a \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{32 \sqrt {2} c^{7/2} f}+\frac {a \cos (e+f x)}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \cos (e+f x)}{24 c f (c-c \sin (e+f x))^{5/2}}-\frac {a \cos (e+f x)}{32 c^2 f (c-c \sin (e+f x))^{3/2}} \] Output:
-1/64*a*arctanh(1/2*c^(1/2)*cos(f*x+e)*2^(1/2)/(c-c*sin(f*x+e))^(1/2))*2^( 1/2)/c^(7/2)/f+1/3*a*cos(f*x+e)/f/(c-c*sin(f*x+e))^(7/2)-1/24*a*cos(f*x+e) /c/f/(c-c*sin(f*x+e))^(5/2)-1/32*a*cos(f*x+e)/c^2/f/(c-c*sin(f*x+e))^(3/2)
Time = 5.04 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.30 \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {a \left (12 \sqrt {2} \arctan \left (\frac {\sqrt {-c (1+\sin (e+f x))}}{\sqrt {2} \sqrt {c}}\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 \sqrt {-c (1+\sin (e+f x))}+2 \sqrt {c} (-14 \cos (2 (e+f x))+131 \sin (e+f x)+3 (38+\sin (3 (e+f x))))\right )}{768 c^{7/2} f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c-c \sin (e+f x)}} \] Input:
Integrate[(a + a*Sin[e + f*x])/(c - c*Sin[e + f*x])^(7/2),x]
Output:
(a*(12*Sqrt[2]*ArcTan[Sqrt[-(c*(1 + Sin[e + f*x]))]/(Sqrt[2]*Sqrt[c])]*(Co s[(e + f*x)/2] - Sin[(e + f*x)/2])^6*Sqrt[-(c*(1 + Sin[e + f*x]))] + 2*Sqr t[c]*(-14*Cos[2*(e + f*x)] + 131*Sin[e + f*x] + 3*(38 + Sin[3*(e + f*x)])) ))/(768*c^(7/2)*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2 ] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]])
Time = 0.67 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {3042, 3215, 3042, 3159, 3042, 3129, 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.
| \(\displaystyle \int \frac {a \sin (e+f x)+a}{(c-c \sin (e+f x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a \sin (e+f x)+a}{(c-c \sin (e+f x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle a c \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^{9/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a c \int \frac {\cos (e+f x)^2}{(c-c \sin (e+f x))^{9/2}}dx\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a c \left (\frac {\cos (e+f x)}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {\int \frac {1}{(c-c \sin (e+f x))^{5/2}}dx}{6 c^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a c \left (\frac {\cos (e+f x)}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {\int \frac {1}{(c-c \sin (e+f x))^{5/2}}dx}{6 c^2}\right )\) |
| \(\Big \downarrow \) 3129 |
| \(\displaystyle a c \left (\frac {\cos (e+f x)}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {\frac {3 \int \frac {1}{(c-c \sin (e+f x))^{3/2}}dx}{8 c}+\frac {\cos (e+f x)}{4 f (c-c \sin (e+f x))^{5/2}}}{6 c^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a c \left (\frac {\cos (e+f x)}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {\frac {3 \int \frac {1}{(c-c \sin (e+f x))^{3/2}}dx}{8 c}+\frac {\cos (e+f x)}{4 f (c-c \sin (e+f x))^{5/2}}}{6 c^2}\right )\) |
| \(\Big \downarrow \) 3129 |
| \(\displaystyle a c \left (\frac {\cos (e+f x)}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {\frac {3 \left (\frac {\int \frac {1}{\sqrt {c-c \sin (e+f x)}}dx}{4 c}+\frac {\cos (e+f x)}{2 f (c-c \sin (e+f x))^{3/2}}\right )}{8 c}+\frac {\cos (e+f x)}{4 f (c-c \sin (e+f x))^{5/2}}}{6 c^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a c \left (\frac {\cos (e+f x)}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {\frac {3 \left (\frac {\int \frac {1}{\sqrt {c-c \sin (e+f x)}}dx}{4 c}+\frac {\cos (e+f x)}{2 f (c-c \sin (e+f x))^{3/2}}\right )}{8 c}+\frac {\cos (e+f x)}{4 f (c-c \sin (e+f x))^{5/2}}}{6 c^2}\right )\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle a c \left (\frac {\cos (e+f x)}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {\frac {3 \left (\frac {\cos (e+f x)}{2 f (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {1}{2 c-\frac {c^2 \cos ^2(e+f x)}{c-c \sin (e+f x)}}d\left (-\frac {c \cos (e+f x)}{\sqrt {c-c \sin (e+f x)}}\right )}{2 c f}\right )}{8 c}+\frac {\cos (e+f x)}{4 f (c-c \sin (e+f x))^{5/2}}}{6 c^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle a c \left (\frac {\cos (e+f x)}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {\frac {3 \left (\frac {\text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{2 \sqrt {2} c^{3/2} f}+\frac {\cos (e+f x)}{2 f (c-c \sin (e+f x))^{3/2}}\right )}{8 c}+\frac {\cos (e+f x)}{4 f (c-c \sin (e+f x))^{5/2}}}{6 c^2}\right )\) |
Input:
Int[(a + a*Sin[e + f*x])/(c - c*Sin[e + f*x])^(7/2),x]
Output:
a*c*(Cos[e + f*x]/(3*c*f*(c - c*Sin[e + f*x])^(7/2)) - (Cos[e + f*x]/(4*f* (c - c*Sin[e + f*x])^(5/2)) + (3*(ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[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))))/(8*c))/(6*c^2))
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 = 0.78 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.68
| method | result | size |
| default | \(\frac {a \left (3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sin \left (f x +e \right )^{3} c^{5}-6 \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {5}{2}} c^{\frac {5}{2}}-9 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sin \left (f x +e \right )^{2} c^{5}+32 \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}} c^{\frac {7}{2}}+9 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sin \left (f x +e \right ) c^{5}+24 \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, c^{\frac {9}{2}}-3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{5}\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}}{192 c^{\frac {17}{2}} \left (-1+\sin \left (f x +e \right )\right )^{2} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(243\) |
| parts | \(\frac {a \left (-15 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sin \left (f x +e \right )^{3} c^{2}+30 \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, c^{\frac {3}{2}} \sin \left (f x +e \right )^{2}+45 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sin \left (f x +e \right )^{2} c^{2}-100 \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, c^{\frac {3}{2}} \sin \left (f x +e \right )-45 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} \sin \left (f x +e \right )+134 \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, c^{\frac {3}{2}}+15 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2}\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}}{384 c^{\frac {11}{2}} \left (-1+\sin \left (f x +e \right )\right )^{2} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {a \left (21 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sin \left (f x +e \right )^{3} c^{5}-42 \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {5}{2}} c^{\frac {5}{2}}-63 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sin \left (f x +e \right )^{2} c^{5}+224 \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}} c^{\frac {7}{2}}+63 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sin \left (f x +e \right ) c^{5}-216 \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, c^{\frac {9}{2}}-21 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{5}\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}}{384 c^{\frac {17}{2}} \left (-1+\sin \left (f x +e \right )\right )^{2} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(500\) |
Input:
int((a+sin(f*x+e)*a)/(c-c*sin(f*x+e))^(7/2),x,method=_RETURNVERBOSE)
Output:
1/192/c^(17/2)*a*(3*2^(1/2)*arctanh(1/2*(c*(1+sin(f*x+e)))^(1/2)*2^(1/2)/c ^(1/2))*sin(f*x+e)^3*c^5-6*(c*(1+sin(f*x+e)))^(5/2)*c^(5/2)-9*2^(1/2)*arct anh(1/2*(c*(1+sin(f*x+e)))^(1/2)*2^(1/2)/c^(1/2))*sin(f*x+e)^2*c^5+32*(c*( 1+sin(f*x+e)))^(3/2)*c^(7/2)+9*2^(1/2)*arctanh(1/2*(c*(1+sin(f*x+e)))^(1/2 )*2^(1/2)/c^(1/2))*sin(f*x+e)*c^5+24*(c*(1+sin(f*x+e)))^(1/2)*c^(9/2)-3*2^ (1/2)*arctanh(1/2*(c*(1+sin(f*x+e)))^(1/2)*2^(1/2)/c^(1/2))*c^5)*(c*(1+sin (f*x+e)))^(1/2)/(-1+sin(f*x+e))^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. 408 vs. \(2 (122) = 244\).
Time = 0.10 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.81 \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {3 \, \sqrt {2} {\left (a \cos \left (f x + e\right )^{4} - 3 \, a \cos \left (f x + e\right )^{3} - 8 \, a \cos \left (f x + e\right )^{2} + 4 \, a \cos \left (f x + e\right ) + {\left (a \cos \left (f x + e\right )^{3} + 4 \, a \cos \left (f x + e\right )^{2} - 4 \, a \cos \left (f x + e\right ) - 8 \, a\right )} \sin \left (f x + e\right ) + 8 \, a\right )} \sqrt {c} \log \left (-\frac {c \cos \left (f x + e\right )^{2} - 2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} \sqrt {c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )} + 3 \, c \cos \left (f x + e\right ) + {\left (c \cos \left (f x + e\right ) - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) + 4 \, {\left (3 \, a \cos \left (f x + e\right )^{3} - 7 \, a \cos \left (f x + e\right )^{2} + 22 \, a \cos \left (f x + e\right ) + {\left (3 \, a \cos \left (f x + e\right )^{2} + 10 \, a \cos \left (f x + e\right ) + 32 \, a\right )} \sin \left (f x + e\right ) + 32 \, a\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{384 \, {\left (c^{4} f \cos \left (f x + e\right )^{4} - 3 \, c^{4} f \cos \left (f x + e\right )^{3} - 8 \, c^{4} f \cos \left (f x + e\right )^{2} + 4 \, c^{4} f \cos \left (f x + e\right ) + 8 \, c^{4} f + {\left (c^{4} f \cos \left (f x + e\right )^{3} + 4 \, c^{4} f \cos \left (f x + e\right )^{2} - 4 \, c^{4} f \cos \left (f x + e\right ) - 8 \, c^{4} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((a+a*sin(f*x+e))/(c-c*sin(f*x+e))^(7/2),x, algorithm="fricas")
Output:
1/384*(3*sqrt(2)*(a*cos(f*x + e)^4 - 3*a*cos(f*x + e)^3 - 8*a*cos(f*x + e) ^2 + 4*a*cos(f*x + e) + (a*cos(f*x + e)^3 + 4*a*cos(f*x + e)^2 - 4*a*cos(f *x + e) - 8*a)*sin(f*x + e) + 8*a)*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)*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*cos(f* x + e)^3 - 7*a*cos(f*x + e)^2 + 22*a*cos(f*x + e) + (3*a*cos(f*x + e)^2 + 10*a*cos(f*x + e) + 32*a)*sin(f*x + e) + 32*a)*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*cos(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)*sin(f*x + e))
Timed out. \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(f*x+e))/(c-c*sin(f*x+e))**(7/2),x)
Output:
Timed out
\[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^{7/2}} \, dx=\int { \frac {a \sin \left (f x + e\right ) + a}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))/(c-c*sin(f*x+e))^(7/2),x, algorithm="maxima")
Output:
integrate((a*sin(f*x + e) + a)/(-c*sin(f*x + e) + c)^(7/2), x)
Exception generated. \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+a*sin(f*x+e))/(c-c*sin(f*x+e))^(7/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^{7/2}} \, dx=\int \frac {a+a\,\sin \left (e+f\,x\right )}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2}} \,d x \] Input:
int((a + a*sin(e + f*x))/(c - c*sin(e + f*x))^(7/2),x)
Output:
int((a + a*sin(e + f*x))/(c - c*sin(e + f*x))^(7/2), x)
\[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {\sqrt {c}\, a \left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}}{\sin \left (f x +e \right )^{4}-4 \sin \left (f x +e \right )^{3}+6 \sin \left (f x +e \right )^{2}-4 \sin \left (f x +e \right )+1}d x +\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{4}-4 \sin \left (f x +e \right )^{3}+6 \sin \left (f x +e \right )^{2}-4 \sin \left (f x +e \right )+1}d x \right )}{c^{4}} \] Input:
int((a+a*sin(f*x+e))/(c-c*sin(f*x+e))^(7/2),x)
Output:
(sqrt(c)*a*(int(sqrt( - sin(e + f*x) + 1)/(sin(e + f*x)**4 - 4*sin(e + f*x )**3 + 6*sin(e + f*x)**2 - 4*sin(e + f*x) + 1),x) + int((sqrt( - sin(e + f *x) + 1)*sin(e + f*x))/(sin(e + f*x)**4 - 4*sin(e + f*x)**3 + 6*sin(e + f* x)**2 - 4*sin(e + f*x) + 1),x)))/c**4