Integrand size = 38, antiderivative size = 200 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {8 \sqrt {2} a^3 (A+B) \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{\sqrt {c} f}-\frac {2 a^3 B c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}-\frac {2 a^3 (A+B) c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}-\frac {4 a^3 (A+B) c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\frac {8 a^3 (A+B) \cos (e+f x)}{f \sqrt {c-c \sin (e+f x)}} \] Output:
8*2^(1/2)*a^3*(A+B)*arctanh(1/2*c^(1/2)*cos(f*x+e)*2^(1/2)/(c-c*sin(f*x+e) )^(1/2))/c^(1/2)/f-2/7*a^3*B*c^3*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^(7/2)-2/5 *a^3*(A+B)*c^2*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^(5/2)-4/3*a^3*(A+B)*c*cos(f *x+e)^3/f/(c-c*sin(f*x+e))^(3/2)-8*a^3*(A+B)*cos(f*x+e)/f/(c-c*sin(f*x+e)) ^(1/2)
Result contains complex when optimal does not.
Time = 12.03 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.96 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^3 \left ((6720+6720 i) \sqrt [4]{-1} (A+B) \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right )-2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (-2086 A-2236 B+6 (7 A+22 B) \cos (2 (e+f x))-(448 A+673 B) \sin (e+f x)+15 B \sin (3 (e+f x)))\right )}{420 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 \sqrt {c-c \sin (e+f x)}} \] Input:
Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/Sqrt[c - c*Sin[e + f*x]],x]
Output:
-1/420*(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*((6 720 + 6720*I)*(-1)^(1/4)*(A + B)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])] - 2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-2086*A - 2236*B + 6*(7*A + 22*B)*Cos[2*(e + f*x)] - (448*A + 673*B)*Sin[e + f*x] + 15*B*Sin [3*(e + f*x)])))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*Sqrt[c - c*Sin [e + f*x]])
Time = 1.10 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.342, Rules used = {3042, 3446, 3042, 3339, 3042, 3158, 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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^3 (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^3 (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}}dx\) |
| \(\Big \downarrow \) 3446 |
| \(\displaystyle a^3 c^3 \int \frac {\cos ^6(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \int \frac {\cos (e+f x)^6 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3339 |
| \(\displaystyle a^3 c^3 \left ((A+B) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{7/2}}dx-\frac {2 B \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left ((A+B) \int \frac {\cos (e+f x)^6}{(c-c \sin (e+f x))^{7/2}}dx-\frac {2 B \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}\right )\) |
| \(\Big \downarrow \) 3158 |
| \(\displaystyle a^3 c^3 \left ((A+B) \left (\frac {2 \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{5/2}}dx}{c}-\frac {2 \cos ^5(e+f x)}{5 c f (c-c \sin (e+f x))^{5/2}}\right )-\frac {2 B \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left ((A+B) \left (\frac {2 \int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^{5/2}}dx}{c}-\frac {2 \cos ^5(e+f x)}{5 c f (c-c \sin (e+f x))^{5/2}}\right )-\frac {2 B \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}\right )\) |
| \(\Big \downarrow \) 3158 |
| \(\displaystyle a^3 c^3 \left ((A+B) \left (\frac {2 \left (\frac {2 \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^{3/2}}dx}{c}-\frac {2 \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{3/2}}\right )}{c}-\frac {2 \cos ^5(e+f x)}{5 c f (c-c \sin (e+f x))^{5/2}}\right )-\frac {2 B \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left ((A+B) \left (\frac {2 \left (\frac {2 \int \frac {\cos (e+f x)^2}{(c-c \sin (e+f x))^{3/2}}dx}{c}-\frac {2 \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{3/2}}\right )}{c}-\frac {2 \cos ^5(e+f x)}{5 c f (c-c \sin (e+f x))^{5/2}}\right )-\frac {2 B \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}\right )\) |
| \(\Big \downarrow \) 3158 |
| \(\displaystyle a^3 c^3 \left ((A+B) \left (\frac {2 \left (\frac {2 \left (\frac {2 \int \frac {1}{\sqrt {c-c \sin (e+f x)}}dx}{c}-\frac {2 \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}}\right )}{c}-\frac {2 \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{3/2}}\right )}{c}-\frac {2 \cos ^5(e+f x)}{5 c f (c-c \sin (e+f x))^{5/2}}\right )-\frac {2 B \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left ((A+B) \left (\frac {2 \left (\frac {2 \left (\frac {2 \int \frac {1}{\sqrt {c-c \sin (e+f x)}}dx}{c}-\frac {2 \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}}\right )}{c}-\frac {2 \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{3/2}}\right )}{c}-\frac {2 \cos ^5(e+f x)}{5 c f (c-c \sin (e+f x))^{5/2}}\right )-\frac {2 B \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}\right )\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle a^3 c^3 \left ((A+B) \left (\frac {2 \left (\frac {2 \left (-\frac {4 \int \frac {1}{2 c-\frac {c^2 \cos ^2(e+f x)}{c-c \sin (e+f x)}}d\left (-\frac {c \cos (e+f x)}{\sqrt {c-c \sin (e+f x)}}\right )}{c f}-\frac {2 \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}}\right )}{c}-\frac {2 \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{3/2}}\right )}{c}-\frac {2 \cos ^5(e+f x)}{5 c f (c-c \sin (e+f x))^{5/2}}\right )-\frac {2 B \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle a^3 c^3 \left ((A+B) \left (\frac {2 \left (\frac {2 \left (\frac {2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{c^{3/2} f}-\frac {2 \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}}\right )}{c}-\frac {2 \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^{3/2}}\right )}{c}-\frac {2 \cos ^5(e+f x)}{5 c f (c-c \sin (e+f x))^{5/2}}\right )-\frac {2 B \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}\right )\) |
Input:
Int[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/Sqrt[c - c*Sin[e + f*x]] ,x]
Output:
a^3*c^3*((-2*B*Cos[e + f*x]^7)/(7*f*(c - c*Sin[e + f*x])^(7/2)) + (A + B)* ((-2*Cos[e + f*x]^5)/(5*c*f*(c - c*Sin[e + f*x])^(5/2)) + (2*((-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*Cos [e + f*x])/(c*f*Sqrt[c - c*Sin[e + f*x]])))/c))/c))
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_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)* (g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x] + S imp[(a*d*m + b*c*(m + p + 1))/(b*(m + p + 1)) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[ a^2 - b^2, 0] && NeQ[m + p + 1, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Si mp[a^m*c^m Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B*Sin [e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a* d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Time = 1.83 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.16
| method | result | size |
| default | \(\frac {2 \left (\sin \left (f x +e \right )-1\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, a^{3} \left (-420 c^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) A -420 c^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) B +15 B \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {7}{2}}+21 A \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {5}{2}} c +21 B \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {5}{2}} c +70 A \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}} c^{2}+70 B \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}} c^{2}+420 A \,c^{3} \sqrt {c \left (1+\sin \left (f x +e \right )\right )}+420 B \,c^{3} \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\right )}{105 c^{4} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(233\) |
| parts | \(-\frac {a^{3} A \left (\sin \left (f x +e \right )-1\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\sqrt {c}\, \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {a^{3} B \left (\sin \left (f x +e \right )-1\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \left (105 c^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-30 \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {7}{2}}+84 c \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {5}{2}}-140 c^{2} \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}}\right )}{105 c^{4} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {a^{3} \left (A +3 B \right ) \left (\sin \left (f x +e \right )-1\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \left (15 c^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-6 \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {5}{2}}+10 c \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}}-30 c^{2} \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\right )}{15 c^{3} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {a^{3} \left (3 A +B \right ) \left (\sin \left (f x +e \right )-1\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \left (\sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-2 \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\right )}{c \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {a^{3} \left (A +B \right ) \left (\sin \left (f x +e \right )-1\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \left (-3 c^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )+2 \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}}\right )}{c^{2} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(539\) |
Input:
int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(1/2),x,method=_R ETURNVERBOSE)
Output:
2/105*(sin(f*x+e)-1)*(c*(1+sin(f*x+e)))^(1/2)*a^3*(-420*c^(7/2)*2^(1/2)*ar ctanh(1/2*(c*(1+sin(f*x+e)))^(1/2)*2^(1/2)/c^(1/2))*A-420*c^(7/2)*2^(1/2)* arctanh(1/2*(c*(1+sin(f*x+e)))^(1/2)*2^(1/2)/c^(1/2))*B+15*B*(c*(1+sin(f*x +e)))^(7/2)+21*A*(c*(1+sin(f*x+e)))^(5/2)*c+21*B*(c*(1+sin(f*x+e)))^(5/2)* c+70*A*(c*(1+sin(f*x+e)))^(3/2)*c^2+70*B*(c*(1+sin(f*x+e)))^(3/2)*c^2+420* A*c^3*(c*(1+sin(f*x+e)))^(1/2)+420*B*c^3*(c*(1+sin(f*x+e)))^(1/2))/c^4/cos (f*x+e)/(c-c*sin(f*x+e))^(1/2)/f
Time = 0.10 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.76 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {2 \, {\left (\frac {210 \, \sqrt {2} {\left ({\left (A + B\right )} a^{3} c \cos \left (f x + e\right ) - {\left (A + B\right )} a^{3} c \sin \left (f x + e\right ) + {\left (A + B\right )} a^{3} c\right )} \log \left (-\frac {\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) - 2\right )} \sin \left (f x + e\right ) + \frac {2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )}}{\sqrt {c}} + 3 \, \cos \left (f x + e\right ) + 2}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right )}{\sqrt {c}} - {\left (15 \, B a^{3} \cos \left (f x + e\right )^{4} - 3 \, {\left (7 \, A + 22 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} - {\left (133 \, A + 253 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 4 \, {\left (133 \, A + 148 \, B\right )} a^{3} \cos \left (f x + e\right ) + 4 \, {\left (161 \, A + 191 \, B\right )} a^{3} - {\left (15 \, B a^{3} \cos \left (f x + e\right )^{3} + 3 \, {\left (7 \, A + 27 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 4 \, {\left (28 \, A + 43 \, B\right )} a^{3} \cos \left (f x + e\right ) - 4 \, {\left (161 \, A + 191 \, B\right )} a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}\right )}}{105 \, {\left (c f \cos \left (f x + e\right ) - c f \sin \left (f x + e\right ) + c f\right )}} \] Input:
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(1/2),x, al gorithm="fricas")
Output:
2/105*(210*sqrt(2)*((A + B)*a^3*c*cos(f*x + e) - (A + B)*a^3*c*sin(f*x + e ) + (A + B)*a^3*c)*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)/sq rt(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) - (15*B*a^3*cos(f*x + e)^4 - 3*(7*A + 22* B)*a^3*cos(f*x + e)^3 - (133*A + 253*B)*a^3*cos(f*x + e)^2 + 4*(133*A + 14 8*B)*a^3*cos(f*x + e) + 4*(161*A + 191*B)*a^3 - (15*B*a^3*cos(f*x + e)^3 + 3*(7*A + 27*B)*a^3*cos(f*x + e)^2 - 4*(28*A + 43*B)*a^3*cos(f*x + e) - 4* (161*A + 191*B)*a^3)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c))/(c*f*cos(f*x + e) - c*f*sin(f*x + e) + c*f)
\[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=a^{3} \left (\int \frac {A}{\sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {3 A \sin {\left (e + f x \right )}}{\sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {3 A \sin ^{2}{\left (e + f x \right )}}{\sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {A \sin ^{3}{\left (e + f x \right )}}{\sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {B \sin {\left (e + f x \right )}}{\sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {3 B \sin ^{2}{\left (e + f x \right )}}{\sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {3 B \sin ^{3}{\left (e + f x \right )}}{\sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {B \sin ^{4}{\left (e + f x \right )}}{\sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx\right ) \] Input:
integrate((a+a*sin(f*x+e))**3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**(1/2),x)
Output:
a**3*(Integral(A/sqrt(-c*sin(e + f*x) + c), x) + Integral(3*A*sin(e + f*x) /sqrt(-c*sin(e + f*x) + c), x) + Integral(3*A*sin(e + f*x)**2/sqrt(-c*sin( e + f*x) + c), x) + Integral(A*sin(e + f*x)**3/sqrt(-c*sin(e + f*x) + c), x) + Integral(B*sin(e + f*x)/sqrt(-c*sin(e + f*x) + c), x) + Integral(3*B* sin(e + f*x)**2/sqrt(-c*sin(e + f*x) + c), x) + Integral(3*B*sin(e + f*x)* *3/sqrt(-c*sin(e + f*x) + c), x) + Integral(B*sin(e + f*x)**4/sqrt(-c*sin( e + f*x) + c), x))
\[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{3}}{\sqrt {-c \sin \left (f x + e\right ) + c}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(1/2),x, al gorithm="maxima")
Output:
integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^3/sqrt(-c*sin(f*x + e) + c), x)
Exception generated. \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(1/2),x, al gorithm="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. \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3}{\sqrt {c-c\,\sin \left (e+f\,x\right )}} \,d x \] Input:
int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3)/(c - c*sin(e + f*x))^(1/ 2),x)
Output:
int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3)/(c - c*sin(e + f*x))^(1/ 2), x)
\[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {\sqrt {c}\, a^{3} \left (-\left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}}{\sin \left (f x +e \right )-1}d x \right ) a -\left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{4}}{\sin \left (f x +e \right )-1}d x \right ) b -\left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}}{\sin \left (f x +e \right )-1}d x \right ) a -3 \left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}}{\sin \left (f x +e \right )-1}d x \right ) b -3 \left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )-1}d x \right ) a -3 \left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )-1}d x \right ) b -3 \left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )-1}d x \right ) a -\left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )-1}d x \right ) b \right )}{c} \] Input:
int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(1/2),x)
Output:
(sqrt(c)*a**3*( - int(sqrt( - sin(e + f*x) + 1)/(sin(e + f*x) - 1),x)*a - int((sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**4)/(sin(e + f*x) - 1),x)*b - int((sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**3)/(sin(e + f*x) - 1),x)*a - 3*int((sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**3)/(sin(e + f*x) - 1),x)*b - 3*int((sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**2)/(sin(e + f*x) - 1),x)* a - 3*int((sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**2)/(sin(e + f*x) - 1),x )*b - 3*int((sqrt( - sin(e + f*x) + 1)*sin(e + f*x))/(sin(e + f*x) - 1),x) *a - int((sqrt( - sin(e + f*x) + 1)*sin(e + f*x))/(sin(e + f*x) - 1),x)*b) )/c