Integrand size = 33, antiderivative size = 371 \[ \int (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x) \, dx=\frac {8 b (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right ) \sin ^{\frac {3}{2}}(d+e x)}{3 e (b+c \cos (d+e x)+a \sin (d+e x)) \sqrt {\frac {b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt {a^2+c^2}}}}+\frac {2 \left (a^2-b^2+c^2\right ) (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right ),\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right ) \sin ^{\frac {3}{2}}(d+e x) \sqrt {\frac {b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt {a^2+c^2}}}}{3 e (b+c \cos (d+e x)+a \sin (d+e x))^2}-\frac {2 (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x) (a \cos (d+e x)-c \sin (d+e x))}{3 e (b+c \cos (d+e x)+a \sin (d+e x))} \]
-2/3*(a+c*cot(e*x+d)+b*csc(e*x+d))^(3/2)*sin(e*x+d)^(3/2)*(a*cos(e*x+d)-c* sin(e*x+d))/e/(b+c*cos(e*x+d)+a*sin(e*x+d))+8/3*b*(a+c*cot(e*x+d)+b*csc(e* x+d))^(3/2)*(cos(1/2*d+1/2*e*x-1/2*arctan(c,a))^2)^(1/2)/cos(1/2*d+1/2*e*x -1/2*arctan(c,a))*EllipticE(sin(1/2*d+1/2*e*x-1/2*arctan(c,a)),2^(1/2)*((a ^2+c^2)^(1/2)/(b+(a^2+c^2)^(1/2)))^(1/2))*sin(e*x+d)^(3/2)/e/(b+c*cos(e*x+ d)+a*sin(e*x+d))/((b+c*cos(e*x+d)+a*sin(e*x+d))/(b+(a^2+c^2)^(1/2)))^(1/2) +2/3*(a^2-b^2+c^2)*(a+c*cot(e*x+d)+b*csc(e*x+d))^(3/2)*(cos(1/2*d+1/2*e*x- 1/2*arctan(c,a))^2)^(1/2)/cos(1/2*d+1/2*e*x-1/2*arctan(c,a))*EllipticF(sin (1/2*d+1/2*e*x-1/2*arctan(c,a)),2^(1/2)*((a^2+c^2)^(1/2)/(b+(a^2+c^2)^(1/2 )))^(1/2))*sin(e*x+d)^(3/2)*((b+c*cos(e*x+d)+a*sin(e*x+d))/(b+(a^2+c^2)^(1 /2)))^(1/2)/e/(b+c*cos(e*x+d)+a*sin(e*x+d))^2
Result contains complex when optimal does not.
Time = 26.31 (sec) , antiderivative size = 5904, normalized size of antiderivative = 15.91 \[ \int (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x) \, dx=\text {Result too large to show} \]
Time = 1.36 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.91, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.424, Rules used = {3042, 3643, 3042, 3599, 27, 3042, 3628, 3042, 3598, 3042, 3132, 3606, 3042, 3140}
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.
| \(\displaystyle \int \sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (d+e x)^{3/2} (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}dx\) |
| \(\Big \downarrow \) 3643 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2} \int (b+c \cos (d+e x)+a \sin (d+e x))^{3/2}dx}{(a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2} \int (b+c \cos (d+e x)+a \sin (d+e x))^{3/2}dx}{(a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\) |
| \(\Big \downarrow \) 3599 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2} \left (\frac {2}{3} \int \frac {a^2+4 b \sin (d+e x) a+3 b^2+c^2+4 b c \cos (d+e x)}{2 \sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}dx-\frac {2 (a \cos (d+e x)-c \sin (d+e x)) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}{3 e}\right )}{(a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2} \left (\frac {1}{3} \int \frac {a^2+4 b \sin (d+e x) a+3 b^2+c^2+4 b c \cos (d+e x)}{\sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}dx-\frac {2 (a \cos (d+e x)-c \sin (d+e x)) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}{3 e}\right )}{(a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2} \left (\frac {1}{3} \int \frac {a^2+4 b \sin (d+e x) a+3 b^2+c^2+4 b c \cos (d+e x)}{\sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}dx-\frac {2 (a \cos (d+e x)-c \sin (d+e x)) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}{3 e}\right )}{(a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\) |
| \(\Big \downarrow \) 3628 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2} \left (\frac {1}{3} \left (\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}dx+4 b \int \sqrt {b+c \cos (d+e x)+a \sin (d+e x)}dx\right )-\frac {2 (a \cos (d+e x)-c \sin (d+e x)) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}{3 e}\right )}{(a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2} \left (\frac {1}{3} \left (\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}dx+4 b \int \sqrt {b+c \cos (d+e x)+a \sin (d+e x)}dx\right )-\frac {2 (a \cos (d+e x)-c \sin (d+e x)) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}{3 e}\right )}{(a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\) |
| \(\Big \downarrow \) 3598 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2} \left (\frac {1}{3} \left (\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}dx+\frac {4 b \sqrt {a \sin (d+e x)+b+c \cos (d+e x)} \int \sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \cos \left (d+e x-\tan ^{-1}(c,a)\right )}{b+\sqrt {a^2+c^2}}}dx}{\sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}}}\right )-\frac {2 (a \cos (d+e x)-c \sin (d+e x)) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}{3 e}\right )}{(a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2} \left (\frac {1}{3} \left (\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}dx+\frac {4 b \sqrt {a \sin (d+e x)+b+c \cos (d+e x)} \int \sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \sin \left (d+e x-\tan ^{-1}(c,a)+\frac {\pi }{2}\right )}{b+\sqrt {a^2+c^2}}}dx}{\sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}}}\right )-\frac {2 (a \cos (d+e x)-c \sin (d+e x)) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}{3 e}\right )}{(a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2} \left (\frac {1}{3} \left (\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}dx+\frac {8 b \sqrt {a \sin (d+e x)+b+c \cos (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}}}\right )-\frac {2 (a \cos (d+e x)-c \sin (d+e x)) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}{3 e}\right )}{(a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\) |
| \(\Big \downarrow \) 3606 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2+c^2\right ) \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}} \int \frac {1}{\sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \cos \left (d+e x-\tan ^{-1}(c,a)\right )}{b+\sqrt {a^2+c^2}}}}dx}{\sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}+\frac {8 b \sqrt {a \sin (d+e x)+b+c \cos (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}}}\right )-\frac {2 (a \cos (d+e x)-c \sin (d+e x)) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}{3 e}\right )}{(a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2+c^2\right ) \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}} \int \frac {1}{\sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \sin \left (d+e x-\tan ^{-1}(c,a)+\frac {\pi }{2}\right )}{b+\sqrt {a^2+c^2}}}}dx}{\sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}+\frac {8 b \sqrt {a \sin (d+e x)+b+c \cos (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}}}\right )-\frac {2 (a \cos (d+e x)-c \sin (d+e x)) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}{3 e}\right )}{(a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2} \left (\frac {1}{3} \left (\frac {2 \left (a^2-b^2+c^2\right ) \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right ),\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}+\frac {8 b \sqrt {a \sin (d+e x)+b+c \cos (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}}}\right )-\frac {2 (a \cos (d+e x)-c \sin (d+e x)) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}{3 e}\right )}{(a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\) |
((a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sin[d + e*x]^(3/2)*((-2*Sqrt[ b + c*Cos[d + e*x] + a*Sin[d + e*x]]*(a*Cos[d + e*x] - c*Sin[d + e*x]))/(3 *e) + ((8*b*EllipticE[(d + e*x - ArcTan[c, a])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sqrt[b + c*Cos[d + e*x] + a*Sin[d + e*x]])/(e*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]) + (2*(a^2 - b^2 + c^2)*EllipticF[(d + e*x - ArcTan[c, a])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt [a^2 + c^2])]*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c ^2])])/(e*Sqrt[b + c*Cos[d + e*x] + a*Sin[d + e*x]]))/3))/(b + c*Cos[d + e *x] + a*Sin[d + e*x])^(3/2)
3.5.67.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_ )]], x_Symbol] :> Simp[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])] Int[Sqrt[a/(a + S qrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - Arc Tan[b, c]]], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] && !GtQ[a + Sqrt[b^2 + c^2], 0]
Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ (n_), x_Symbol] :> Simp[(-(c*Cos[d + e*x] - b*Sin[d + e*x]))*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)/(e*n)), x] + Simp[1/n Int[Simp[n*a^2 + ( n - 1)*(b^2 + c^2) + a*b*(2*n - 1)*Cos[d + e*x] + a*c*(2*n - 1)*Sin[d + e*x ], x]*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 2), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && GtQ[n, 1]
Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*( x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sq rt[b^2 + c^2])]/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]] Int[1/Sqrt[a/(a + Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - ArcTan[b, c]]], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2 , 0] && NeQ[b^2 + c^2, 0] && !GtQ[a + Sqrt[b^2 + c^2], 0]
Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)]) /Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]] , x_Symbol] :> Simp[B/b Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x] , x] + Simp[(A*b - a*B)/b Int[1/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]] , x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && EqQ[B*c - b*C, 0] && NeQ[ A*b - a*B, 0]
Int[((a_.) + csc[(d_.) + (e_.)*(x_)]*(b_.) + cot[(d_.) + (e_.)*(x_)]*(c_.)) ^(n_)*sin[(d_.) + (e_.)*(x_)]^(n_), x_Symbol] :> Simp[Sin[d + e*x]^n*((a + b*Csc[d + e*x] + c*Cot[d + e*x])^n/(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n) Int[(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n, x], x] /; FreeQ[{a, b, c, d , e}, x] && !IntegerQ[n]
Result contains complex when optimal does not.
Time = 28.39 (sec) , antiderivative size = 21719, normalized size of antiderivative = 58.54
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 1500, normalized size of antiderivative = 4.04 \[ \int (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x) \, dx=\text {Too large to display} \]
1/9*(sqrt(2)*(3*a^3 + a*b^2 + 3*a*c^2 + 3*I*c^3 + I*(3*a^2 + b^2)*c)*sqrt( I*a + c)*weierstrassPInverse(4/3*(3*a^4 - 4*a^2*b^2 + 4*b^2*c^2 + 6*I*a*c^ 3 - 3*c^4 + 2*I*(3*a^3 - 4*a*b^2)*c)/(a^4 + 2*a^2*c^2 + c^4), -8/27*(-9*I* a^5*b + 8*I*a^3*b^3 + 27*I*a*b*c^4 - 9*b*c^5 + 2*(9*a^2*b + 4*b^3)*c^3 + 6 *I*(3*a^3*b - 4*a*b^3)*c^2 + 3*(9*a^4*b - 8*a^2*b^3)*c)/(a^6 + 3*a^4*c^2 + 3*a^2*c^4 + c^6), 1/3*(-2*I*a*b + 2*b*c + 3*(a^2 + c^2)*cos(e*x + d) - 3* (I*a^2 + I*c^2)*sin(e*x + d))/(a^2 + c^2)) + sqrt(2)*(3*a^3 + a*b^2 + 3*a* c^2 - 3*I*c^3 - I*(3*a^2 + b^2)*c)*sqrt(-I*a + c)*weierstrassPInverse(4/3* (3*a^4 - 4*a^2*b^2 + 4*b^2*c^2 - 6*I*a*c^3 - 3*c^4 - 2*I*(3*a^3 - 4*a*b^2) *c)/(a^4 + 2*a^2*c^2 + c^4), -8/27*(9*I*a^5*b - 8*I*a^3*b^3 - 27*I*a*b*c^4 - 9*b*c^5 + 2*(9*a^2*b + 4*b^3)*c^3 - 6*I*(3*a^3*b - 4*a*b^3)*c^2 + 3*(9* a^4*b - 8*a^2*b^3)*c)/(a^6 + 3*a^4*c^2 + 3*a^2*c^4 + c^6), 1/3*(2*I*a*b + 2*b*c + 3*(a^2 + c^2)*cos(e*x + d) - 3*(-I*a^2 - I*c^2)*sin(e*x + d))/(a^2 + c^2)) - 12*sqrt(2)*(I*a^2*b + I*b*c^2)*sqrt(I*a + c)*weierstrassZeta(4/ 3*(3*a^4 - 4*a^2*b^2 + 4*b^2*c^2 + 6*I*a*c^3 - 3*c^4 + 2*I*(3*a^3 - 4*a*b^ 2)*c)/(a^4 + 2*a^2*c^2 + c^4), -8/27*(-9*I*a^5*b + 8*I*a^3*b^3 + 27*I*a*b* c^4 - 9*b*c^5 + 2*(9*a^2*b + 4*b^3)*c^3 + 6*I*(3*a^3*b - 4*a*b^3)*c^2 + 3* (9*a^4*b - 8*a^2*b^3)*c)/(a^6 + 3*a^4*c^2 + 3*a^2*c^4 + c^6), weierstrassP Inverse(4/3*(3*a^4 - 4*a^2*b^2 + 4*b^2*c^2 + 6*I*a*c^3 - 3*c^4 + 2*I*(3*a^ 3 - 4*a*b^2)*c)/(a^4 + 2*a^2*c^2 + c^4), -8/27*(-9*I*a^5*b + 8*I*a^3*b^...
Timed out. \[ \int (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x) \, dx=\text {Timed out} \]
\[ \int (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x) \, dx=\int { {\left (c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a\right )}^{\frac {3}{2}} \sin \left (e x + d\right )^{\frac {3}{2}} \,d x } \]
\[ \int (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x) \, dx=\int { {\left (c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a\right )}^{\frac {3}{2}} \sin \left (e x + d\right )^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x) \, dx=\int {\sin \left (d+e\,x\right )}^{3/2}\,{\left (a+c\,\mathrm {cot}\left (d+e\,x\right )+\frac {b}{\sin \left (d+e\,x\right )}\right )}^{3/2} \,d x \]