Integrand size = 33, antiderivative size = 240 \[ \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)} \, dx=-\frac {2 E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right ) (b+c \cos (d+e x)+a \sin (d+e x))^2}{\left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x) \sqrt {\frac {b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt {a^2+c^2}}}}-\frac {2 (b+c \cos (d+e x)+a \sin (d+e x)) (a \cos (d+e x)-c \sin (d+e x))}{\left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)} \]
-2*(b+c*cos(e*x+d)+a*sin(e*x+d))*(a*cos(e*x+d)-c*sin(e*x+d))/(a^2-b^2+c^2) /e/(a+c*cot(e*x+d)+b*csc(e*x+d))^(3/2)/sin(e*x+d)^(3/2)-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))*(b+c*cos(e*x+d)+a*sin(e*x+d))^2/(a^2-b^2+c^2)/e/(a+c*cot(e*x +d)+b*csc(e*x+d))^(3/2)/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)
Result contains complex when optimal does not.
Time = 17.46 (sec) , antiderivative size = 5959, normalized size of antiderivative = 24.83 \[ \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)} \, dx=\text {Result too large to show} \]
Time = 0.79 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3042, 3643, 3042, 3607, 3042, 3598, 3042, 3132}
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.
| \(\displaystyle \int \frac {1}{\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (d+e x)^{3/2} (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3643 |
| \(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{3/2} \int \frac {1}{(b+c \cos (d+e x)+a \sin (d+e x))^{3/2}}dx}{\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{3/2} \int \frac {1}{(b+c \cos (d+e x)+a \sin (d+e x))^{3/2}}dx}{\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}\) |
| \(\Big \downarrow \) 3607 |
| \(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{3/2} \left (-\frac {\int \sqrt {b+c \cos (d+e x)+a \sin (d+e x)}dx}{a^2-b^2+c^2}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}\right )}{\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{3/2} \left (-\frac {\int \sqrt {b+c \cos (d+e x)+a \sin (d+e x)}dx}{a^2-b^2+c^2}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}\right )}{\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}\) |
| \(\Big \downarrow \) 3598 |
| \(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{3/2} \left (-\frac {\sqrt {a \sin (d+e x)+b+c \cos (d+e x)} \int \sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \cos \left (d+e x-\tan ^{-1}(c,a)\right )}{b+\sqrt {a^2+c^2}}}dx}{\left (a^2-b^2+c^2\right ) \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}}}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}\right )}{\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{3/2} \left (-\frac {\sqrt {a \sin (d+e x)+b+c \cos (d+e x)} \int \sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \sin \left (d+e x-\tan ^{-1}(c,a)+\frac {\pi }{2}\right )}{b+\sqrt {a^2+c^2}}}dx}{\left (a^2-b^2+c^2\right ) \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}}}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}\right )}{\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{3/2} \left (-\frac {2 \sqrt {a \sin (d+e x)+b+c \cos (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \left (a^2-b^2+c^2\right ) \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}}}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}\right )}{\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}\) |
((b + c*Cos[d + e*x] + a*Sin[d + e*x])^(3/2)*((-2*EllipticE[(d + e*x - Arc Tan[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]])/((a^2 - b^2 + c^2)*e*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]) - (2*(a*Cos[d + e*x] - c*Sin[d + e *x]))/((a^2 - b^2 + c^2)*e*Sqrt[b + c*Cos[d + e*x] + a*Sin[d + e*x]])))/(( a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sin[d + e*x]^(3/2))
3.5.70.3.1 Defintions of rubi rules used
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[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_)])^ (-3/2), x_Symbol] :> Simp[2*((c*Cos[d + e*x] - b*Sin[d + e*x])/(e*(a^2 - b^ 2 - c^2)*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])), x] + Simp[1/(a^2 - b^ 2 - c^2) Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x], x] /; FreeQ[{ a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 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 = 26.13 (sec) , antiderivative size = 12838, normalized size of antiderivative = 53.49
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 1746, normalized size of antiderivative = 7.28 \[ \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)} \, dx=\text {Too large to display} \]
-1/3*((sqrt(2)*(a*b*c + I*b*c^2)*cos(e*x + d) + sqrt(2)*(a^2*b + I*a*b*c)* sin(e*x + d) + sqrt(2)*(a*b^2 + I*b^2*c))*sqrt(I*a + c)*weierstrassPInvers e(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)*(a*b*c - I*b*c^2)*cos(e*x + d) + sqrt(2)*(a^2*b - I*a*b*c)*sin(e*x + d) + sqrt(2)*(a*b^2 - I*b^2*c))*sqrt(-I*a + c)*weier strassPInverse(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)) + 3*(sqrt(2)*(-I*a^2*c - I*c^3)*cos(e*x + d) + sqrt(2)*(-I*a^3 - I*a*c^2)*sin(e*x + d) + sqrt(2)*(-I*a^2*b - I*b*c^2))*s qrt(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^...
Timed out. \[ \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)} \, dx=\int { \frac {1}{{\left (c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a\right )}^{\frac {3}{2}} \sin \left (e x + d\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)} \, dx=\int { \frac {1}{{\left (c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a\right )}^{\frac {3}{2}} \sin \left (e x + d\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)} \, dx=\int \frac {1}{{\sin \left (d+e\,x\right )}^{3/2}\,{\left (a+c\,\mathrm {cot}\left (d+e\,x\right )+\frac {b}{\sin \left (d+e\,x\right )}\right )}^{3/2}} \,d x \]