Integrand size = 33, antiderivative size = 492 \[ \int \frac {\csc ^{\frac {5}{2}}(d+e x)}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2}} \, dx=\frac {8 b \csc ^{\frac {5}{2}}(d+e x) E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right ) (b+c \cos (d+e x)+a \sin (d+e x))^3}{3 \left (a^2-b^2+c^2\right )^2 e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sqrt {\frac {b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt {a^2+c^2}}}}+\frac {2 \csc ^{\frac {5}{2}}(d+e x) \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right ),\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right ) (b+c \cos (d+e x)+a \sin (d+e x))^2 \sqrt {\frac {b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt {a^2+c^2}}}}{3 \left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2}}-\frac {2 \csc ^{\frac {5}{2}}(d+e x) (b+c \cos (d+e x)+a \sin (d+e x)) (a \cos (d+e x)-c \sin (d+e x))}{3 \left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2}}+\frac {8 \csc ^{\frac {5}{2}}(d+e x) (b+c \cos (d+e x)+a \sin (d+e x))^2 (a b \cos (d+e x)-b c \sin (d+e x))}{3 \left (a^2-b^2+c^2\right )^2 e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2}} \]
-2/3*csc(e*x+d)^(5/2)*(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))^(5/2)+8/3*csc(e*x+d)^( 5/2)*(b+c*cos(e*x+d)+a*sin(e*x+d))^2*(a*b*cos(e*x+d)-b*c*sin(e*x+d))/(a^2- b^2+c^2)^2/e/(a+c*cot(e*x+d)+b*csc(e*x+d))^(5/2)+8/3*b*csc(e*x+d)^(5/2)*(c os(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))^3/(a^2-b^2+c^2)^ 2/e/(a+c*cot(e*x+d)+b*csc(e*x+d))^(5/2)/((b+c*cos(e*x+d)+a*sin(e*x+d))/(b+ (a^2+c^2)^(1/2)))^(1/2)+2/3*csc(e*x+d)^(5/2)*(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)) *(b+c*cos(e*x+d)+a*sin(e*x+d))^2*((b+c*cos(e*x+d)+a*sin(e*x+d))/(b+(a^2+c^ 2)^(1/2)))^(1/2)/(a^2-b^2+c^2)/e/(a+c*cot(e*x+d)+b*csc(e*x+d))^(5/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 7.19 (sec) , antiderivative size = 2708, normalized size of antiderivative = 5.50 \[ \int \frac {\csc ^{\frac {5}{2}}(d+e x)}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2}} \, dx=\text {Result too large to show} \]
(Csc[d + e*x]^(5/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^3*((8*b*(a^2 + c ^2))/(3*a*c*(-a^2 + b^2 - c^2)^2) + (2*(a*b + a^2*Sin[d + e*x] + c^2*Sin[d + e*x]))/(3*c*(a^2 - b^2 + c^2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^2) - (2*(a^3 + 3*a*b^2 + a*c^2 + 4*a^2*b*Sin[d + e*x] + 4*b*c^2*Sin[d + e*x]) )/(3*c*(a^2 - b^2 + c^2)^2*(b + c*Cos[d + e*x] + a*Sin[d + e*x]))))/(e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)) + (4*a*b*Csc[d + e*x]^(5/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^(5/2)*(-((a*AppellF1[-1/2, -1/2, -1/2, 1 /2, -((b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c ^2]*(1 - b/(Sqrt[1 + a^2/c^2]*c))*c)), -((b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(-1 - b/(Sqrt[1 + a^2/c^2]*c))*c))] *Sin[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*c*Sqrt[(c*Sqrt[(a^2 + c^2) /c^2] - c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]])/(b + c*Sqrt[(a ^2 + c^2)/c^2])]*Sqrt[b + c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c ]]]*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] + c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]])/(-b + c*Sqrt[(a^2 + c^2)/c^2])])) - ((2*c*(b + Sqrt[1 + a^2/ c^2]*c*Cos[d + e*x - ArcTan[a/c]]))/(a^2 + c^2) - (a*Sin[d + e*x - ArcTan[ a/c]])/(Sqrt[1 + a^2/c^2]*c))/Sqrt[b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - A rcTan[a/c]]]))/(3*(a^2 - b^2 + c^2)^2*e*(a + c*Cot[d + e*x] + b*Csc[d + e* x])^(5/2)) + (4*b*c^2*Csc[d + e*x]^(5/2)*(b + c*Cos[d + e*x] + a*Sin[d + e *x])^(5/2)*(-((a*AppellF1[-1/2, -1/2, -1/2, 1/2, -((b + Sqrt[1 + a^2/c^...
Time = 1.83 (sec) , antiderivative size = 443, normalized size of antiderivative = 0.90, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.515, Rules used = {3042, 3647, 3042, 3608, 27, 3042, 3635, 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.
| \(\displaystyle \int \frac {\csc ^{\frac {5}{2}}(d+e x)}{(a+b \csc (d+e x)+c \cot (d+e x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc (d+e x)^{5/2}}{(a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3647 |
| \(\displaystyle \frac {\csc ^{\frac {5}{2}}(d+e x) (a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \int \frac {1}{(b+c \cos (d+e x)+a \sin (d+e x))^{5/2}}dx}{(a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\csc ^{\frac {5}{2}}(d+e x) (a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \int \frac {1}{(b+c \cos (d+e x)+a \sin (d+e x))^{5/2}}dx}{(a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3608 |
| \(\displaystyle \frac {\csc ^{\frac {5}{2}}(d+e x) (a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (\frac {2 \int -\frac {3 b-c \cos (d+e x)-a \sin (d+e x)}{2 (b+c \cos (d+e x)+a \sin (d+e x))^{3/2}}dx}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{(a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\csc ^{\frac {5}{2}}(d+e x) (a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {\int \frac {3 b-c \cos (d+e x)-a \sin (d+e x)}{(b+c \cos (d+e x)+a \sin (d+e x))^{3/2}}dx}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{(a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\csc ^{\frac {5}{2}}(d+e x) (a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {\int \frac {3 b-c \cos (d+e x)-a \sin (d+e x)}{(b+c \cos (d+e x)+a \sin (d+e x))^{3/2}}dx}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{(a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3635 |
| \(\displaystyle \frac {\csc ^{\frac {5}{2}}(d+e x) (a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {\frac {2 \int -\frac {a^2+4 b \sin (d+e x) a+3 b^2+c^2+4 b c \cos (d+e x)}{2 \sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}dx}{a^2-b^2+c^2}-\frac {8 (a b \cos (d+e x)-b c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{(a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\csc ^{\frac {5}{2}}(d+e x) (a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {-\frac {\int \frac {a^2+4 b \sin (d+e x) a+3 b^2+c^2+4 b c \cos (d+e x)}{\sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}dx}{a^2-b^2+c^2}-\frac {8 (a b \cos (d+e x)-b c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{(a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\csc ^{\frac {5}{2}}(d+e x) (a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {-\frac {\int \frac {a^2+4 b \sin (d+e x) a+3 b^2+c^2+4 b c \cos (d+e x)}{\sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}dx}{a^2-b^2+c^2}-\frac {8 (a b \cos (d+e x)-b c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{(a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3628 |
| \(\displaystyle \frac {\csc ^{\frac {5}{2}}(d+e x) (a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {-\frac {\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}dx+4 b \int \sqrt {b+c \cos (d+e x)+a \sin (d+e x)}dx}{a^2-b^2+c^2}-\frac {8 (a b \cos (d+e x)-b c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{(a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\csc ^{\frac {5}{2}}(d+e x) (a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {-\frac {\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}dx+4 b \int \sqrt {b+c \cos (d+e x)+a \sin (d+e x)}dx}{a^2-b^2+c^2}-\frac {8 (a b \cos (d+e x)-b c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{(a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3598 |
| \(\displaystyle \frac {\csc ^{\frac {5}{2}}(d+e x) (a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {-\frac {\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}dx+\frac {4 b \sqrt {a \sin (d+e x)+b+c \cos (d+e x)} \int \sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \cos \left (d+e x-\tan ^{-1}(c,a)\right )}{b+\sqrt {a^2+c^2}}}dx}{\sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}}}}{a^2-b^2+c^2}-\frac {8 (a b \cos (d+e x)-b c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{(a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\csc ^{\frac {5}{2}}(d+e x) (a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {-\frac {\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}dx+\frac {4 b \sqrt {a \sin (d+e x)+b+c \cos (d+e x)} \int \sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \sin \left (d+e x-\tan ^{-1}(c,a)+\frac {\pi }{2}\right )}{b+\sqrt {a^2+c^2}}}dx}{\sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}}}}{a^2-b^2+c^2}-\frac {8 (a b \cos (d+e x)-b c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{(a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\csc ^{\frac {5}{2}}(d+e x) (a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {-\frac {\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}dx+\frac {8 b \sqrt {a \sin (d+e x)+b+c \cos (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}}}}{a^2-b^2+c^2}-\frac {8 (a b \cos (d+e x)-b c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{(a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3606 |
| \(\displaystyle \frac {\csc ^{\frac {5}{2}}(d+e x) (a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {-\frac {\frac {\left (a^2-b^2+c^2\right ) \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}} \int \frac {1}{\sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \cos \left (d+e x-\tan ^{-1}(c,a)\right )}{b+\sqrt {a^2+c^2}}}}dx}{\sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}+\frac {8 b \sqrt {a \sin (d+e x)+b+c \cos (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}}}}{a^2-b^2+c^2}-\frac {8 (a b \cos (d+e x)-b c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{(a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\csc ^{\frac {5}{2}}(d+e x) (a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {-\frac {\frac {\left (a^2-b^2+c^2\right ) \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}} \int \frac {1}{\sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \sin \left (d+e x-\tan ^{-1}(c,a)+\frac {\pi }{2}\right )}{b+\sqrt {a^2+c^2}}}}dx}{\sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}+\frac {8 b \sqrt {a \sin (d+e x)+b+c \cos (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}}}}{a^2-b^2+c^2}-\frac {8 (a b \cos (d+e x)-b c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{(a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\csc ^{\frac {5}{2}}(d+e x) (a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {-\frac {\frac {2 \left (a^2-b^2+c^2\right ) \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right ),\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}+\frac {8 b \sqrt {a \sin (d+e x)+b+c \cos (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}}}}{a^2-b^2+c^2}-\frac {8 (a b \cos (d+e x)-b c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{(a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\) |
(Csc[d + e*x]^(5/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^(5/2)*((-2*(a*Co s[d + e*x] - c*Sin[d + e*x]))/(3*(a^2 - b^2 + c^2)*e*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^(3/2)) - ((-8*(a*b*Cos[d + e*x] - b*c*Sin[d + e*x]))/((a^ 2 - b^2 + c^2)*e*Sqrt[b + c*Cos[d + e*x] + a*Sin[d + e*x]]) - ((8*b*Ellipt icE[(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]]))/(a^2 - b^2 + c^2))/(3*(a^2 - b^2 + c^2 ))))/(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)
3.5.66.3.1 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[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[(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 + 1)*(a^2 - b^2 - c^2))), x] + Simp[ 1/((n + 1)*(a^2 - b^2 - c^2)) Int[(a*(n + 1) - b*(n + 2)*Cos[d + e*x] - c *(n + 2)*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1), x], x ] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && LtQ[n, -1] && NeQ[n, -3/2]
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_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]) ^(n_)*((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_) ]), x_Symbol] :> Simp[(-(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A) *Sin[d + e*x]))*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*( a^2 - b^2 - c^2))), x] + Simp[1/((n + 1)*(a^2 - b^2 - c^2)) Int[(a + b*Co s[d + e*x] + c*Sin[d + e*x])^(n + 1)*Simp[(n + 1)*(a*A - b*B - c*C) + (n + 2)*(a*B - b*A)*Cos[d + e*x] + (n + 2)*(a*C - c*A)*Sin[d + e*x], x], x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[n, -2]
Int[csc[(d_.) + (e_.)*(x_)]^(n_.)*((a_.) + csc[(d_.) + (e_.)*(x_)]*(b_.) + cot[(d_.) + (e_.)*(x_)]*(c_.))^(m_), x_Symbol] :> Simp[Csc[d + e*x]^n*((b + a*Sin[d + e*x] + c*Cos[d + e*x])^n/(a + b*Csc[d + e*x] + c*Cot[d + e*x])^n ) Int[1/(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n, x], x] /; FreeQ[{a, b, c , d, e}, x] && EqQ[m + n, 0] && !IntegerQ[n]
Result contains complex when optimal does not.
Time = 29.43 (sec) , antiderivative size = 59351, normalized size of antiderivative = 120.63
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.30 (sec) , antiderivative size = 2777, normalized size of antiderivative = 5.64 \[ \int \frac {\csc ^{\frac {5}{2}}(d+e x)}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2}} \, dx=\text {Too large to display} \]
-1/9*((3*I*a^5 + 4*I*a^3*b^2 + I*a*b^4 - 3*(a^2 + b^2)*c^3 + 3*I*(a^3 + a* b^2)*c^2 + (-3*I*a^5 - I*a^3*b^2 + I*a*b^2*c^2 - b^2*c^3 + 3*I*a*c^4 - 3*c ^5 + (3*a^4 + a^2*b^2)*c)*cos(e*x + d)^2 - (3*a^4 + 4*a^2*b^2 + b^4)*c - 2 *(-3*I*a*b*c^3 + 3*b*c^4 + (3*a^2*b + b^3)*c^2 - I*(3*a^3*b + a*b^3)*c)*co s(e*x + d) - 2*(-3*I*a^4*b - I*a^2*b^3 - 3*I*a^2*b*c^2 + 3*a*b*c^3 + (3*a^ 3*b + a*b^3)*c + (-3*I*a^2*c^3 + 3*a*c^4 + (3*a^3 + a*b^2)*c^2 - I*(3*a^4 + a^2*b^2)*c)*cos(e*x + d))*sin(e*x + d))*sqrt(-2*I*a - 2*c)*weierstrassPI nverse(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*I*a^5 - 4*I*a^3*b^2 - I*a*b^4 - 3*(a^2 + b^2)*c ^3 - 3*I*(a^3 + a*b^2)*c^2 + (3*I*a^5 + I*a^3*b^2 - I*a*b^2*c^2 - b^2*c^3 - 3*I*a*c^4 - 3*c^5 + (3*a^4 + a^2*b^2)*c)*cos(e*x + d)^2 - (3*a^4 + 4*a^2 *b^2 + b^4)*c - 2*(3*I*a*b*c^3 + 3*b*c^4 + (3*a^2*b + b^3)*c^2 + I*(3*a^3* b + a*b^3)*c)*cos(e*x + d) - 2*(3*I*a^4*b + I*a^2*b^3 + 3*I*a^2*b*c^2 + 3* a*b*c^3 + (3*a^3*b + a*b^3)*c + (3*I*a^2*c^3 + 3*a*c^4 + (3*a^3 + a*b^2)*c ^2 + I*(3*a^4 + a^2*b^2)*c)*cos(e*x + d))*sin(e*x + d))*sqrt(2*I*a - 2*c)* weierstrassPInverse(4/3*(3*a^4 - 4*a^2*b^2 + 4*b^2*c^2 - 6*I*a*c^3 - 3*...
Timed out. \[ \int \frac {\csc ^{\frac {5}{2}}(d+e x)}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\csc ^{\frac {5}{2}}(d+e x)}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2}} \, dx=\int { \frac {\csc \left (e x + d\right )^{\frac {5}{2}}}{{\left (c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\csc ^{\frac {5}{2}}(d+e x)}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2}} \, dx=\int { \frac {\csc \left (e x + d\right )^{\frac {5}{2}}}{{\left (c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\csc ^{\frac {5}{2}}(d+e x)}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2}} \, dx=\int \frac {{\left (\frac {1}{\sin \left (d+e\,x\right )}\right )}^{5/2}}{{\left (a+c\,\mathrm {cot}\left (d+e\,x\right )+\frac {b}{\sin \left (d+e\,x\right )}\right )}^{5/2}} \,d x \]