Integrand size = 21, antiderivative size = 98 \[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{5/2}} \, dx=-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{3/2}}+\frac {5 \sqrt {\cos (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )}{3 b d^2 \sqrt {d \cos (a+b x)}}+\frac {5 \sin (a+b x)}{3 b d (d \cos (a+b x))^{3/2}} \]
-csc(b*x+a)/b/d/(d*cos(b*x+a))^(3/2)+5/3*sin(b*x+a)/b/d/(d*cos(b*x+a))^(3/ 2)+5/3*(cos(1/2*a+1/2*b*x)^2)^(1/2)/cos(1/2*a+1/2*b*x)*EllipticF(sin(1/2*a +1/2*b*x),2^(1/2))*cos(b*x+a)^(1/2)/b/d^2/(d*cos(b*x+a))^(1/2)
Time = 0.31 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.63 \[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{5/2}} \, dx=\frac {-3 \cot (a+b x)+5 \sqrt {\cos (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )+2 \tan (a+b x)}{3 b d^2 \sqrt {d \cos (a+b x)}} \]
(-3*Cot[a + b*x] + 5*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2] + 2*Tan[ a + b*x])/(3*b*d^2*Sqrt[d*Cos[a + b*x]])
Time = 0.45 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 3050, 3042, 3116, 3042, 3121, 3042, 3120}
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.
\(\displaystyle \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{5/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sin (a+b x)^2 (d \cos (a+b x))^{5/2}}dx\) |
\(\Big \downarrow \) 3050 |
\(\displaystyle \frac {5}{2} \int \frac {1}{(d \cos (a+b x))^{5/2}}dx-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{2} \int \frac {1}{\left (d \sin \left (a+b x+\frac {\pi }{2}\right )\right )^{5/2}}dx-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{3/2}}\) |
\(\Big \downarrow \) 3116 |
\(\displaystyle \frac {5}{2} \left (\frac {\int \frac {1}{\sqrt {d \cos (a+b x)}}dx}{3 d^2}+\frac {2 \sin (a+b x)}{3 b d (d \cos (a+b x))^{3/2}}\right )-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{2} \left (\frac {\int \frac {1}{\sqrt {d \sin \left (a+b x+\frac {\pi }{2}\right )}}dx}{3 d^2}+\frac {2 \sin (a+b x)}{3 b d (d \cos (a+b x))^{3/2}}\right )-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{3/2}}\) |
\(\Big \downarrow \) 3121 |
\(\displaystyle \frac {5}{2} \left (\frac {\sqrt {\cos (a+b x)} \int \frac {1}{\sqrt {\cos (a+b x)}}dx}{3 d^2 \sqrt {d \cos (a+b x)}}+\frac {2 \sin (a+b x)}{3 b d (d \cos (a+b x))^{3/2}}\right )-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{2} \left (\frac {\sqrt {\cos (a+b x)} \int \frac {1}{\sqrt {\sin \left (a+b x+\frac {\pi }{2}\right )}}dx}{3 d^2 \sqrt {d \cos (a+b x)}}+\frac {2 \sin (a+b x)}{3 b d (d \cos (a+b x))^{3/2}}\right )-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{3/2}}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {5}{2} \left (\frac {2 \sqrt {\cos (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )}{3 b d^2 \sqrt {d \cos (a+b x)}}+\frac {2 \sin (a+b x)}{3 b d (d \cos (a+b x))^{3/2}}\right )-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{3/2}}\) |
-(Csc[a + b*x]/(b*d*(d*Cos[a + b*x])^(3/2))) + (5*((2*Sqrt[Cos[a + b*x]]*E llipticF[(a + b*x)/2, 2])/(3*b*d^2*Sqrt[d*Cos[a + b*x]]) + (2*Sin[a + b*x] )/(3*b*d*(d*Cos[a + b*x])^(3/2))))/2
3.3.40.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m + 1)/(a *b*f*(m + 1))), x] + Simp[(m + n + 2)/(a^2*(m + 1)) Int[(b*Cos[e + f*x])^ n*(a*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, - 1] && IntegersQ[2*m, 2*n]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Time = 0.70 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.94
method | result | size |
default | \(\frac {\sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (10 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) {\left (2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}^{\frac {3}{2}} F\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}-20 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+20 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-3\right ) \sin \left (\frac {b x}{2}+\frac {a}{2}\right )}{6 d {\left (-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d +d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right )}^{\frac {3}{2}} \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b}\) | \(190\) |
1/6*(d*(2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(1/2)/d/(-2*sin(1/ 2*b*x+1/2*a)^4*d+d*sin(1/2*b*x+1/2*a)^2)^(3/2)/cos(1/2*b*x+1/2*a)*(10*cos( 1/2*b*x+1/2*a)*(2*sin(1/2*b*x+1/2*a)^2-1)^(3/2)*EllipticF(cos(1/2*b*x+1/2* a),2^(1/2))*(sin(1/2*b*x+1/2*a)^2)^(1/2)-20*sin(1/2*b*x+1/2*a)^4+20*sin(1/ 2*b*x+1/2*a)^2-3)*sin(1/2*b*x+1/2*a)/(d*(2*cos(1/2*b*x+1/2*a)^2-1))^(1/2)/ b
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.32 \[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{5/2}} \, dx=\frac {-5 i \, \sqrt {2} \sqrt {d} \cos \left (b x + a\right )^{2} \sin \left (b x + a\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 5 i \, \sqrt {2} \sqrt {d} \cos \left (b x + a\right )^{2} \sin \left (b x + a\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - 2 \, \sqrt {d \cos \left (b x + a\right )} {\left (5 \, \cos \left (b x + a\right )^{2} - 2\right )}}{6 \, b d^{3} \cos \left (b x + a\right )^{2} \sin \left (b x + a\right )} \]
1/6*(-5*I*sqrt(2)*sqrt(d)*cos(b*x + a)^2*sin(b*x + a)*weierstrassPInverse( -4, 0, cos(b*x + a) + I*sin(b*x + a)) + 5*I*sqrt(2)*sqrt(d)*cos(b*x + a)^2 *sin(b*x + a)*weierstrassPInverse(-4, 0, cos(b*x + a) - I*sin(b*x + a)) - 2*sqrt(d*cos(b*x + a))*(5*cos(b*x + a)^2 - 2))/(b*d^3*cos(b*x + a)^2*sin(b *x + a))
Timed out. \[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{5/2}} \, dx=\int { \frac {\csc \left (b x + a\right )^{2}}{\left (d \cos \left (b x + a\right )\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{5/2}} \, dx=\int { \frac {\csc \left (b x + a\right )^{2}}{\left (d \cos \left (b x + a\right )\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{5/2}} \, dx=\int \frac {1}{{\sin \left (a+b\,x\right )}^2\,{\left (d\,\cos \left (a+b\,x\right )\right )}^{5/2}} \,d x \]