Integrand size = 21, antiderivative size = 126 \[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}-\frac {21 \sqrt {d \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b d^4 \sqrt {\cos (a+b x)}}+\frac {7 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}+\frac {21 \sin (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}} \]
-csc(b*x+a)/b/d/(d*cos(b*x+a))^(5/2)+7/5*sin(b*x+a)/b/d/(d*cos(b*x+a))^(5/ 2)+21/5*sin(b*x+a)/b/d^3/(d*cos(b*x+a))^(1/2)-21/5*(cos(1/2*a+1/2*b*x)^2)^ (1/2)/cos(1/2*a+1/2*b*x)*EllipticE(sin(1/2*a+1/2*b*x),2^(1/2))*(d*cos(b*x+ a))^(1/2)/b/d^4/cos(b*x+a)^(1/2)
Time = 0.34 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.65 \[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\frac {-5 \cos (a+b x) \cot (a+b x)-21 \sqrt {\cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )+16 \sin (a+b x)+2 \sec (a+b x) \tan (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}} \]
(-5*Cos[a + b*x]*Cot[a + b*x] - 21*Sqrt[Cos[a + b*x]]*EllipticE[(a + b*x)/ 2, 2] + 16*Sin[a + b*x] + 2*Sec[a + b*x]*Tan[a + b*x])/(5*b*d^3*Sqrt[d*Cos [a + b*x]])
Time = 0.56 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 3050, 3042, 3116, 3042, 3116, 3042, 3121, 3042, 3119}
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.
| \(\displaystyle \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (a+b x)^2 (d \cos (a+b x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3050 |
| \(\displaystyle \frac {7}{2} \int \frac {1}{(d \cos (a+b x))^{7/2}}dx-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{2} \int \frac {1}{\left (d \sin \left (a+b x+\frac {\pi }{2}\right )\right )^{7/2}}dx-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {7}{2} \left (\frac {3 \int \frac {1}{(d \cos (a+b x))^{3/2}}dx}{5 d^2}+\frac {2 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}\right )-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{2} \left (\frac {3 \int \frac {1}{\left (d \sin \left (a+b x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{5 d^2}+\frac {2 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}\right )-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {7}{2} \left (\frac {3 \left (\frac {2 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {\int \sqrt {d \cos (a+b x)}dx}{d^2}\right )}{5 d^2}+\frac {2 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}\right )-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{2} \left (\frac {3 \left (\frac {2 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {\int \sqrt {d \sin \left (a+b x+\frac {\pi }{2}\right )}dx}{d^2}\right )}{5 d^2}+\frac {2 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}\right )-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {7}{2} \left (\frac {3 \left (\frac {2 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {\sqrt {d \cos (a+b x)} \int \sqrt {\cos (a+b x)}dx}{d^2 \sqrt {\cos (a+b x)}}\right )}{5 d^2}+\frac {2 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}\right )-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{2} \left (\frac {3 \left (\frac {2 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {\sqrt {d \cos (a+b x)} \int \sqrt {\sin \left (a+b x+\frac {\pi }{2}\right )}dx}{d^2 \sqrt {\cos (a+b x)}}\right )}{5 d^2}+\frac {2 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}\right )-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {7}{2} \left (\frac {3 \left (\frac {2 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {d \cos (a+b x)}}{b d^2 \sqrt {\cos (a+b x)}}\right )}{5 d^2}+\frac {2 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}\right )-\frac {\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}\) |
-(Csc[a + b*x]/(b*d*(d*Cos[a + b*x])^(5/2))) + (7*((2*Sin[a + b*x])/(5*b*d *(d*Cos[a + b*x])^(5/2)) + (3*((-2*Sqrt[d*Cos[a + b*x]]*EllipticE[(a + b*x )/2, 2])/(b*d^2*Sqrt[Cos[a + b*x]]) + (2*Sin[a + b*x])/(b*d*Sqrt[d*Cos[a + b*x]])))/(5*d^2)))/2
3.3.41.3.1 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[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(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]
Leaf count of result is larger than twice the leaf count of optimal. \(407\) vs. \(2(136)=272\).
Time = 1.06 (sec) , antiderivative size = 408, normalized size of antiderivative = 3.24
| 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 (168 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) E\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}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+336 \left (\sin ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-168 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) E\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}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-672 \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+42 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) E\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}+448 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-112 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+5\right ) {\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}}}{10 d^{5} \left (2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{5} \left (8 \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-12 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+6 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b}\) | \(408\) |
-1/10*(d*(2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(1/2)/d^5/(2*sin (1/2*b*x+1/2*a)^2-1)/cos(1/2*b*x+1/2*a)/sin(1/2*b*x+1/2*a)^5/(8*sin(1/2*b* x+1/2*a)^6-12*sin(1/2*b*x+1/2*a)^4+6*sin(1/2*b*x+1/2*a)^2-1)*(168*cos(1/2* b*x+1/2*a)*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2))*(sin(1/2*b*x+1/2*a)^2)^(1 /2)*(2*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*sin(1/2*b*x+1/2*a)^4+336*sin(1/2*b*x+ 1/2*a)^8-168*cos(1/2*b*x+1/2*a)*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2))*(sin (1/2*b*x+1/2*a)^2)^(1/2)*(2*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*sin(1/2*b*x+1/2* a)^2-672*sin(1/2*b*x+1/2*a)^6+42*cos(1/2*b*x+1/2*a)*EllipticE(cos(1/2*b*x+ 1/2*a),2^(1/2))*(2*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*(sin(1/2*b*x+1/2*a)^2)^(1 /2)+448*sin(1/2*b*x+1/2*a)^4-112*sin(1/2*b*x+1/2*a)^2+5)*(-2*sin(1/2*b*x+1 /2*a)^4*d+d*sin(1/2*b*x+1/2*a)^2)^(3/2)/(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 = 145, normalized size of antiderivative = 1.15 \[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\frac {-21 i \, \sqrt {2} \sqrt {d} \cos \left (b x + a\right )^{3} \sin \left (b x + a\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 21 i \, \sqrt {2} \sqrt {d} \cos \left (b x + a\right )^{3} \sin \left (b x + a\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) - 2 \, {\left (21 \, \cos \left (b x + a\right )^{4} - 14 \, \cos \left (b x + a\right )^{2} - 2\right )} \sqrt {d \cos \left (b x + a\right )}}{10 \, b d^{4} \cos \left (b x + a\right )^{3} \sin \left (b x + a\right )} \]
1/10*(-21*I*sqrt(2)*sqrt(d)*cos(b*x + a)^3*sin(b*x + a)*weierstrassZeta(-4 , 0, weierstrassPInverse(-4, 0, cos(b*x + a) + I*sin(b*x + a))) + 21*I*sqr t(2)*sqrt(d)*cos(b*x + a)^3*sin(b*x + a)*weierstrassZeta(-4, 0, weierstras sPInverse(-4, 0, cos(b*x + a) - I*sin(b*x + a))) - 2*(21*cos(b*x + a)^4 - 14*cos(b*x + a)^2 - 2)*sqrt(d*cos(b*x + a)))/(b*d^4*cos(b*x + a)^3*sin(b*x + a))
Timed out. \[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\int { \frac {\csc \left (b x + a\right )^{2}}{\left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\int { \frac {\csc \left (b x + a\right )^{2}}{\left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\int \frac {1}{{\sin \left (a+b\,x\right )}^2\,{\left (d\,\cos \left (a+b\,x\right )\right )}^{7/2}} \,d x \]