Integrand size = 21, antiderivative size = 140 \[ \int \frac {\csc ^3(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx=\frac {2 \csc (a+b x)}{15 b d (d \tan (a+b x))^{3/2}}-\frac {2 \csc ^3(a+b x)}{9 b d (d \tan (a+b x))^{3/2}}+\frac {4 \cos (a+b x)}{15 b d^2 \sqrt {d \tan (a+b x)}}+\frac {4 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sin (a+b x)}{15 b d^2 \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}} \]
4/15*cos(b*x+a)/b/d^2/(d*tan(b*x+a))^(1/2)-4/15*(sin(a+1/4*Pi+b*x)^2)^(1/2 )/sin(a+1/4*Pi+b*x)*EllipticE(cos(a+1/4*Pi+b*x),2^(1/2))*sin(b*x+a)/b/d^2/ sin(2*b*x+2*a)^(1/2)/(d*tan(b*x+a))^(1/2)+2/15*csc(b*x+a)/b/d/(d*tan(b*x+a ))^(3/2)-2/9*csc(b*x+a)^3/b/d/(d*tan(b*x+a))^(3/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.91 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.83 \[ \int \frac {\csc ^3(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx=\frac {2 \left (4 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\tan ^2(a+b x)\right ) \sec ^2(a+b x)+\left (-6+3 \csc ^2(a+b x)+8 \csc ^4(a+b x)-5 \csc ^6(a+b x)\right ) \sqrt {\sec ^2(a+b x)}\right ) \sin (a+b x) \sqrt {d \tan (a+b x)}}{45 b d^3 \sqrt {\sec ^2(a+b x)}} \]
(2*(4*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sec[a + b*x]^2 + ( -6 + 3*Csc[a + b*x]^2 + 8*Csc[a + b*x]^4 - 5*Csc[a + b*x]^6)*Sqrt[Sec[a + b*x]^2])*Sin[a + b*x]*Sqrt[d*Tan[a + b*x]])/(45*b*d^3*Sqrt[Sec[a + b*x]^2] )
Time = 0.75 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.26, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 3077, 3042, 3079, 3042, 3081, 3042, 3050, 3042, 3052, 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.
| \(\displaystyle \int \frac {\csc ^3(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (a+b x)^3 (d \tan (a+b x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3077 |
| \(\displaystyle -\frac {\int \frac {\csc ^3(a+b x)}{\sqrt {d \tan (a+b x)}}dx}{3 d^2}-\frac {2 \csc ^3(a+b x)}{9 b d (d \tan (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {1}{\sin (a+b x)^3 \sqrt {d \tan (a+b x)}}dx}{3 d^2}-\frac {2 \csc ^3(a+b x)}{9 b d (d \tan (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3079 |
| \(\displaystyle -\frac {\frac {2}{5} \int \frac {\csc (a+b x)}{\sqrt {d \tan (a+b x)}}dx-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}}{3 d^2}-\frac {2 \csc ^3(a+b x)}{9 b d (d \tan (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2}{5} \int \frac {1}{\sin (a+b x) \sqrt {d \tan (a+b x)}}dx-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}}{3 d^2}-\frac {2 \csc ^3(a+b x)}{9 b d (d \tan (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3081 |
| \(\displaystyle -\frac {\frac {2 \sqrt {\sin (a+b x)} \int \frac {\sqrt {\cos (a+b x)}}{\sin ^{\frac {3}{2}}(a+b x)}dx}{5 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}}{3 d^2}-\frac {2 \csc ^3(a+b x)}{9 b d (d \tan (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \sqrt {\sin (a+b x)} \int \frac {\sqrt {\cos (a+b x)}}{\sin (a+b x)^{3/2}}dx}{5 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}}{3 d^2}-\frac {2 \csc ^3(a+b x)}{9 b d (d \tan (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3050 |
| \(\displaystyle -\frac {\frac {2 \sqrt {\sin (a+b x)} \left (-2 \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}dx-\frac {2 \cos ^{\frac {3}{2}}(a+b x)}{b \sqrt {\sin (a+b x)}}\right )}{5 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}}{3 d^2}-\frac {2 \csc ^3(a+b x)}{9 b d (d \tan (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \sqrt {\sin (a+b x)} \left (-2 \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}dx-\frac {2 \cos ^{\frac {3}{2}}(a+b x)}{b \sqrt {\sin (a+b x)}}\right )}{5 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}}{3 d^2}-\frac {2 \csc ^3(a+b x)}{9 b d (d \tan (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3052 |
| \(\displaystyle -\frac {\frac {2 \sqrt {\sin (a+b x)} \left (-\frac {2 \sqrt {\sin (a+b x)} \sqrt {\cos (a+b x)} \int \sqrt {\sin (2 a+2 b x)}dx}{\sqrt {\sin (2 a+2 b x)}}-\frac {2 \cos ^{\frac {3}{2}}(a+b x)}{b \sqrt {\sin (a+b x)}}\right )}{5 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}}{3 d^2}-\frac {2 \csc ^3(a+b x)}{9 b d (d \tan (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {2 \sqrt {\sin (a+b x)} \left (-\frac {2 \sqrt {\sin (a+b x)} \sqrt {\cos (a+b x)} \int \sqrt {\sin (2 a+2 b x)}dx}{\sqrt {\sin (2 a+2 b x)}}-\frac {2 \cos ^{\frac {3}{2}}(a+b x)}{b \sqrt {\sin (a+b x)}}\right )}{5 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}}{3 d^2}-\frac {2 \csc ^3(a+b x)}{9 b d (d \tan (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle -\frac {\frac {2 \sqrt {\sin (a+b x)} \left (-\frac {2 \cos ^{\frac {3}{2}}(a+b x)}{b \sqrt {\sin (a+b x)}}-\frac {2 \sqrt {\sin (a+b x)} \sqrt {\cos (a+b x)} E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{b \sqrt {\sin (2 a+2 b x)}}\right )}{5 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}-\frac {2 d \csc (a+b x)}{5 b (d \tan (a+b x))^{3/2}}}{3 d^2}-\frac {2 \csc ^3(a+b x)}{9 b d (d \tan (a+b x))^{3/2}}\) |
(-2*Csc[a + b*x]^3)/(9*b*d*(d*Tan[a + b*x])^(3/2)) - ((-2*d*Csc[a + b*x])/ (5*b*(d*Tan[a + b*x])^(3/2)) + (2*Sqrt[Sin[a + b*x]]*((-2*Cos[a + b*x]^(3/ 2))/(b*Sqrt[Sin[a + b*x]]) - (2*Sqrt[Cos[a + b*x]]*EllipticE[a - Pi/4 + b* x, 2]*Sqrt[Sin[a + b*x]])/(b*Sqrt[Sin[2*a + 2*b*x]])))/(5*Sqrt[Cos[a + b*x ]]*Sqrt[d*Tan[a + b*x]]))/(3*d^2)
3.2.13.3.1 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[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]] , x_Symbol] :> Simp[Sqrt[a*Sin[e + f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]) Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f}, x]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[(a*Sin[e + f*x])^m*((b*Tan[e + f*x])^(n + 1)/(b*f*(m + n + 1))), x] - Simp[(n + 1)/(b^2*(m + n + 1)) Int[(a*Sin[e + f*x])^m*( b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && LtQ[n, -1] && NeQ[m + n + 1, 0] && IntegersQ[2*m, 2*n] && !(EqQ[n, -3/2] && EqQ[m, 1] )
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n _.), x_Symbol] :> Simp[b*(a*Sin[e + f*x])^(m + 2)*((b*Tan[e + f*x])^(n - 1) /(a^2*f*(m + n + 1))), x] + Simp[(m + 2)/(a^2*(m + n + 1)) Int[(a*Sin[e + f*x])^(m + 2)*(b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && L tQ[m, -1] && NeQ[m + n + 1, 0] && IntegersQ[2*m, 2*n]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[Cos[e + f*x]^n*((b*Tan[e + f*x])^n/(a*Sin[e + f*x])^ n) Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x], x] /; FreeQ[{a, b, e, f, m, n}, x] && !IntegerQ[n] && (ILtQ[m, 0] || (EqQ[m, 1] && EqQ[n, -2^(- 1)]) || IntegersQ[m - 1/2, n - 1/2])
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]
Leaf count of result is larger than twice the leaf count of optimal. \(391\) vs. \(2(147)=294\).
Time = 1.00 (sec) , antiderivative size = 392, normalized size of antiderivative = 2.80
| method | result | size |
| default | \(\frac {\left (-12 \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+6 \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-12 \sec \left (b x +a \right ) \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+6 \sec \left (b x +a \right ) \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+6 \sqrt {2}+3 \cot \left (b x +a \right ) \csc \left (b x +a \right ) \sqrt {2}-5 \cot \left (b x +a \right ) \left (\csc ^{3}\left (b x +a \right )\right ) \sqrt {2}\right ) \sqrt {2}}{45 b \sqrt {d \tan \left (b x +a \right )}\, d^{2}}\) | \(392\) |
1/45/b/(d*tan(b*x+a))^(1/2)/d^2*(-12*(1+csc(b*x+a)-cot(b*x+a))^(1/2)*(-csc (b*x+a)+1+cot(b*x+a))^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticE((1+csc (b*x+a)-cot(b*x+a))^(1/2),1/2*2^(1/2))+6*(cot(b*x+a)-csc(b*x+a))^(1/2)*(-c sc(b*x+a)+1+cot(b*x+a))^(1/2)*(1+csc(b*x+a)-cot(b*x+a))^(1/2)*EllipticF((1 +csc(b*x+a)-cot(b*x+a))^(1/2),1/2*2^(1/2))-12*sec(b*x+a)*(1+csc(b*x+a)-cot (b*x+a))^(1/2)*(-csc(b*x+a)+1+cot(b*x+a))^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1 /2)*EllipticE((1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2*2^(1/2))+6*sec(b*x+a)*(1 +csc(b*x+a)-cot(b*x+a))^(1/2)*(-csc(b*x+a)+1+cot(b*x+a))^(1/2)*(cot(b*x+a) -csc(b*x+a))^(1/2)*EllipticF((1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2*2^(1/2))+ 6*2^(1/2)+3*cot(b*x+a)*csc(b*x+a)*2^(1/2)-5*cot(b*x+a)*csc(b*x+a)^3*2^(1/2 ))*2^(1/2)
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 309, normalized size of antiderivative = 2.21 \[ \int \frac {\csc ^3(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx=-\frac {2 \, {\left (3 \, {\left (-i \, \cos \left (b x + a\right )^{4} + 2 i \, \cos \left (b x + a\right )^{2} - i\right )} \sqrt {i \, d} E(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) + 3 \, {\left (i \, \cos \left (b x + a\right )^{4} - 2 i \, \cos \left (b x + a\right )^{2} + i\right )} \sqrt {-i \, d} E(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) + 3 \, {\left (i \, \cos \left (b x + a\right )^{4} - 2 i \, \cos \left (b x + a\right )^{2} + i\right )} \sqrt {i \, d} F(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) + 3 \, {\left (-i \, \cos \left (b x + a\right )^{4} + 2 i \, \cos \left (b x + a\right )^{2} - i\right )} \sqrt {-i \, d} F(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) - {\left (6 \, \cos \left (b x + a\right )^{6} - 15 \, \cos \left (b x + a\right )^{4} + 4 \, \cos \left (b x + a\right )^{2}\right )} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}\right )}}{45 \, {\left (b d^{3} \cos \left (b x + a\right )^{4} - 2 \, b d^{3} \cos \left (b x + a\right )^{2} + b d^{3}\right )} \sin \left (b x + a\right )} \]
-2/45*(3*(-I*cos(b*x + a)^4 + 2*I*cos(b*x + a)^2 - I)*sqrt(I*d)*elliptic_e (arcsin(cos(b*x + a) + I*sin(b*x + a)), -1)*sin(b*x + a) + 3*(I*cos(b*x + a)^4 - 2*I*cos(b*x + a)^2 + I)*sqrt(-I*d)*elliptic_e(arcsin(cos(b*x + a) - I*sin(b*x + a)), -1)*sin(b*x + a) + 3*(I*cos(b*x + a)^4 - 2*I*cos(b*x + a )^2 + I)*sqrt(I*d)*elliptic_f(arcsin(cos(b*x + a) + I*sin(b*x + a)), -1)*s in(b*x + a) + 3*(-I*cos(b*x + a)^4 + 2*I*cos(b*x + a)^2 - I)*sqrt(-I*d)*el liptic_f(arcsin(cos(b*x + a) - I*sin(b*x + a)), -1)*sin(b*x + a) - (6*cos( b*x + a)^6 - 15*cos(b*x + a)^4 + 4*cos(b*x + a)^2)*sqrt(d*sin(b*x + a)/cos (b*x + a)))/((b*d^3*cos(b*x + a)^4 - 2*b*d^3*cos(b*x + a)^2 + b*d^3)*sin(b *x + a))
\[ \int \frac {\csc ^3(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx=\int \frac {\csc ^{3}{\left (a + b x \right )}}{\left (d \tan {\left (a + b x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\csc ^3(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx=\int { \frac {\csc \left (b x + a\right )^{3}}{\left (d \tan \left (b x + a\right )\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\csc ^3(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx=\int { \frac {\csc \left (b x + a\right )^{3}}{\left (d \tan \left (b x + a\right )\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\csc ^3(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx=\int \frac {1}{{\sin \left (a+b\,x\right )}^3\,{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{5/2}} \,d x \]