Integrand size = 19, antiderivative size = 72 \[ \int \frac {\csc (a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=-\frac {2 \cos (a+b x)}{b \sqrt {d \tan (a+b x)}}-\frac {2 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sin (a+b x)}{b \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}} \]
-2*cos(b*x+a)/b/(d*tan(b*x+a))^(1/2)+2*(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/sin(2*b*x+2*a )^(1/2)/(d*tan(b*x+a))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.32 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.96 \[ \int \frac {\csc (a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=-\frac {2 \cos (a+b x) \left (3+2 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\tan ^2(a+b x)\right ) \sqrt {\sec ^2(a+b x)} \tan ^2(a+b x)\right )}{3 b \sqrt {d \tan (a+b x)}} \]
(-2*Cos[a + b*x]*(3 + 2*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]* Sqrt[Sec[a + b*x]^2]*Tan[a + b*x]^2))/(3*b*Sqrt[d*Tan[a + b*x]])
Time = 0.49 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.51, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {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.
\(\displaystyle \int \frac {\csc (a+b x)}{\sqrt {d \tan (a+b x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sin (a+b x) \sqrt {d \tan (a+b x)}}dx\) |
\(\Big \downarrow \) 3081 |
\(\displaystyle \frac {\sqrt {\sin (a+b x)} \int \frac {\sqrt {\cos (a+b x)}}{\sin ^{\frac {3}{2}}(a+b x)}dx}{\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {\sin (a+b x)} \int \frac {\sqrt {\cos (a+b x)}}{\sin (a+b x)^{3/2}}dx}{\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}\) |
\(\Big \downarrow \) 3050 |
\(\displaystyle \frac {\sqrt {\sin (a+b x)} \left (-2 \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}dx-\frac {2 \cos ^{\frac {3}{2}}(a+b x)}{b \sqrt {\sin (a+b x)}}\right )}{\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {\sin (a+b x)} \left (-2 \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}dx-\frac {2 \cos ^{\frac {3}{2}}(a+b x)}{b \sqrt {\sin (a+b x)}}\right )}{\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}\) |
\(\Big \downarrow \) 3052 |
\(\displaystyle \frac {\sqrt {\sin (a+b x)} \left (-\frac {2 \sqrt {\sin (a+b x)} \sqrt {\cos (a+b x)} \int \sqrt {\sin (2 a+2 b x)}dx}{\sqrt {\sin (2 a+2 b x)}}-\frac {2 \cos ^{\frac {3}{2}}(a+b x)}{b \sqrt {\sin (a+b x)}}\right )}{\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {\sin (a+b x)} \left (-\frac {2 \sqrt {\sin (a+b x)} \sqrt {\cos (a+b x)} \int \sqrt {\sin (2 a+2 b x)}dx}{\sqrt {\sin (2 a+2 b x)}}-\frac {2 \cos ^{\frac {3}{2}}(a+b x)}{b \sqrt {\sin (a+b x)}}\right )}{\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {\sqrt {\sin (a+b x)} \left (-\frac {2 \cos ^{\frac {3}{2}}(a+b x)}{b \sqrt {\sin (a+b x)}}-\frac {2 \sqrt {\sin (a+b x)} \sqrt {\cos (a+b x)} E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{b \sqrt {\sin (2 a+2 b x)}}\right )}{\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}\) |
(Sqrt[Sin[a + b*x]]*((-2*Cos[a + b*x]^(3/2))/(b*Sqrt[Sin[a + b*x]]) - (2*S qrt[Cos[a + b*x]]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[Sin[a + b*x]])/(b*Sqrt [Sin[2*a + 2*b*x]])))/(Sqrt[Cos[a + b*x]]*Sqrt[d*Tan[a + b*x]])
3.1.92.3.1 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[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. \(348\) vs. \(2(91)=182\).
Time = 0.88 (sec) , antiderivative size = 349, normalized size of antiderivative = 4.85
method | result | size |
default | \(-\frac {\left (-2 \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-2 \sec \left (b x +a \right ) \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+\sec \left (b x +a \right ) \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\right ) \sqrt {2}}{b \sqrt {d \tan \left (b x +a \right )}}\) | \(349\) |
-1/b/(d*tan(b*x+a))^(1/2)*(-2*(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))+(cot(b*x+a)-csc(b*x+a))^(1/2)*(-csc(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))-2*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)*Ellipt icE((1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2*2^(1/2))+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))+2^(1/2))*2^( 1/2)
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.35 \[ \int \frac {\csc (a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=-\frac {2 \, \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \cos \left (b x + a\right )^{2} + i \, \sqrt {i \, d} E(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) - i \, \sqrt {-i \, d} E(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) - i \, \sqrt {i \, d} F(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) + i \, \sqrt {-i \, d} F(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right )}{b d \sin \left (b x + a\right )} \]
-(2*sqrt(d*sin(b*x + a)/cos(b*x + a))*cos(b*x + a)^2 + I*sqrt(I*d)*ellipti c_e(arcsin(cos(b*x + a) + I*sin(b*x + a)), -1)*sin(b*x + a) - I*sqrt(-I*d) *elliptic_e(arcsin(cos(b*x + a) - I*sin(b*x + a)), -1)*sin(b*x + a) - I*sq rt(I*d)*elliptic_f(arcsin(cos(b*x + a) + I*sin(b*x + a)), -1)*sin(b*x + a) + I*sqrt(-I*d)*elliptic_f(arcsin(cos(b*x + a) - I*sin(b*x + a)), -1)*sin( b*x + a))/(b*d*sin(b*x + a))
\[ \int \frac {\csc (a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=\int \frac {\csc {\left (a + b x \right )}}{\sqrt {d \tan {\left (a + b x \right )}}}\, dx \]
\[ \int \frac {\csc (a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=\int { \frac {\csc \left (b x + a\right )}{\sqrt {d \tan \left (b x + a\right )}} \,d x } \]
\[ \int \frac {\csc (a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=\int { \frac {\csc \left (b x + a\right )}{\sqrt {d \tan \left (b x + a\right )}} \,d x } \]
Timed out. \[ \int \frac {\csc (a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=\int \frac {1}{\sin \left (a+b\,x\right )\,\sqrt {d\,\mathrm {tan}\left (a+b\,x\right )}} \,d x \]