Integrand size = 19, antiderivative size = 76 \[ \int \csc (a+b x) (d \tan (a+b x))^{3/2} \, dx=-\frac {2 d^2 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sin (a+b x)}{b \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}+\frac {2 d \sin (a+b x) \sqrt {d \tan (a+b x)}}{b} \]
2*d^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)+2*d* sin(b*x+a)*(d*tan(b*x+a))^(1/2)/b
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.34 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.80 \[ \int \csc (a+b x) (d \tan (a+b x))^{3/2} \, dx=-\frac {2 \cos (a+b x) \left (-3+2 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\tan ^2(a+b x)\right ) \sqrt {\sec ^2(a+b x)}\right ) (d \tan (a+b x))^{3/2}}{3 b} \]
(-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])*(d*Tan[a + b*x])^(3/2))/(3*b)
Time = 0.46 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {3042, 3073, 3042, 3081, 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.
\(\displaystyle \int \csc (a+b x) (d \tan (a+b x))^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(d \tan (a+b x))^{3/2}}{\sin (a+b x)}dx\) |
\(\Big \downarrow \) 3073 |
\(\displaystyle \frac {2 d \sin (a+b x) \sqrt {d \tan (a+b x)}}{b}-2 d^2 \int \frac {\sin (a+b x)}{\sqrt {d \tan (a+b x)}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d \sin (a+b x) \sqrt {d \tan (a+b x)}}{b}-2 d^2 \int \frac {\sin (a+b x)}{\sqrt {d \tan (a+b x)}}dx\) |
\(\Big \downarrow \) 3081 |
\(\displaystyle \frac {2 d \sin (a+b x) \sqrt {d \tan (a+b x)}}{b}-\frac {2 d^2 \sqrt {\sin (a+b x)} \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}dx}{\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d \sin (a+b x) \sqrt {d \tan (a+b x)}}{b}-\frac {2 d^2 \sqrt {\sin (a+b x)} \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}dx}{\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}\) |
\(\Big \downarrow \) 3052 |
\(\displaystyle \frac {2 d \sin (a+b x) \sqrt {d \tan (a+b x)}}{b}-\frac {2 d^2 \sin (a+b x) \int \sqrt {\sin (2 a+2 b x)}dx}{\sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d \sin (a+b x) \sqrt {d \tan (a+b x)}}{b}-\frac {2 d^2 \sin (a+b x) \int \sqrt {\sin (2 a+2 b x)}dx}{\sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {2 d \sin (a+b x) \sqrt {d \tan (a+b x)}}{b}-\frac {2 d^2 \sin (a+b x) E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{b \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}\) |
(-2*d^2*EllipticE[a - Pi/4 + b*x, 2]*Sin[a + b*x])/(b*Sqrt[Sin[2*a + 2*b*x ]]*Sqrt[d*Tan[a + b*x]]) + (2*d*Sin[a + b*x]*Sqrt[d*Tan[a + b*x]])/b
3.1.71.3.1 Defintions of rubi rules used
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[b*(a*Sin[e + f*x])^(m + 2)*((b*Tan[e + f*x])^(n - 1)/ (a^2*f*(n - 1))), x] - Simp[b^2*((m + 2)/(a^2*(n - 1))) Int[(a*Sin[e + f* x])^(m + 2)*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f}, x] && G tQ[n, 1] && (LtQ[m, -1] || (EqQ[m, -1] && EqQ[n, 3/2])) && 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. \(380\) vs. \(2(95)=190\).
Time = 0.62 (sec) , antiderivative size = 381, normalized size of antiderivative = 5.01
method | result | size |
default | \(-\frac {\sin \left (b x +a \right ) \left (2 \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )-\sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )+2 \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {2}\, \cos \left (b x +a \right )+\sqrt {2}\right ) \sqrt {d \tan \left (b x +a \right )}\, d \sqrt {2}}{b \left (\cos ^{2}\left (b x +a \right )-1\right )}\) | \(381\) |
-1/b*sin(b*x+a)*(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))*cos(b*x+a)-(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))*cos(b*x+a)+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)*Ellipt icE((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)*Elli pticF((1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2*2^(1/2))-2^(1/2)*cos(b*x+a)+2^(1 /2))*(d*tan(b*x+a))^(1/2)*d/(cos(b*x+a)^2-1)*2^(1/2)
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.79 \[ \int \csc (a+b x) (d \tan (a+b x))^{3/2} \, dx=\frac {-i \, \sqrt {i \, d} d E(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) + i \, \sqrt {-i \, d} d E(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) + i \, \sqrt {i \, d} d F(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) - i \, \sqrt {-i \, d} d F(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) + 2 \, d \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \sin \left (b x + a\right )}{b} \]
(-I*sqrt(I*d)*d*elliptic_e(arcsin(cos(b*x + a) + I*sin(b*x + a)), -1) + I* sqrt(-I*d)*d*elliptic_e(arcsin(cos(b*x + a) - I*sin(b*x + a)), -1) + I*sqr t(I*d)*d*elliptic_f(arcsin(cos(b*x + a) + I*sin(b*x + a)), -1) - I*sqrt(-I *d)*d*elliptic_f(arcsin(cos(b*x + a) - I*sin(b*x + a)), -1) + 2*d*sqrt(d*s in(b*x + a)/cos(b*x + a))*sin(b*x + a))/b
\[ \int \csc (a+b x) (d \tan (a+b x))^{3/2} \, dx=\int \left (d \tan {\left (a + b x \right )}\right )^{\frac {3}{2}} \csc {\left (a + b x \right )}\, dx \]
\[ \int \csc (a+b x) (d \tan (a+b x))^{3/2} \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}} \csc \left (b x + a\right ) \,d x } \]
\[ \int \csc (a+b x) (d \tan (a+b x))^{3/2} \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}} \csc \left (b x + a\right ) \,d x } \]
Timed out. \[ \int \csc (a+b x) (d \tan (a+b x))^{3/2} \, dx=\int \frac {{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2}}{\sin \left (a+b\,x\right )} \,d x \]