Integrand size = 21, antiderivative size = 102 \[ \int \csc ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=-\frac {4 d^2 \cos (a+b x)}{b \sqrt {d \tan (a+b x)}}-\frac {4 d^2 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sin (a+b x)}{b \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}+\frac {2 d \csc (a+b x) \sqrt {d \tan (a+b x)}}{b} \]
-4*d^2*cos(b*x+a)/b/(d*tan(b*x+a))^(1/2)+4*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*csc(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 = 0.64 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.70 \[ \int \csc ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=-\frac {2 \cos (a+b x) \left (-6+3 \csc ^2(a+b x)+4 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\tan ^2(a+b x)\right ) \sqrt {\sec ^2(a+b x)}\right ) (d \tan (a+b x))^{3/2}}{3 b} \]
(-2*Cos[a + b*x]*(-6 + 3*Csc[a + b*x]^2 + 4*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 = 0.61 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.35, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 3073, 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.
\(\displaystyle \int \csc ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(d \tan (a+b x))^{3/2}}{\sin (a+b x)^3}dx\) |
\(\Big \downarrow \) 3073 |
\(\displaystyle 2 d^2 \int \frac {\csc (a+b x)}{\sqrt {d \tan (a+b x)}}dx+\frac {2 d \csc (a+b x) \sqrt {d \tan (a+b x)}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 d^2 \int \frac {1}{\sin (a+b x) \sqrt {d \tan (a+b x)}}dx+\frac {2 d \csc (a+b x) \sqrt {d \tan (a+b x)}}{b}\) |
\(\Big \downarrow \) 3081 |
\(\displaystyle \frac {2 d^2 \sqrt {\sin (a+b x)} \int \frac {\sqrt {\cos (a+b x)}}{\sin ^{\frac {3}{2}}(a+b x)}dx}{\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}+\frac {2 d \csc (a+b x) \sqrt {d \tan (a+b x)}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d^2 \sqrt {\sin (a+b x)} \int \frac {\sqrt {\cos (a+b x)}}{\sin (a+b x)^{3/2}}dx}{\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}+\frac {2 d \csc (a+b x) \sqrt {d \tan (a+b x)}}{b}\) |
\(\Big \downarrow \) 3050 |
\(\displaystyle \frac {2 d^2 \sqrt {\sin (a+b x)} \left (-2 \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}dx-\frac {2 \cos ^{\frac {3}{2}}(a+b x)}{b \sqrt {\sin (a+b x)}}\right )}{\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}+\frac {2 d \csc (a+b x) \sqrt {d \tan (a+b x)}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d^2 \sqrt {\sin (a+b x)} \left (-2 \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}dx-\frac {2 \cos ^{\frac {3}{2}}(a+b x)}{b \sqrt {\sin (a+b x)}}\right )}{\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}+\frac {2 d \csc (a+b x) \sqrt {d \tan (a+b x)}}{b}\) |
\(\Big \downarrow \) 3052 |
\(\displaystyle \frac {2 d^2 \sqrt {\sin (a+b x)} \left (-\frac {2 \sqrt {\sin (a+b x)} \sqrt {\cos (a+b x)} \int \sqrt {\sin (2 a+2 b x)}dx}{\sqrt {\sin (2 a+2 b x)}}-\frac {2 \cos ^{\frac {3}{2}}(a+b x)}{b \sqrt {\sin (a+b x)}}\right )}{\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}+\frac {2 d \csc (a+b x) \sqrt {d \tan (a+b x)}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d^2 \sqrt {\sin (a+b x)} \left (-\frac {2 \sqrt {\sin (a+b x)} \sqrt {\cos (a+b x)} \int \sqrt {\sin (2 a+2 b x)}dx}{\sqrt {\sin (2 a+2 b x)}}-\frac {2 \cos ^{\frac {3}{2}}(a+b x)}{b \sqrt {\sin (a+b x)}}\right )}{\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}+\frac {2 d \csc (a+b x) \sqrt {d \tan (a+b x)}}{b}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {2 d^2 \sqrt {\sin (a+b x)} \left (-\frac {2 \cos ^{\frac {3}{2}}(a+b x)}{b \sqrt {\sin (a+b x)}}-\frac {2 \sqrt {\sin (a+b x)} \sqrt {\cos (a+b x)} E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{b \sqrt {\sin (2 a+2 b x)}}\right )}{\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}+\frac {2 d \csc (a+b x) \sqrt {d \tan (a+b x)}}{b}\) |
(2*d^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]])))/(Sqrt[Cos[a + b*x]]*Sqrt[d*Tan[a + b*x]]) + (2 *d*Csc[a + b*x]*Sqrt[d*Tan[a + b*x]])/b
3.1.72.3.1 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[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. \(370\) vs. \(2(119)=238\).
Time = 0.67 (sec) , antiderivative size = 371, normalized size of antiderivative = 3.64
method | result | size |
default | \(-\frac {\csc \left (b x +a \right ) \sqrt {d \tan \left (b x +a \right )}\, d \left (-4 \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )+2 \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )-4 \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+2 \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+2 \sqrt {2}\, \cos \left (b x +a \right )-\sqrt {2}\right ) \sqrt {2}}{b}\) | \(371\) |
-1/b*csc(b*x+a)*(d*tan(b*x+a))^(1/2)*d*(-4*(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)+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))*cos(b*x+a)-4*( 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)) +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*2^(1/2)*cos(b*x+a)-2^(1/2))*2^(1/2)
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.73 \[ \int \csc ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=-\frac {2 \, {\left (i \, \sqrt {i \, d} d E(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) - i \, \sqrt {-i \, d} d E(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) - i \, \sqrt {i \, d} d F(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) + i \, \sqrt {-i \, d} d F(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) + {\left (2 \, d \cos \left (b x + a\right )^{2} - d\right )} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}\right )}}{b \sin \left (b x + a\right )} \]
-2*(I*sqrt(I*d)*d*elliptic_e(arcsin(cos(b*x + a) + I*sin(b*x + a)), -1)*si n(b*x + a) - I*sqrt(-I*d)*d*elliptic_e(arcsin(cos(b*x + a) - I*sin(b*x + a )), -1)*sin(b*x + a) - I*sqrt(I*d)*d*elliptic_f(arcsin(cos(b*x + a) + I*si n(b*x + a)), -1)*sin(b*x + a) + I*sqrt(-I*d)*d*elliptic_f(arcsin(cos(b*x + a) - I*sin(b*x + a)), -1)*sin(b*x + a) + (2*d*cos(b*x + a)^2 - d)*sqrt(d* sin(b*x + a)/cos(b*x + a)))/(b*sin(b*x + a))
Timed out. \[ \int \csc ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\text {Timed out} \]
\[ \int \csc ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}} \csc \left (b x + a\right )^{3} \,d x } \]
\[ \int \csc ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}} \csc \left (b x + a\right )^{3} \,d x } \]
Timed out. \[ \int \csc ^3(a+b x) (d \tan (a+b x))^{3/2} \, dx=\int \frac {{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2}}{{\sin \left (a+b\,x\right )}^3} \,d x \]