Integrand size = 25, antiderivative size = 89 \[ \int \frac {(c \sec (a+b x))^{3/2}}{\sqrt {d \csc (a+b x)}} \, dx=\frac {2 c d \sqrt {c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}}-\frac {2 c^2 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{b \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {\sin (2 a+2 b x)}} \]
2*c*d*(c*sec(b*x+a))^(1/2)/b/(d*csc(b*x+a))^(3/2)+2*c^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))/b/(d*csc( b*x+a))^(1/2)/(c*sec(b*x+a))^(1/2)/sin(2*b*x+2*a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.85 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.74 \[ \int \frac {(c \sec (a+b x))^{3/2}}{\sqrt {d \csc (a+b x)}} \, dx=-\frac {2 c d \left (-1+\sqrt [4]{-\cot ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {1}{2},\csc ^2(a+b x)\right )\right ) \sqrt {c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}} \]
(-2*c*d*(-1 + (-Cot[a + b*x]^2)^(1/4)*Hypergeometric2F1[-1/2, 1/4, 1/2, Cs c[a + b*x]^2])*Sqrt[c*Sec[a + b*x]])/(b*(d*Csc[a + b*x])^(3/2))
Time = 0.51 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 3106, 3042, 3110, 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.
\(\displaystyle \int \frac {(c \sec (a+b x))^{3/2}}{\sqrt {d \csc (a+b x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(c \sec (a+b x))^{3/2}}{\sqrt {d \csc (a+b x)}}dx\) |
\(\Big \downarrow \) 3106 |
\(\displaystyle \frac {2 c d \sqrt {c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}}-2 c^2 \int \frac {1}{\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 c d \sqrt {c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}}-2 c^2 \int \frac {1}{\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}}dx\) |
\(\Big \downarrow \) 3110 |
\(\displaystyle \frac {2 c d \sqrt {c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}}-\frac {2 c^2 \int \sqrt {c \cos (a+b x)} \sqrt {d \sin (a+b x)}dx}{\sqrt {c \cos (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {d \sin (a+b x)} \sqrt {d \csc (a+b x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 c d \sqrt {c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}}-\frac {2 c^2 \int \sqrt {c \cos (a+b x)} \sqrt {d \sin (a+b x)}dx}{\sqrt {c \cos (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {d \sin (a+b x)} \sqrt {d \csc (a+b x)}}\) |
\(\Big \downarrow \) 3052 |
\(\displaystyle \frac {2 c d \sqrt {c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}}-\frac {2 c^2 \int \sqrt {\sin (2 a+2 b x)}dx}{\sqrt {\sin (2 a+2 b x)} \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 c d \sqrt {c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}}-\frac {2 c^2 \int \sqrt {\sin (2 a+2 b x)}dx}{\sqrt {\sin (2 a+2 b x)} \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {2 c d \sqrt {c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}}-\frac {2 c^2 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{b \sqrt {\sin (2 a+2 b x)} \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}\) |
(2*c*d*Sqrt[c*Sec[a + b*x]])/(b*(d*Csc[a + b*x])^(3/2)) - (2*c^2*EllipticE [a - Pi/4 + b*x, 2])/(b*Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin [2*a + 2*b*x]])
3.3.43.3.1 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[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*((b_.)*sec[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[a*b*(a*Csc[e + f*x])^(m - 1)*((b*Sec[e + f*x])^(n - 1)/(f*(n - 1))), x] + Simp[b^2*((m + n - 2)/(n - 1)) Int[(a*Csc[e + f*x]) ^m*(b*Sec[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && IntegersQ[2*m, 2*n]
Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[(a*Csc[e + f*x])^m*(b*Sec[e + f*x])^n*(a*Sin[e + f*x] )^m*(b*Cos[e + f*x])^n Int[1/((a*Sin[e + f*x])^m*(b*Cos[e + f*x])^n), x], x] /; FreeQ[{a, b, e, f, m, n}, x] && IntegerQ[m - 1/2] && IntegerQ[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. \(354\) vs. \(2(104)=208\).
Time = 1.22 (sec) , antiderivative size = 355, normalized size of antiderivative = 3.99
method | result | size |
default | \(-\frac {\sqrt {2}\, {\left (-\frac {c \left (\left (1-\cos \left (b x +a \right )\right )^{2} \csc \left (b x +a \right )^{2}+1\right )}{\left (1-\cos \left (b x +a \right )\right )^{2} \csc \left (b x +a \right )^{2}-1}\right )}^{\frac {3}{2}} \left (\left (1-\cos \left (b x +a \right )\right )^{2} \csc \left (b x +a \right )^{2}-1\right ) \left (2 \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {2-2 \csc \left (b x +a \right )+2 \cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \operatorname {EllipticE}\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {2-2 \csc \left (b x +a \right )+2 \cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+2 \left (1-\cos \left (b x +a \right )\right )^{2} \csc \left (b x +a \right )^{2}\right ) \sin \left (b x +a \right )}{b \left (\left (1-\cos \left (b x +a \right )\right )^{2} \csc \left (b x +a \right )^{2}+1\right ) \sqrt {\frac {d \left (\left (1-\cos \left (b x +a \right )\right )^{2} \csc \left (b x +a \right )+\sin \left (b x +a \right )\right )}{1-\cos \left (b x +a \right )}}\, \left (1-\cos \left (b x +a \right )\right )}\) | \(355\) |
-1/b*2^(1/2)*(-c*((1-cos(b*x+a))^2*csc(b*x+a)^2+1)/((1-cos(b*x+a))^2*csc(b *x+a)^2-1))^(3/2)*((1-cos(b*x+a))^2*csc(b*x+a)^2-1)*(2*(1+csc(b*x+a)-cot(b *x+a))^(1/2)*(2-2*csc(b*x+a)+2*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))-(1+csc(b*x+a)- cot(b*x+a))^(1/2)*(2-2*csc(b*x+a)+2*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-cos( b*x+a))^2*csc(b*x+a)^2)/((1-cos(b*x+a))^2*csc(b*x+a)^2+1)/(d/(1-cos(b*x+a) )*((1-cos(b*x+a))^2*csc(b*x+a)+sin(b*x+a)))^(1/2)/(1-cos(b*x+a))*sin(b*x+a )
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.74 \[ \int \frac {(c \sec (a+b x))^{3/2}}{\sqrt {d \csc (a+b x)}} \, dx=\frac {\sqrt {-4 i \, c d} c E(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) + \sqrt {4 i \, c d} c E(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) - \sqrt {-4 i \, c d} c F(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) - \sqrt {4 i \, c d} c F(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) - 4 \, {\left (c \cos \left (b x + a\right )^{2} - c\right )} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}}}{2 \, b d} \]
1/2*(sqrt(-4*I*c*d)*c*elliptic_e(arcsin(cos(b*x + a) + I*sin(b*x + a)), -1 ) + sqrt(4*I*c*d)*c*elliptic_e(arcsin(cos(b*x + a) - I*sin(b*x + a)), -1) - sqrt(-4*I*c*d)*c*elliptic_f(arcsin(cos(b*x + a) + I*sin(b*x + a)), -1) - sqrt(4*I*c*d)*c*elliptic_f(arcsin(cos(b*x + a) - I*sin(b*x + a)), -1) - 4 *(c*cos(b*x + a)^2 - c)*sqrt(c/cos(b*x + a))*sqrt(d/sin(b*x + a)))/(b*d)
\[ \int \frac {(c \sec (a+b x))^{3/2}}{\sqrt {d \csc (a+b x)}} \, dx=\int \frac {\left (c \sec {\left (a + b x \right )}\right )^{\frac {3}{2}}}{\sqrt {d \csc {\left (a + b x \right )}}}\, dx \]
\[ \int \frac {(c \sec (a+b x))^{3/2}}{\sqrt {d \csc (a+b x)}} \, dx=\int { \frac {\left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}}}{\sqrt {d \csc \left (b x + a\right )}} \,d x } \]
\[ \int \frac {(c \sec (a+b x))^{3/2}}{\sqrt {d \csc (a+b x)}} \, dx=\int { \frac {\left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}}}{\sqrt {d \csc \left (b x + a\right )}} \,d x } \]
Timed out. \[ \int \frac {(c \sec (a+b x))^{3/2}}{\sqrt {d \csc (a+b x)}} \, dx=\int \frac {{\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{3/2}}{\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}} \,d x \]