Integrand size = 25, antiderivative size = 92 \[ \int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{3/2}} \, dx=\frac {d}{b c \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}}+\frac {\sqrt {d \csc (a+b x)} \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {c \sec (a+b x)} \sqrt {\sin (2 a+2 b x)}}{2 b c^2} \]
d/b/c/(d*csc(b*x+a))^(1/2)/(c*sec(b*x+a))^(1/2)-1/2*(sin(a+1/4*Pi+b*x)^2)^ (1/2)/sin(a+1/4*Pi+b*x)*EllipticF(cos(a+1/4*Pi+b*x),2^(1/2))*(d*csc(b*x+a) )^(1/2)*(c*sec(b*x+a))^(1/2)*sin(2*b*x+2*a)^(1/2)/b/c^2
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 1.12 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{3/2}} \, dx=\frac {d \left (1+\cos (2 (a+b x))-\left (-\cot ^2(a+b x)\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\csc ^2(a+b x)\right )\right ) \sec ^3(a+b x)}{2 b \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}} \]
(d*(1 + Cos[2*(a + b*x)] - (-Cot[a + b*x]^2)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, Csc[a + b*x]^2])*Sec[a + b*x]^3)/(2*b*Sqrt[d*Csc[a + b*x]]*(c*Se c[a + b*x])^(3/2))
Time = 0.50 (sec) , antiderivative size = 92, 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, 3108, 3042, 3110, 3042, 3053, 3042, 3120}
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.
\(\displaystyle \int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{3/2}}dx\) |
\(\Big \downarrow \) 3108 |
\(\displaystyle \frac {\int \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}dx}{2 c^2}+\frac {d}{b c \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}dx}{2 c^2}+\frac {d}{b c \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}\) |
\(\Big \downarrow \) 3110 |
\(\displaystyle \frac {\sqrt {c \cos (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {d \sin (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\sqrt {c \cos (a+b x)} \sqrt {d \sin (a+b x)}}dx}{2 c^2}+\frac {d}{b c \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {c \cos (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {d \sin (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\sqrt {c \cos (a+b x)} \sqrt {d \sin (a+b x)}}dx}{2 c^2}+\frac {d}{b c \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}\) |
\(\Big \downarrow \) 3053 |
\(\displaystyle \frac {\sqrt {\sin (2 a+2 b x)} \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}}dx}{2 c^2}+\frac {d}{b c \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {\sin (2 a+2 b x)} \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}}dx}{2 c^2}+\frac {d}{b c \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {\sqrt {\sin (2 a+2 b x)} \operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right ) \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}{2 b c^2}+\frac {d}{b c \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}\) |
d/(b*c*Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]]) + (Sqrt[d*Csc[a + b*x]]* EllipticF[a - Pi/4 + b*x, 2]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])/ (2*b*c^2)
3.3.67.3.1 Defintions of rubi rules used
Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_ )]]), x_Symbol] :> Simp[Sqrt[Sin[2*e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b *Cos[e + f*x]]) Int[1/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)*(a*Csc[e + f*x])^(m - 1)*((b*Sec[e + f*x])^(n + 1)/(b*f*(m + n))), x] + Simp[(n + 1)/(b^2*(m + n)) Int[(a*Csc[e + f*x])^ m*(b*Sec[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && LtQ[n, - 1] && NeQ[m + n, 0] && 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[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Time = 8.12 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.23
method | result | size |
default | \(\frac {\sqrt {2}\, \sqrt {d \csc \left (b x +a \right )}\, \left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \sec \left (b x +a \right )+\sqrt {2}\, \sin \left (b x +a \right )\right )}{2 b \sqrt {c \sec \left (b x +a \right )}\, c}\) | \(205\) |
1/2/b*2^(1/2)*(d*csc(b*x+a))^(1/2)/(c*sec(b*x+a))^(1/2)/c*((1+csc(b*x+a)-c ot(b*x+a))^(1/2)*(cot(b*x+a)-csc(b*x+a)+1)^(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))+(1+csc(b*x+a)- cot(b*x+a))^(1/2)*(cot(b*x+a)-csc(b*x+a)+1)^(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))*sec(b*x+a)+2^ (1/2)*sin(b*x+a))
\[ \int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{3/2}} \, dx=\int { \frac {\sqrt {d \csc \left (b x + a\right )}}{\left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{3/2}} \, dx=\int \frac {\sqrt {d \csc {\left (a + b x \right )}}}{\left (c \sec {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{3/2}} \, dx=\int { \frac {\sqrt {d \csc \left (b x + a\right )}}{\left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{3/2}} \, dx=\int { \frac {\sqrt {d \csc \left (b x + a\right )}}{\left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{3/2}} \, dx=\int \frac {\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}}{{\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{3/2}} \,d x \]