Integrand size = 25, antiderivative size = 98 \[ \int \frac {(c \sec (a+b x))^{5/2}}{(d \csc (a+b x))^{3/2}} \, dx=\frac {2 c (c \sec (a+b x))^{3/2}}{3 b d \sqrt {d \csc (a+b x)}}-\frac {c^2 \sqrt {d \csc (a+b x)} \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {c \sec (a+b x)} \sqrt {\sin (2 a+2 b x)}}{3 b d^2} \]
2/3*c*(c*sec(b*x+a))^(3/2)/b/d/(d*csc(b*x+a))^(1/2)+1/3*c^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*cs c(b*x+a))^(1/2)*(c*sec(b*x+a))^(1/2)*sin(2*b*x+2*a)^(1/2)/b/d^2
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 1.21 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.71 \[ \int \frac {(c \sec (a+b x))^{5/2}}{(d \csc (a+b x))^{3/2}} \, dx=\frac {c \left (2+\left (-\cot ^2(a+b x)\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\csc ^2(a+b x)\right )\right ) (c \sec (a+b x))^{3/2}}{3 b d \sqrt {d \csc (a+b x)}} \]
(c*(2 + (-Cot[a + b*x]^2)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, Csc[a + b *x]^2])*(c*Sec[a + b*x])^(3/2))/(3*b*d*Sqrt[d*Csc[a + b*x]])
Time = 0.52 (sec) , antiderivative size = 98, 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, 3104, 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.
\(\displaystyle \int \frac {(c \sec (a+b x))^{5/2}}{(d \csc (a+b x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(c \sec (a+b x))^{5/2}}{(d \csc (a+b x))^{3/2}}dx\) |
\(\Big \downarrow \) 3104 |
\(\displaystyle \frac {2 c (c \sec (a+b x))^{3/2}}{3 b d \sqrt {d \csc (a+b x)}}-\frac {c^2 \int \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}dx}{3 d^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 c (c \sec (a+b x))^{3/2}}{3 b d \sqrt {d \csc (a+b x)}}-\frac {c^2 \int \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}dx}{3 d^2}\) |
\(\Big \downarrow \) 3110 |
\(\displaystyle \frac {2 c (c \sec (a+b x))^{3/2}}{3 b d \sqrt {d \csc (a+b x)}}-\frac {c^2 \sqrt {c \cos (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {d \sin (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\sqrt {c \cos (a+b x)} \sqrt {d \sin (a+b x)}}dx}{3 d^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 c (c \sec (a+b x))^{3/2}}{3 b d \sqrt {d \csc (a+b x)}}-\frac {c^2 \sqrt {c \cos (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {d \sin (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\sqrt {c \cos (a+b x)} \sqrt {d \sin (a+b x)}}dx}{3 d^2}\) |
\(\Big \downarrow \) 3053 |
\(\displaystyle \frac {2 c (c \sec (a+b x))^{3/2}}{3 b d \sqrt {d \csc (a+b x)}}-\frac {c^2 \sqrt {\sin (2 a+2 b x)} \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}}dx}{3 d^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 c (c \sec (a+b x))^{3/2}}{3 b d \sqrt {d \csc (a+b x)}}-\frac {c^2 \sqrt {\sin (2 a+2 b x)} \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}}dx}{3 d^2}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {2 c (c \sec (a+b x))^{3/2}}{3 b d \sqrt {d \csc (a+b x)}}-\frac {c^2 \sqrt {\sin (2 a+2 b x)} \operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right ) \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}{3 b d^2}\) |
(2*c*(c*Sec[a + b*x])^(3/2))/(3*b*d*Sqrt[d*Csc[a + b*x]]) - (c^2*Sqrt[d*Cs c[a + b*x]]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])/(3*b*d^2)
3.3.52.3.1 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[b*(a*Csc[e + f*x])^(m + 1)*((b*Sec[e + f*x])^(n - 1)/ (f*a*(n - 1))), x] + Simp[b^2*((m + 1)/(a^2*(n - 1))) Int[(a*Csc[e + f*x] )^(m + 2)*(b*Sec[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ [n, 1] && LtQ[m, -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[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]
Leaf count of result is larger than twice the leaf count of optimal. \(242\) vs. \(2(109)=218\).
Time = 1.23 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.48
method | result | size |
default | \(\frac {\sqrt {2}\, \left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )^{2}+\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )-\sqrt {2}\, \sin \left (b x +a \right )\right ) \sqrt {c \sec \left (b x +a \right )}\, c^{2} \tan \left (b x +a \right )}{3 b d \sqrt {d \csc \left (b x +a \right )}\, \left (\cos \left (b x +a \right )-1\right ) \left (\cos \left (b x +a \right )+1\right )}\) | \(243\) |
1/3/b*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))*cos(b*x+a)^2+(1+csc(b*x+a)-cot(b*x+a))^(1/2)*(cot(b*x+a)-c sc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticF((1+csc(b*x+a)-c ot(b*x+a))^(1/2),1/2*2^(1/2))*cos(b*x+a)-2^(1/2)*sin(b*x+a))*(c*sec(b*x+a) )^(1/2)*c^2/d/(d*csc(b*x+a))^(1/2)/(cos(b*x+a)-1)/(cos(b*x+a)+1)*tan(b*x+a )
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.24 \[ \int \frac {(c \sec (a+b x))^{5/2}}{(d \csc (a+b x))^{3/2}} \, dx=\frac {i \, \sqrt {-4 i \, c d} c^{2} \cos \left (b x + a\right ) F(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) - i \, \sqrt {4 i \, c d} c^{2} \cos \left (b x + a\right ) F(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) + 4 \, c^{2} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}} \sin \left (b x + a\right )}{6 \, b d^{2} \cos \left (b x + a\right )} \]
1/6*(I*sqrt(-4*I*c*d)*c^2*cos(b*x + a)*elliptic_f(arcsin(cos(b*x + a) + I* sin(b*x + a)), -1) - I*sqrt(4*I*c*d)*c^2*cos(b*x + a)*elliptic_f(arcsin(co s(b*x + a) - I*sin(b*x + a)), -1) + 4*c^2*sqrt(c/cos(b*x + a))*sqrt(d/sin( b*x + a))*sin(b*x + a))/(b*d^2*cos(b*x + a))
Timed out. \[ \int \frac {(c \sec (a+b x))^{5/2}}{(d \csc (a+b x))^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(c \sec (a+b x))^{5/2}}{(d \csc (a+b x))^{3/2}} \, dx=\int { \frac {\left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}}}{\left (d \csc \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(c \sec (a+b x))^{5/2}}{(d \csc (a+b x))^{3/2}} \, dx=\int { \frac {\left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}}}{\left (d \csc \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(c \sec (a+b x))^{5/2}}{(d \csc (a+b x))^{3/2}} \, dx=\int \frac {{\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{5/2}}{{\left (\frac {d}{\sin \left (a+b\,x\right )}\right )}^{3/2}} \,d x \]