Integrand size = 25, antiderivative size = 166 \[ \int (d \csc (a+b x))^{9/2} (c \sec (a+b x))^{5/2} \, dx=\frac {40 c d^5 (c \sec (a+b x))^{3/2}}{21 b \sqrt {d \csc (a+b x)}}-\frac {20 c d^3 (d \csc (a+b x))^{3/2} (c \sec (a+b x))^{3/2}}{21 b}-\frac {2 c d (d \csc (a+b x))^{7/2} (c \sec (a+b x))^{3/2}}{7 b}+\frac {40 c^2 d^4 \sqrt {d \csc (a+b x)} \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {c \sec (a+b x)} \sqrt {\sin (2 a+2 b x)}}{21 b} \]
-20/21*c*d^3*(d*csc(b*x+a))^(3/2)*(c*sec(b*x+a))^(3/2)/b-2/7*c*d*(d*csc(b* x+a))^(7/2)*(c*sec(b*x+a))^(3/2)/b+40/21*c*d^5*(c*sec(b*x+a))^(3/2)/b/(d*c sc(b*x+a))^(1/2)-40/21*c^2*d^4*(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
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 2.39 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.55 \[ \int (d \csc (a+b x))^{9/2} (c \sec (a+b x))^{5/2} \, dx=-\frac {2 c d^5 \left (-7+\cot ^2(a+b x) \left (13+3 \csc ^2(a+b x)\right )+20 \left (-\cot ^2(a+b x)\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\csc ^2(a+b x)\right )\right ) (c \sec (a+b x))^{3/2}}{21 b \sqrt {d \csc (a+b x)}} \]
(-2*c*d^5*(-7 + Cot[a + b*x]^2*(13 + 3*Csc[a + b*x]^2) + 20*(-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))/(21*b*Sqrt[d*Csc[a + b*x]])
Time = 0.83 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 3105, 3042, 3105, 3042, 3106, 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.
| \(\displaystyle \int (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{9/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c \sec (a+b x))^{5/2} (d \csc (a+b x))^{9/2}dx\) |
| \(\Big \downarrow \) 3105 |
| \(\displaystyle \frac {10}{7} d^2 \int (d \csc (a+b x))^{5/2} (c \sec (a+b x))^{5/2}dx-\frac {2 c d (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{7/2}}{7 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {10}{7} d^2 \int (d \csc (a+b x))^{5/2} (c \sec (a+b x))^{5/2}dx-\frac {2 c d (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{7/2}}{7 b}\) |
| \(\Big \downarrow \) 3105 |
| \(\displaystyle \frac {10}{7} d^2 \left (2 d^2 \int \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{5/2}dx-\frac {2 c d (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{3/2}}{3 b}\right )-\frac {2 c d (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{7/2}}{7 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {10}{7} d^2 \left (2 d^2 \int \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{5/2}dx-\frac {2 c d (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{3/2}}{3 b}\right )-\frac {2 c d (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{7/2}}{7 b}\) |
| \(\Big \downarrow \) 3106 |
| \(\displaystyle \frac {10}{7} d^2 \left (2 d^2 \left (\frac {2}{3} c^2 \int \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}dx+\frac {2 c d (c \sec (a+b x))^{3/2}}{3 b \sqrt {d \csc (a+b x)}}\right )-\frac {2 c d (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{3/2}}{3 b}\right )-\frac {2 c d (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{7/2}}{7 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {10}{7} d^2 \left (2 d^2 \left (\frac {2}{3} c^2 \int \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}dx+\frac {2 c d (c \sec (a+b x))^{3/2}}{3 b \sqrt {d \csc (a+b x)}}\right )-\frac {2 c d (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{3/2}}{3 b}\right )-\frac {2 c d (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{7/2}}{7 b}\) |
| \(\Big \downarrow \) 3110 |
| \(\displaystyle \frac {10}{7} d^2 \left (2 d^2 \left (\frac {2}{3} c^2 \sqrt {c \cos (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {d \sin (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\sqrt {c \cos (a+b x)} \sqrt {d \sin (a+b x)}}dx+\frac {2 c d (c \sec (a+b x))^{3/2}}{3 b \sqrt {d \csc (a+b x)}}\right )-\frac {2 c d (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{3/2}}{3 b}\right )-\frac {2 c d (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{7/2}}{7 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {10}{7} d^2 \left (2 d^2 \left (\frac {2}{3} c^2 \sqrt {c \cos (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {d \sin (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\sqrt {c \cos (a+b x)} \sqrt {d \sin (a+b x)}}dx+\frac {2 c d (c \sec (a+b x))^{3/2}}{3 b \sqrt {d \csc (a+b x)}}\right )-\frac {2 c d (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{3/2}}{3 b}\right )-\frac {2 c d (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{7/2}}{7 b}\) |
| \(\Big \downarrow \) 3053 |
| \(\displaystyle \frac {10}{7} d^2 \left (2 d^2 \left (\frac {2}{3} c^2 \sqrt {\sin (2 a+2 b x)} \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}}dx+\frac {2 c d (c \sec (a+b x))^{3/2}}{3 b \sqrt {d \csc (a+b x)}}\right )-\frac {2 c d (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{3/2}}{3 b}\right )-\frac {2 c d (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{7/2}}{7 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {10}{7} d^2 \left (2 d^2 \left (\frac {2}{3} c^2 \sqrt {\sin (2 a+2 b x)} \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}}dx+\frac {2 c d (c \sec (a+b x))^{3/2}}{3 b \sqrt {d \csc (a+b x)}}\right )-\frac {2 c d (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{3/2}}{3 b}\right )-\frac {2 c d (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{7/2}}{7 b}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {10}{7} d^2 \left (2 d^2 \left (\frac {2 c^2 \sqrt {\sin (2 a+2 b x)} \operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right ) \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}{3 b}+\frac {2 c d (c \sec (a+b x))^{3/2}}{3 b \sqrt {d \csc (a+b x)}}\right )-\frac {2 c d (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{3/2}}{3 b}\right )-\frac {2 c d (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{7/2}}{7 b}\) |
(-2*c*d*(d*Csc[a + b*x])^(7/2)*(c*Sec[a + b*x])^(3/2))/(7*b) + (10*d^2*((- 2*c*d*(d*Csc[a + b*x])^(3/2)*(c*Sec[a + b*x])^(3/2))/(3*b) + 2*d^2*((2*c*d *(c*Sec[a + b*x])^(3/2))/(3*b*Sqrt[d*Csc[a + b*x]]) + (2*c^2*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]])/(3*b))))/7
3.3.46.3.1 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)*b*(a*Csc[e + f*x])^(m - 1)*((b*Sec[e + f*x])^(n - 1)/(f*(m - 1))), x] + Simp[a^2*((m + n - 2)/(m - 1)) Int[(a*Csc[e + f* x])^(m - 2)*(b*Sec[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[ m, 1] && IntegersQ[2*m, 2*n] && !GtQ[n, m]
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[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 = 1.88 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.99
\[\frac {\sqrt {2}\, d^{4} c^{2} \sqrt {d \csc \left (b x +a \right )}\, \sqrt {c \sec \left (b x +a \right )}\, \left (\left (40 \cos \left (b x +a \right )+40\right ) \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+\left (20 \cos \left (b x +a \right )^{4}-30 \cos \left (b x +a \right )^{2}+7\right ) \sqrt {2}\, \sec \left (b x +a \right ) \csc \left (b x +a \right )^{3}\right )}{21 b}\]
1/21/b*2^(1/2)*d^4*c^2*(d*csc(b*x+a))^(1/2)*(c*sec(b*x+a))^(1/2)*((40*cos( b*x+a)+40)*(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))+(20*cos(b*x+a)^4-30*cos(b*x+a)^2+7)*2^(1/2)*sec(b*x+a)*csc(b*x +a)^3)
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.36 \[ \int (d \csc (a+b x))^{9/2} (c \sec (a+b x))^{5/2} \, dx=-\frac {2 \, {\left (10 \, {\left (i \, c^{2} d^{4} \cos \left (b x + a\right )^{3} - i \, c^{2} d^{4} \cos \left (b x + a\right )\right )} \sqrt {-4 i \, c d} F(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) + 10 \, {\left (-i \, c^{2} d^{4} \cos \left (b x + a\right )^{3} + i \, c^{2} d^{4} \cos \left (b x + a\right )\right )} \sqrt {4 i \, c d} F(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) \sin \left (b x + a\right ) + {\left (20 \, c^{2} d^{4} \cos \left (b x + a\right )^{4} - 30 \, c^{2} d^{4} \cos \left (b x + a\right )^{2} + 7 \, c^{2} d^{4}\right )} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}}\right )}}{21 \, {\left (b \cos \left (b x + a\right )^{3} - b \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )} \]
-2/21*(10*(I*c^2*d^4*cos(b*x + a)^3 - I*c^2*d^4*cos(b*x + a))*sqrt(-4*I*c* d)*elliptic_f(arcsin(cos(b*x + a) + I*sin(b*x + a)), -1)*sin(b*x + a) + 10 *(-I*c^2*d^4*cos(b*x + a)^3 + I*c^2*d^4*cos(b*x + a))*sqrt(4*I*c*d)*ellipt ic_f(arcsin(cos(b*x + a) - I*sin(b*x + a)), -1)*sin(b*x + a) + (20*c^2*d^4 *cos(b*x + a)^4 - 30*c^2*d^4*cos(b*x + a)^2 + 7*c^2*d^4)*sqrt(c/cos(b*x + a))*sqrt(d/sin(b*x + a)))/((b*cos(b*x + a)^3 - b*cos(b*x + a))*sin(b*x + a ))
Timed out. \[ \int (d \csc (a+b x))^{9/2} (c \sec (a+b x))^{5/2} \, dx=\text {Timed out} \]
\[ \int (d \csc (a+b x))^{9/2} (c \sec (a+b x))^{5/2} \, dx=\int { \left (d \csc \left (b x + a\right )\right )^{\frac {9}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}} \,d x } \]
\[ \int (d \csc (a+b x))^{9/2} (c \sec (a+b x))^{5/2} \, dx=\int { \left (d \csc \left (b x + a\right )\right )^{\frac {9}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (d \csc (a+b x))^{9/2} (c \sec (a+b x))^{5/2} \, dx=\int {\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{5/2}\,{\left (\frac {d}{\sin \left (a+b\,x\right )}\right )}^{9/2} \,d x \]