Integrand size = 25, antiderivative size = 135 \[ \int \frac {1}{(d \csc (a+b x))^{5/2} (c \sec (a+b x))^{5/2}} \, dx=-\frac {c}{5 b d (d \csc (a+b x))^{3/2} (c \sec (a+b x))^{7/2}}+\frac {1}{10 b c d (d \csc (a+b x))^{3/2} (c \sec (a+b x))^{3/2}}+\frac {3 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{20 b c^2 d^2 \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {\sin (2 a+2 b x)}} \]
-1/5*c/b/d/(d*csc(b*x+a))^(3/2)/(c*sec(b*x+a))^(7/2)+1/10/b/c/d/(d*csc(b*x +a))^(3/2)/(c*sec(b*x+a))^(3/2)-3/20*(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/c^2/d^2/(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 = 1.45 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(d \csc (a+b x))^{5/2} (c \sec (a+b x))^{5/2}} \, dx=\frac {\left (-2 \cos ^2(a+b x) \cos (2 (a+b x))+3 \sqrt [4]{-\cot ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {1}{2},\csc ^2(a+b x)\right )\right ) \sqrt {c \sec (a+b x)}}{20 b c^3 d (d \csc (a+b x))^{3/2}} \]
((-2*Cos[a + b*x]^2*Cos[2*(a + b*x)] + 3*(-Cot[a + b*x]^2)^(1/4)*Hypergeom etric2F1[-1/2, 1/4, 1/2, Csc[a + b*x]^2])*Sqrt[c*Sec[a + b*x]])/(20*b*c^3* d*(d*Csc[a + b*x])^(3/2))
Time = 0.67 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3107, 3042, 3108, 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.
| \(\displaystyle \int \frac {1}{(c \sec (a+b x))^{5/2} (d \csc (a+b x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(c \sec (a+b x))^{5/2} (d \csc (a+b x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3107 |
| \(\displaystyle \frac {3 \int \frac {1}{\sqrt {d \csc (a+b x)} (c \sec (a+b x))^{5/2}}dx}{10 d^2}-\frac {c}{5 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \int \frac {1}{\sqrt {d \csc (a+b x)} (c \sec (a+b x))^{5/2}}dx}{10 d^2}-\frac {c}{5 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3108 |
| \(\displaystyle \frac {3 \left (\frac {\int \frac {1}{\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}}dx}{2 c^2}+\frac {d}{3 b c (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{3/2}}\right )}{10 d^2}-\frac {c}{5 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {\int \frac {1}{\sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)}}dx}{2 c^2}+\frac {d}{3 b c (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{3/2}}\right )}{10 d^2}-\frac {c}{5 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3110 |
| \(\displaystyle \frac {3 \left (\frac {\int \sqrt {c \cos (a+b x)} \sqrt {d \sin (a+b x)}dx}{2 c^2 \sqrt {c \cos (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {d \sin (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{3 b c (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{3/2}}\right )}{10 d^2}-\frac {c}{5 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {\int \sqrt {c \cos (a+b x)} \sqrt {d \sin (a+b x)}dx}{2 c^2 \sqrt {c \cos (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {d \sin (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{3 b c (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{3/2}}\right )}{10 d^2}-\frac {c}{5 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3052 |
| \(\displaystyle \frac {3 \left (\frac {\int \sqrt {\sin (2 a+2 b x)}dx}{2 c^2 \sqrt {\sin (2 a+2 b x)} \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{3 b c (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{3/2}}\right )}{10 d^2}-\frac {c}{5 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {\int \sqrt {\sin (2 a+2 b x)}dx}{2 c^2 \sqrt {\sin (2 a+2 b x)} \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{3 b c (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{3/2}}\right )}{10 d^2}-\frac {c}{5 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{3/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {3 \left (\frac {E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{2 b c^2 \sqrt {\sin (2 a+2 b x)} \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{3 b c (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{3/2}}\right )}{10 d^2}-\frac {c}{5 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{3/2}}\) |
-1/5*c/(b*d*(d*Csc[a + b*x])^(3/2)*(c*Sec[a + b*x])^(7/2)) + (3*(d/(3*b*c* (d*Csc[a + b*x])^(3/2)*(c*Sec[a + b*x])^(3/2)) + EllipticE[a - Pi/4 + b*x, 2]/(2*b*c^2*Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b* x]])))/(10*d^2)
3.3.78.3.1 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[b*(a*Csc[e + f*x])^(m + 1)*((b*Sec[e + f*x])^(n - 1) /(a*f*(m + n))), x] + Simp[(m + 1)/(a^2*(m + n)) Int[(a*Csc[e + f*x])^(m + 2)*(b*Sec[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -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)*(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[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. \(424\) vs. \(2(140)=280\).
Time = 8.06 (sec) , antiderivative size = 425, normalized size of antiderivative = 3.15
| method | result | size |
| default | \(-\frac {\sqrt {2}\, \left (-4 \cos \left (b x +a \right )^{6} \sqrt {2}+6 \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \operatorname {EllipticE}\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )-3 \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )+6 \cos \left (b x +a \right )^{4} \sqrt {2}+6 \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \operatorname {EllipticE}\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \cos \left (b x +a \right )^{2}-3 \sqrt {2}\, \cos \left (b x +a \right )\right ) \sec \left (b x +a \right ) \csc \left (b x +a \right )}{40 b \sqrt {d \csc \left (b x +a \right )}\, \sqrt {c \sec \left (b x +a \right )}\, c^{2} d^{2}}\) | \(425\) |
-1/40/b*2^(1/2)*(-4*cos(b*x+a)^6*2^(1/2)+6*(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)*EllipticE(( 1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2*2^(1/2))*cos(b*x+a)-3*(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)+6*cos (b*x+a)^4*2^(1/2)+6*(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)*EllipticE((1+csc(b*x+a)-cot(b*x+a) )^(1/2),1/2*2^(1/2))-3*(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))+2^(1/2)*cos(b*x+a)^2-3*2^(1/2)*cos(b*x+a))/(d*csc( b*x+a))^(1/2)/(c*sec(b*x+a))^(1/2)/c^2/d^2*sec(b*x+a)*csc(b*x+a)
\[ \int \frac {1}{(d \csc (a+b x))^{5/2} (c \sec (a+b x))^{5/2}} \, dx=\int { \frac {1}{\left (d \csc \left (b x + a\right )\right )^{\frac {5}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}}} \,d x } \]
integral(sqrt(d*csc(b*x + a))*sqrt(c*sec(b*x + a))/(c^3*d^3*csc(b*x + a)^3 *sec(b*x + a)^3), x)
Timed out. \[ \int \frac {1}{(d \csc (a+b x))^{5/2} (c \sec (a+b x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(d \csc (a+b x))^{5/2} (c \sec (a+b x))^{5/2}} \, dx=\int { \frac {1}{\left (d \csc \left (b x + a\right )\right )^{\frac {5}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{(d \csc (a+b x))^{5/2} (c \sec (a+b x))^{5/2}} \, dx=\int { \frac {1}{\left (d \csc \left (b x + a\right )\right )^{\frac {5}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(d \csc (a+b x))^{5/2} (c \sec (a+b x))^{5/2}} \, dx=\int \frac {1}{{\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{5/2}\,{\left (\frac {d}{\sin \left (a+b\,x\right )}\right )}^{5/2}} \,d x \]