Integrand size = 19, antiderivative size = 92 \[ \int \cos ^4(a+b x) \sqrt {\csc (a+b x)} \, dx=\frac {4 \cos (a+b x)}{7 b \sqrt {\csc (a+b x)}}+\frac {2 \cos ^3(a+b x)}{7 b \sqrt {\csc (a+b x)}}+\frac {8 \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{7 b} \]
4/7*cos(b*x+a)/b/csc(b*x+a)^(1/2)+2/7*cos(b*x+a)^3/b/csc(b*x+a)^(1/2)-8/7* (sin(1/2*a+1/4*Pi+1/2*b*x)^2)^(1/2)/sin(1/2*a+1/4*Pi+1/2*b*x)*EllipticF(co s(1/2*a+1/4*Pi+1/2*b*x),2^(1/2))*csc(b*x+a)^(1/2)*sin(b*x+a)^(1/2)/b
Time = 0.31 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.68 \[ \int \cos ^4(a+b x) \sqrt {\csc (a+b x)} \, dx=\frac {\sqrt {\csc (a+b x)} \left (-32 \operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right ) \sqrt {\sin (a+b x)}+10 \sin (2 (a+b x))+\sin (4 (a+b x))\right )}{28 b} \]
(Sqrt[Csc[a + b*x]]*(-32*EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]] + 10*Sin[2*(a + b*x)] + Sin[4*(a + b*x)]))/(28*b)
Time = 0.43 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {3042, 3108, 3042, 3108, 3042, 4258, 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.
\(\displaystyle \int \cos ^4(a+b x) \sqrt {\csc (a+b x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {\csc (a+b x)}}{\sec (a+b x)^4}dx\) |
\(\Big \downarrow \) 3108 |
\(\displaystyle \frac {6}{7} \int \cos ^2(a+b x) \sqrt {\csc (a+b x)}dx+\frac {2 \cos ^3(a+b x)}{7 b \sqrt {\csc (a+b x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {6}{7} \int \frac {\sqrt {\csc (a+b x)}}{\sec (a+b x)^2}dx+\frac {2 \cos ^3(a+b x)}{7 b \sqrt {\csc (a+b x)}}\) |
\(\Big \downarrow \) 3108 |
\(\displaystyle \frac {6}{7} \left (\frac {2}{3} \int \sqrt {\csc (a+b x)}dx+\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}\right )+\frac {2 \cos ^3(a+b x)}{7 b \sqrt {\csc (a+b x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {6}{7} \left (\frac {2}{3} \int \sqrt {\csc (a+b x)}dx+\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}\right )+\frac {2 \cos ^3(a+b x)}{7 b \sqrt {\csc (a+b x)}}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {6}{7} \left (\frac {2}{3} \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \int \frac {1}{\sqrt {\sin (a+b x)}}dx+\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}\right )+\frac {2 \cos ^3(a+b x)}{7 b \sqrt {\csc (a+b x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {6}{7} \left (\frac {2}{3} \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \int \frac {1}{\sqrt {\sin (a+b x)}}dx+\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}\right )+\frac {2 \cos ^3(a+b x)}{7 b \sqrt {\csc (a+b x)}}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {2 \cos ^3(a+b x)}{7 b \sqrt {\csc (a+b x)}}+\frac {6}{7} \left (\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}+\frac {4 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right )}{3 b}\right )\) |
(2*Cos[a + b*x]^3)/(7*b*Sqrt[Csc[a + b*x]]) + (6*((2*Cos[a + b*x])/(3*b*Sq rt[Csc[a + b*x]]) + (4*Sqrt[Csc[a + b*x]]*EllipticF[(a - Pi/2 + b*x)/2, 2] *Sqrt[Sin[a + b*x]])/(3*b)))/7
3.3.68.3.1 Defintions of rubi rules used
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[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]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Time = 1.22 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.09
method | result | size |
default | \(\frac {\frac {4 \sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, F\left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{7}+\frac {2 \left (\sin ^{5}\left (b x +a \right )\right )}{7}-\frac {8 \left (\sin ^{3}\left (b x +a \right )\right )}{7}+\frac {6 \sin \left (b x +a \right )}{7}}{\cos \left (b x +a \right ) \sqrt {\sin \left (b x +a \right )}\, b}\) | \(100\) |
(4/7*(sin(b*x+a)+1)^(1/2)*(-2*sin(b*x+a)+2)^(1/2)*(-sin(b*x+a))^(1/2)*Elli pticF((sin(b*x+a)+1)^(1/2),1/2*2^(1/2))+2/7*sin(b*x+a)^5-8/7*sin(b*x+a)^3+ 6/7*sin(b*x+a))/cos(b*x+a)/sin(b*x+a)^(1/2)/b
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.85 \[ \int \cos ^4(a+b x) \sqrt {\csc (a+b x)} \, dx=\frac {2 \, {\left ({\left (\cos \left (b x + a\right )^{3} + 2 \, \cos \left (b x + a\right )\right )} \sqrt {\sin \left (b x + a\right )} - 2 i \, \sqrt {2 i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 2 i \, \sqrt {-2 i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )}}{7 \, b} \]
2/7*((cos(b*x + a)^3 + 2*cos(b*x + a))*sqrt(sin(b*x + a)) - 2*I*sqrt(2*I)* weierstrassPInverse(4, 0, cos(b*x + a) + I*sin(b*x + a)) + 2*I*sqrt(-2*I)* weierstrassPInverse(4, 0, cos(b*x + a) - I*sin(b*x + a)))/b
\[ \int \cos ^4(a+b x) \sqrt {\csc (a+b x)} \, dx=\int \cos ^{4}{\left (a + b x \right )} \sqrt {\csc {\left (a + b x \right )}}\, dx \]
\[ \int \cos ^4(a+b x) \sqrt {\csc (a+b x)} \, dx=\int { \cos \left (b x + a\right )^{4} \sqrt {\csc \left (b x + a\right )} \,d x } \]
\[ \int \cos ^4(a+b x) \sqrt {\csc (a+b x)} \, dx=\int { \cos \left (b x + a\right )^{4} \sqrt {\csc \left (b x + a\right )} \,d x } \]
Timed out. \[ \int \cos ^4(a+b x) \sqrt {\csc (a+b x)} \, dx=\int {\cos \left (a+b\,x\right )}^4\,\sqrt {\frac {1}{\sin \left (a+b\,x\right )}} \,d x \]