Integrand size = 19, antiderivative size = 92 \[ \int \frac {\cos ^4(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\frac {4 \cos (a+b x)}{15 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {2 \cos ^3(a+b x)}{9 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {8 \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{15 b} \] Output:
4/15*cos(b*x+a)/b/csc(b*x+a)^(3/2)+2/9*cos(b*x+a)^3/b/csc(b*x+a)^(3/2)-8/1 5*csc(b*x+a)^(1/2)*EllipticE(cos(1/2*a+1/4*Pi+1/2*b*x),2^(1/2))*sin(b*x+a) ^(1/2)/b
Time = 0.57 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.68 \[ \int \frac {\cos ^4(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\frac {39 \cos (a+b x)+5 \cos (3 (a+b x))-\frac {48 E\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right )}{\sin ^{\frac {3}{2}}(a+b x)}}{90 b \csc ^{\frac {3}{2}}(a+b x)} \] Input:
Integrate[Cos[a + b*x]^4/Sqrt[Csc[a + b*x]],x]
Output:
(39*Cos[a + b*x] + 5*Cos[3*(a + b*x)] - (48*EllipticE[(-2*a + Pi - 2*b*x)/ 4, 2])/Sin[a + b*x]^(3/2))/(90*b*Csc[a + b*x]^(3/2))
Time = 0.45 (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, 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.
| \(\displaystyle \int \frac {\cos ^4(a+b x)}{\sqrt {\csc (a+b x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sqrt {\csc (a+b x)} \sec (a+b x)^4}dx\) |
| \(\Big \downarrow \) 3108 |
| \(\displaystyle \frac {2}{3} \int \frac {\cos ^2(a+b x)}{\sqrt {\csc (a+b x)}}dx+\frac {2 \cos ^3(a+b x)}{9 b \csc ^{\frac {3}{2}}(a+b x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} \int \frac {1}{\sqrt {\csc (a+b x)} \sec (a+b x)^2}dx+\frac {2 \cos ^3(a+b x)}{9 b \csc ^{\frac {3}{2}}(a+b x)}\) |
| \(\Big \downarrow \) 3108 |
| \(\displaystyle \frac {2}{3} \left (\frac {2}{5} \int \frac {1}{\sqrt {\csc (a+b x)}}dx+\frac {2 \cos (a+b x)}{5 b \csc ^{\frac {3}{2}}(a+b x)}\right )+\frac {2 \cos ^3(a+b x)}{9 b \csc ^{\frac {3}{2}}(a+b x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} \left (\frac {2}{5} \int \frac {1}{\sqrt {\csc (a+b x)}}dx+\frac {2 \cos (a+b x)}{5 b \csc ^{\frac {3}{2}}(a+b x)}\right )+\frac {2 \cos ^3(a+b x)}{9 b \csc ^{\frac {3}{2}}(a+b x)}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {2}{3} \left (\frac {2}{5} \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \int \sqrt {\sin (a+b x)}dx+\frac {2 \cos (a+b x)}{5 b \csc ^{\frac {3}{2}}(a+b x)}\right )+\frac {2 \cos ^3(a+b x)}{9 b \csc ^{\frac {3}{2}}(a+b x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} \left (\frac {2}{5} \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \int \sqrt {\sin (a+b x)}dx+\frac {2 \cos (a+b x)}{5 b \csc ^{\frac {3}{2}}(a+b x)}\right )+\frac {2 \cos ^3(a+b x)}{9 b \csc ^{\frac {3}{2}}(a+b x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {2 \cos ^3(a+b x)}{9 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {2}{3} \left (\frac {2 \cos (a+b x)}{5 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {4 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{5 b}\right )\) |
Input:
Int[Cos[a + b*x]^4/Sqrt[Csc[a + b*x]],x]
Output:
(2*Cos[a + b*x]^3)/(9*b*Csc[a + b*x]^(3/2)) + (2*((2*Cos[a + b*x])/(5*b*Cs c[a + b*x]^(3/2)) + (4*Sqrt[Csc[a + b*x]]*EllipticE[(a - Pi/2 + b*x)/2, 2] *Sqrt[Sin[a + b*x]])/(5*b)))/3
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[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(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 = 0.91 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.65
| method | result | size |
| default | \(\frac {-\frac {2 \cos \left (b x +a \right )^{6}}{9}-\frac {8 \sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \operatorname {EllipticE}\left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{15}+\frac {4 \sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{15}-\frac {2 \cos \left (b x +a \right )^{4}}{45}+\frac {4 \cos \left (b x +a \right )^{2}}{15}}{\cos \left (b x +a \right ) \sqrt {\sin \left (b x +a \right )}\, b}\) | \(152\) |
Input:
int(cos(b*x+a)^4/csc(b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
(-2/9*cos(b*x+a)^6-8/15*(sin(b*x+a)+1)^(1/2)*(-2*sin(b*x+a)+2)^(1/2)*(-sin (b*x+a))^(1/2)*EllipticE((sin(b*x+a)+1)^(1/2),1/2*2^(1/2))+4/15*(sin(b*x+a )+1)^(1/2)*(-2*sin(b*x+a)+2)^(1/2)*(-sin(b*x+a))^(1/2)*EllipticF((sin(b*x+ a)+1)^(1/2),1/2*2^(1/2))-2/45*cos(b*x+a)^4+4/15*cos(b*x+a)^2)/cos(b*x+a)/s in(b*x+a)^(1/2)/b
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.03 \[ \int \frac {\cos ^4(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\frac {2 \, {\left (6 \, \sqrt {2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 6 \, \sqrt {-2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) - \frac {5 \, \cos \left (b x + a\right )^{5} + \cos \left (b x + a\right )^{3} - 6 \, \cos \left (b x + a\right )}{\sqrt {\sin \left (b x + a\right )}}\right )}}{45 \, b} \] Input:
integrate(cos(b*x+a)^4/csc(b*x+a)^(1/2),x, algorithm="fricas")
Output:
2/45*(6*sqrt(2*I)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(b*x + a) + I*sin(b*x + a))) + 6*sqrt(-2*I)*weierstrassZeta(4, 0, weierstrassPI nverse(4, 0, cos(b*x + a) - I*sin(b*x + a))) - (5*cos(b*x + a)^5 + cos(b*x + a)^3 - 6*cos(b*x + a))/sqrt(sin(b*x + a)))/b
\[ \int \frac {\cos ^4(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\int \frac {\cos ^{4}{\left (a + b x \right )}}{\sqrt {\csc {\left (a + b x \right )}}}\, dx \] Input:
integrate(cos(b*x+a)**4/csc(b*x+a)**(1/2),x)
Output:
Integral(cos(a + b*x)**4/sqrt(csc(a + b*x)), x)
\[ \int \frac {\cos ^4(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\int { \frac {\cos \left (b x + a\right )^{4}}{\sqrt {\csc \left (b x + a\right )}} \,d x } \] Input:
integrate(cos(b*x+a)^4/csc(b*x+a)^(1/2),x, algorithm="maxima")
Output:
integrate(cos(b*x + a)^4/sqrt(csc(b*x + a)), x)
\[ \int \frac {\cos ^4(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\int { \frac {\cos \left (b x + a\right )^{4}}{\sqrt {\csc \left (b x + a\right )}} \,d x } \] Input:
integrate(cos(b*x+a)^4/csc(b*x+a)^(1/2),x, algorithm="giac")
Output:
integrate(cos(b*x + a)^4/sqrt(csc(b*x + a)), x)
Timed out. \[ \int \frac {\cos ^4(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^4}{\sqrt {\frac {1}{\sin \left (a+b\,x\right )}}} \,d x \] Input:
int(cos(a + b*x)^4/(1/sin(a + b*x))^(1/2),x)
Output:
int(cos(a + b*x)^4/(1/sin(a + b*x))^(1/2), x)
\[ \int \frac {\cos ^4(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\int \frac {\sqrt {\csc \left (b x +a \right )}\, \cos \left (b x +a \right )^{4}}{\csc \left (b x +a \right )}d x \] Input:
int(cos(b*x+a)^4/csc(b*x+a)^(1/2),x)
Output:
int((sqrt(csc(a + b*x))*cos(a + b*x)**4)/csc(a + b*x),x)