Integrand size = 12, antiderivative size = 71 \[ \int (c \csc (a+b x))^{3/2} \, dx=-\frac {2 c \cos (a+b x) \sqrt {c \csc (a+b x)}}{b}-\frac {2 c^2 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{b \sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}} \] Output:
-2*c*cos(b*x+a)*(c*csc(b*x+a))^(1/2)/b+2*c^2*EllipticE(cos(1/2*a+1/4*Pi+1/ 2*b*x),2^(1/2))/b/(c*csc(b*x+a))^(1/2)/sin(b*x+a)^(1/2)
Time = 0.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.76 \[ \int (c \csc (a+b x))^{3/2} \, dx=\frac {(c \csc (a+b x))^{3/2} \left (2 E\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right ) \sin ^{\frac {3}{2}}(a+b x)-\sin (2 (a+b x))\right )}{b} \] Input:
Integrate[(c*Csc[a + b*x])^(3/2),x]
Output:
((c*Csc[a + b*x])^(3/2)*(2*EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sin[a + b*x ]^(3/2) - Sin[2*(a + b*x)]))/b
Time = 0.33 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 4255, 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.
\(\displaystyle \int (c \csc (a+b x))^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (c \csc (a+b x))^{3/2}dx\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle c^2 \left (-\int \frac {1}{\sqrt {c \csc (a+b x)}}dx\right )-\frac {2 c \cos (a+b x) \sqrt {c \csc (a+b x)}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle c^2 \left (-\int \frac {1}{\sqrt {c \csc (a+b x)}}dx\right )-\frac {2 c \cos (a+b x) \sqrt {c \csc (a+b x)}}{b}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle -\frac {c^2 \int \sqrt {\sin (a+b x)}dx}{\sqrt {\sin (a+b x)} \sqrt {c \csc (a+b x)}}-\frac {2 c \cos (a+b x) \sqrt {c \csc (a+b x)}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {c^2 \int \sqrt {\sin (a+b x)}dx}{\sqrt {\sin (a+b x)} \sqrt {c \csc (a+b x)}}-\frac {2 c \cos (a+b x) \sqrt {c \csc (a+b x)}}{b}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle -\frac {2 c^2 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b \sqrt {\sin (a+b x)} \sqrt {c \csc (a+b x)}}-\frac {2 c \cos (a+b x) \sqrt {c \csc (a+b x)}}{b}\) |
Input:
Int[(c*Csc[a + b*x])^(3/2),x]
Output:
(-2*c*Cos[a + b*x]*Sqrt[c*Csc[a + b*x]])/b - (2*c^2*EllipticE[(a - Pi/2 + b*x)/2, 2])/(b*Sqrt[c*Csc[a + b*x]]*Sqrt[Sin[a + b*x]])
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)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
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]
Result contains complex when optimal does not.
Time = 0.20 (sec) , antiderivative size = 223, normalized size of antiderivative = 3.14
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (\sqrt {2}-\sqrt {1+i \left (-\csc \left (b x +a \right )+\cot \left (b x +a \right )\right )}\, \sqrt {1-i \left (-\csc \left (b x +a \right )+\cot \left (b x +a \right )\right )}\, \sqrt {-i \left (-\csc \left (b x +a \right )+\cot \left (b x +a \right )\right )}\, \left (2 \operatorname {EllipticE}\left (\sqrt {1+i \cot \left (b x +a \right )-i \csc \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )-\operatorname {EllipticF}\left (\sqrt {1+i \cot \left (b x +a \right )-i \csc \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )+2 \operatorname {EllipticE}\left (\sqrt {1+i \cot \left (b x +a \right )-i \csc \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-\operatorname {EllipticF}\left (\sqrt {1+i \cot \left (b x +a \right )-i \csc \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )\right )\right ) c \sqrt {c \csc \left (b x +a \right )}}{b}\) | \(223\) |
Input:
int((c*csc(b*x+a))^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/b*2^(1/2)*(2^(1/2)-(1+I*(-csc(b*x+a)+cot(b*x+a)))^(1/2)*(1-I*(-csc(b*x+ a)+cot(b*x+a)))^(1/2)*(-I*(-csc(b*x+a)+cot(b*x+a)))^(1/2)*(2*EllipticE((1+ I*cot(b*x+a)-I*csc(b*x+a))^(1/2),1/2*2^(1/2))*cos(b*x+a)-EllipticF((1+I*co t(b*x+a)-I*csc(b*x+a))^(1/2),1/2*2^(1/2))*cos(b*x+a)+2*EllipticE((1+I*cot( b*x+a)-I*csc(b*x+a))^(1/2),1/2*2^(1/2))-EllipticF((1+I*cot(b*x+a)-I*csc(b* x+a))^(1/2),1/2*2^(1/2))))*c*(c*csc(b*x+a))^(1/2)
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.17 \[ \int (c \csc (a+b x))^{3/2} \, dx=-\frac {2 \, c \sqrt {\frac {c}{\sin \left (b x + a\right )}} \cos \left (b x + a\right ) + \sqrt {2 i \, c} c {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + \sqrt {-2 i \, c} c {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )}{b} \] Input:
integrate((c*csc(b*x+a))^(3/2),x, algorithm="fricas")
Output:
-(2*c*sqrt(c/sin(b*x + a))*cos(b*x + a) + sqrt(2*I*c)*c*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(b*x + a) + I*sin(b*x + a))) + sqrt(-2*I* c)*c*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(b*x + a) - I*sin( b*x + a))))/b
\[ \int (c \csc (a+b x))^{3/2} \, dx=\int \left (c \csc {\left (a + b x \right )}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((c*csc(b*x+a))**(3/2),x)
Output:
Integral((c*csc(a + b*x))**(3/2), x)
\[ \int (c \csc (a+b x))^{3/2} \, dx=\int { \left (c \csc \left (b x + a\right )\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((c*csc(b*x+a))^(3/2),x, algorithm="maxima")
Output:
integrate((c*csc(b*x + a))^(3/2), x)
\[ \int (c \csc (a+b x))^{3/2} \, dx=\int { \left (c \csc \left (b x + a\right )\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((c*csc(b*x+a))^(3/2),x, algorithm="giac")
Output:
integrate((c*csc(b*x + a))^(3/2), x)
Timed out. \[ \int (c \csc (a+b x))^{3/2} \, dx=\int {\left (\frac {c}{\sin \left (a+b\,x\right )}\right )}^{3/2} \,d x \] Input:
int((c/sin(a + b*x))^(3/2),x)
Output:
int((c/sin(a + b*x))^(3/2), x)
\[ \int (c \csc (a+b x))^{3/2} \, dx=\sqrt {c}\, \left (\int \sqrt {\csc \left (b x +a \right )}\, \csc \left (b x +a \right )d x \right ) c \] Input:
int((c*csc(b*x+a))^(3/2),x)
Output:
sqrt(c)*int(sqrt(csc(a + b*x))*csc(a + b*x),x)*c