\(\int \frac {\sin ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx\) [524]

Optimal result

Integrand size = 21, antiderivative size = 72 \[ \int \frac {\sin ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=-\frac {2 d \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}}+\frac {6 E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right )}{5 f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \] Output:

-2/5*d*cos(f*x+e)/f/(d*csc(f*x+e))^(3/2)-6/5*EllipticE(cos(1/2*e+1/4*Pi+1/ 
2*f*x),2^(1/2))/f/(d*csc(f*x+e))^(1/2)/sin(f*x+e)^(1/2)
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.79 \[ \int \frac {\sin ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\frac {-\frac {12 E\left (\left .\frac {1}{4} (-2 e+\pi -2 f x)\right |2\right )}{\sqrt {\sin (e+f x)}}-2 \sin (2 (e+f x))}{10 f \sqrt {d \csc (e+f x)}} \] Input:

Integrate[Sin[e + f*x]^2/Sqrt[d*Csc[e + f*x]],x]
 

Output:

((-12*EllipticE[(-2*e + Pi - 2*f*x)/4, 2])/Sqrt[Sin[e + f*x]] - 2*Sin[2*(e 
 + f*x)])/(10*f*Sqrt[d*Csc[e + f*x]])
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 2030, 4256, 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.

\(\displaystyle \int \frac {\sin ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\csc (e+f x)^2 \sqrt {d \csc (e+f x)}}dx\)

\(\Big \downarrow \) 2030

\(\displaystyle d^2 \int \frac {1}{(d \csc (e+f x))^{5/2}}dx\)

\(\Big \downarrow \) 4256

\(\displaystyle d^2 \left (\frac {3 \int \frac {1}{\sqrt {d \csc (e+f x)}}dx}{5 d^2}-\frac {2 \cos (e+f x)}{5 d f (d \csc (e+f x))^{3/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle d^2 \left (\frac {3 \int \frac {1}{\sqrt {d \csc (e+f x)}}dx}{5 d^2}-\frac {2 \cos (e+f x)}{5 d f (d \csc (e+f x))^{3/2}}\right )\)

\(\Big \downarrow \) 4258

\(\displaystyle d^2 \left (\frac {3 \int \sqrt {\sin (e+f x)}dx}{5 d^2 \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}-\frac {2 \cos (e+f x)}{5 d f (d \csc (e+f x))^{3/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle d^2 \left (\frac {3 \int \sqrt {\sin (e+f x)}dx}{5 d^2 \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}-\frac {2 \cos (e+f x)}{5 d f (d \csc (e+f x))^{3/2}}\right )\)

\(\Big \downarrow \) 3119

\(\displaystyle d^2 \left (\frac {6 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{5 d^2 f \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}-\frac {2 \cos (e+f x)}{5 d f (d \csc (e+f x))^{3/2}}\right )\)

Input:

Int[Sin[e + f*x]^2/Sqrt[d*Csc[e + f*x]],x]
 

Output:

d^2*((-2*Cos[e + f*x])/(5*d*f*(d*Csc[e + f*x])^(3/2)) + (6*EllipticE[(e - 
Pi/2 + f*x)/2, 2])/(5*d^2*f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]]))
 

Defintions of rubi rules used

rule 2030

Int[(Fx_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Simp[1/b^m   Int[(b*v) 
^(m + n)*Fx, x], x] /; FreeQ[{b, n}, x] && IntegerQ[m]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3119

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]
 

rule 4256

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( 
b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n)   Int[(b*Csc[c 
+ d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2* 
n]
 

rule 4258

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]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.92 (sec) , antiderivative size = 254, normalized size of antiderivative = 3.53

method	result	size
default	\(\frac {\sqrt {2}\, \left (\left (-6 \cos \left (f x +e \right )-6\right ) \sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}\, \sqrt {i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {1-i \cot \left (f x +e \right )+i \csc \left (f x +e \right )}+\left (3 \cos \left (f x +e \right )+3\right ) \sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}\, \sqrt {i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}\, \sqrt {1-i \cot \left (f x +e \right )+i \csc \left (f x +e \right )}\, \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )+\left (\cos \left (f x +e \right )^{3}-4 \cos \left (f x +e \right )+3\right ) \sqrt {2}\right ) \csc \left (f x +e \right )}{5 f \sqrt {d \csc \left (f x +e \right )}}\)	\(254\)

Input:

int(sin(f*x+e)^2/(d*csc(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/5/f*2^(1/2)*((-6*cos(f*x+e)-6)*(1+I*cot(f*x+e)-I*csc(f*x+e))^(1/2)*(I*(c 
sc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticE((1+I*cot(f*x+e)-I*csc(f*x+e))^(1/2) 
,1/2*2^(1/2))*(1-I*cot(f*x+e)+I*csc(f*x+e))^(1/2)+(3*cos(f*x+e)+3)*(1+I*co 
t(f*x+e)-I*csc(f*x+e))^(1/2)*(I*(csc(f*x+e)-cot(f*x+e)))^(1/2)*(1-I*cot(f* 
x+e)+I*csc(f*x+e))^(1/2)*EllipticF((1+I*cot(f*x+e)-I*csc(f*x+e))^(1/2),1/2 
*2^(1/2))+(cos(f*x+e)^3-4*cos(f*x+e)+3)*2^(1/2))/(d*csc(f*x+e))^(1/2)*csc( 
f*x+e)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.33 \[ \int \frac {\sin ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\frac {2 \, {\left (\cos \left (f x + e\right )^{3} - \cos \left (f x + e\right )\right )} \sqrt {\frac {d}{\sin \left (f x + e\right )}} + 3 \, \sqrt {2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 3 \, \sqrt {-2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{5 \, d f} \] Input:

integrate(sin(f*x+e)^2/(d*csc(f*x+e))^(1/2),x, algorithm="fricas")
 

Output:

1/5*(2*(cos(f*x + e)^3 - cos(f*x + e))*sqrt(d/sin(f*x + e)) + 3*sqrt(2*I*d 
)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(f*x + e) + I*sin(f*x 
 + e))) + 3*sqrt(-2*I*d)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, c 
os(f*x + e) - I*sin(f*x + e))))/(d*f)
 

Sympy [F]

\[ \int \frac {\sin ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\int \frac {\sin ^{2}{\left (e + f x \right )}}{\sqrt {d \csc {\left (e + f x \right )}}}\, dx \] Input:

integrate(sin(f*x+e)**2/(d*csc(f*x+e))**(1/2),x)
 

Output:

Integral(sin(e + f*x)**2/sqrt(d*csc(e + f*x)), x)
 

Maxima [F]

\[ \int \frac {\sin ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\int { \frac {\sin \left (f x + e\right )^{2}}{\sqrt {d \csc \left (f x + e\right )}} \,d x } \] Input:

integrate(sin(f*x+e)^2/(d*csc(f*x+e))^(1/2),x, algorithm="maxima")
 

Output:

integrate(sin(f*x + e)^2/sqrt(d*csc(f*x + e)), x)
 

Giac [F]

\[ \int \frac {\sin ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\int { \frac {\sin \left (f x + e\right )^{2}}{\sqrt {d \csc \left (f x + e\right )}} \,d x } \] Input:

integrate(sin(f*x+e)^2/(d*csc(f*x+e))^(1/2),x, algorithm="giac")
 

Output:

integrate(sin(f*x + e)^2/sqrt(d*csc(f*x + e)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^2}{\sqrt {\frac {d}{\sin \left (e+f\,x\right )}}} \,d x \] Input:

int(sin(e + f*x)^2/(d/sin(e + f*x))^(1/2),x)
 

Output:

int(sin(e + f*x)^2/(d/sin(e + f*x))^(1/2), x)
 

Reduce [F]

\[ \int \frac {\sin ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\frac {\sqrt {d}\, \left (\int \frac {\sqrt {\csc \left (f x +e \right )}\, \sin \left (f x +e \right )^{2}}{\csc \left (f x +e \right )}d x \right )}{d} \] Input:

int(sin(f*x+e)^2/(d*csc(f*x+e))^(1/2),x)
 

Output:

(sqrt(d)*int((sqrt(csc(e + f*x))*sin(e + f*x)**2)/csc(e + f*x),x))/d