\(\int \sqrt {d \csc (e+f x)} \sin ^4(e+f x) \, dx\) [507]

Optimal result

Integrand size = 21, antiderivative size = 100 \[ \int \sqrt {d \csc (e+f x)} \sin ^4(e+f x) \, dx=-\frac {2 d^3 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac {10 d \cos (e+f x)}{21 f \sqrt {d \csc (e+f x)}}+\frac {10 \sqrt {d \csc (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),2\right ) \sqrt {\sin (e+f x)}}{21 f} \] Output:

-2/7*d^3*cos(f*x+e)/f/(d*csc(f*x+e))^(5/2)-10/21*d*cos(f*x+e)/f/(d*csc(f*x 
+e))^(1/2)+10/21*(d*csc(f*x+e))^(1/2)*InverseJacobiAM(1/2*e-1/4*Pi+1/2*f*x 
,2^(1/2))*sin(f*x+e)^(1/2)/f
 

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.67 \[ \int \sqrt {d \csc (e+f x)} \sin ^4(e+f x) \, dx=-\frac {\sqrt {d \csc (e+f x)} \left (40 \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),2\right ) \sqrt {\sin (e+f x)}+26 \sin (2 (e+f x))-3 \sin (4 (e+f x))\right )}{84 f} \] Input:

Integrate[Sqrt[d*Csc[e + f*x]]*Sin[e + f*x]^4,x]
 

Output:

-1/84*(Sqrt[d*Csc[e + f*x]]*(40*EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt[S 
in[e + f*x]] + 26*Sin[2*(e + f*x)] - 3*Sin[4*(e + f*x)]))/f
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.17, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 2030, 4256, 3042, 4256, 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.

\(\displaystyle \int \sin ^4(e+f x) \sqrt {d \csc (e+f x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt {d \csc (e+f x)}}{\csc (e+f x)^4}dx\)

\(\Big \downarrow \) 2030

\(\displaystyle d^4 \int \frac {1}{(d \csc (e+f x))^{7/2}}dx\)

\(\Big \downarrow \) 4256

\(\displaystyle d^4 \left (\frac {5 \int \frac {1}{(d \csc (e+f x))^{3/2}}dx}{7 d^2}-\frac {2 \cos (e+f x)}{7 d f (d \csc (e+f x))^{5/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle d^4 \left (\frac {5 \int \frac {1}{(d \csc (e+f x))^{3/2}}dx}{7 d^2}-\frac {2 \cos (e+f x)}{7 d f (d \csc (e+f x))^{5/2}}\right )\)

\(\Big \downarrow \) 4256

\(\displaystyle d^4 \left (\frac {5 \left (\frac {\int \sqrt {d \csc (e+f x)}dx}{3 d^2}-\frac {2 \cos (e+f x)}{3 d f \sqrt {d \csc (e+f x)}}\right )}{7 d^2}-\frac {2 \cos (e+f x)}{7 d f (d \csc (e+f x))^{5/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle d^4 \left (\frac {5 \left (\frac {\int \sqrt {d \csc (e+f x)}dx}{3 d^2}-\frac {2 \cos (e+f x)}{3 d f \sqrt {d \csc (e+f x)}}\right )}{7 d^2}-\frac {2 \cos (e+f x)}{7 d f (d \csc (e+f x))^{5/2}}\right )\)

\(\Big \downarrow \) 4258

\(\displaystyle d^4 \left (\frac {5 \left (\frac {\sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)} \int \frac {1}{\sqrt {\sin (e+f x)}}dx}{3 d^2}-\frac {2 \cos (e+f x)}{3 d f \sqrt {d \csc (e+f x)}}\right )}{7 d^2}-\frac {2 \cos (e+f x)}{7 d f (d \csc (e+f x))^{5/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle d^4 \left (\frac {5 \left (\frac {\sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)} \int \frac {1}{\sqrt {\sin (e+f x)}}dx}{3 d^2}-\frac {2 \cos (e+f x)}{3 d f \sqrt {d \csc (e+f x)}}\right )}{7 d^2}-\frac {2 \cos (e+f x)}{7 d f (d \csc (e+f x))^{5/2}}\right )\)

\(\Big \downarrow \) 3120

\(\displaystyle d^4 \left (\frac {5 \left (\frac {2 \sqrt {\sin (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),2\right ) \sqrt {d \csc (e+f x)}}{3 d^2 f}-\frac {2 \cos (e+f x)}{3 d f \sqrt {d \csc (e+f x)}}\right )}{7 d^2}-\frac {2 \cos (e+f x)}{7 d f (d \csc (e+f x))^{5/2}}\right )\)

Input:

Int[Sqrt[d*Csc[e + f*x]]*Sin[e + f*x]^4,x]
 

Output:

d^4*((-2*Cos[e + f*x])/(7*d*f*(d*Csc[e + f*x])^(5/2)) + (5*((-2*Cos[e + f* 
x])/(3*d*f*Sqrt[d*Csc[e + f*x]]) + (2*Sqrt[d*Csc[e + f*x]]*EllipticF[(e - 
Pi/2 + f*x)/2, 2]*Sqrt[Sin[e + f*x]])/(3*d^2*f)))/(7*d^2))
 

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 3120

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]
 

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 = 1.00 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.53

method	result	size
default	\(\frac {\sqrt {2}\, \left (\sin \left (f x +e \right ) \cos \left (f x +e \right ) \sqrt {2}\, \left (3 \cos \left (f x +e \right )^{2}-8\right )+i \left (5 \cos \left (f x +e \right )+5\right ) \sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}\, \sqrt {1-i \cot \left (f x +e \right )+i \csc \left (f x +e \right )}\, \sqrt {-i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )\right ) \sqrt {d \csc \left (f x +e \right )}}{21 f}\)	\(153\)

Input:

int((d*csc(f*x+e))^(1/2)*sin(f*x+e)^4,x,method=_RETURNVERBOSE)
 

Output:

1/21/f*2^(1/2)*(sin(f*x+e)*cos(f*x+e)*2^(1/2)*(3*cos(f*x+e)^2-8)+I*(5*cos( 
f*x+e)+5)*(1+I*cot(f*x+e)-I*csc(f*x+e))^(1/2)*(1-I*cot(f*x+e)+I*csc(f*x+e) 
)^(1/2)*(-I*(-csc(f*x+e)+cot(f*x+e)))^(1/2)*EllipticF((1+I*cot(f*x+e)-I*cs 
c(f*x+e))^(1/2),1/2*2^(1/2)))*(d*csc(f*x+e))^(1/2)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.95 \[ \int \sqrt {d \csc (e+f x)} \sin ^4(e+f x) \, dx=\frac {2 \, {\left (3 \, \cos \left (f x + e\right )^{3} - 8 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {d}{\sin \left (f x + e\right )}} \sin \left (f x + e\right ) - 5 i \, \sqrt {2 i \, d} {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + 5 i \, \sqrt {-2 i \, d} {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}{21 \, f} \] Input:

integrate((d*csc(f*x+e))^(1/2)*sin(f*x+e)^4,x, algorithm="fricas")
 

Output:

1/21*(2*(3*cos(f*x + e)^3 - 8*cos(f*x + e))*sqrt(d/sin(f*x + e))*sin(f*x + 
 e) - 5*I*sqrt(2*I*d)*weierstrassPInverse(4, 0, cos(f*x + e) + I*sin(f*x + 
 e)) + 5*I*sqrt(-2*I*d)*weierstrassPInverse(4, 0, cos(f*x + e) - I*sin(f*x 
 + e)))/f
 

Sympy [F]

\[ \int \sqrt {d \csc (e+f x)} \sin ^4(e+f x) \, dx=\int \sqrt {d \csc {\left (e + f x \right )}} \sin ^{4}{\left (e + f x \right )}\, dx \] Input:

integrate((d*csc(f*x+e))**(1/2)*sin(f*x+e)**4,x)
 

Output:

Integral(sqrt(d*csc(e + f*x))*sin(e + f*x)**4, x)
 

Maxima [F]

\[ \int \sqrt {d \csc (e+f x)} \sin ^4(e+f x) \, dx=\int { \sqrt {d \csc \left (f x + e\right )} \sin \left (f x + e\right )^{4} \,d x } \] Input:

integrate((d*csc(f*x+e))^(1/2)*sin(f*x+e)^4,x, algorithm="maxima")
 

Output:

integrate(sqrt(d*csc(f*x + e))*sin(f*x + e)^4, x)
 

Giac [F]

\[ \int \sqrt {d \csc (e+f x)} \sin ^4(e+f x) \, dx=\int { \sqrt {d \csc \left (f x + e\right )} \sin \left (f x + e\right )^{4} \,d x } \] Input:

integrate((d*csc(f*x+e))^(1/2)*sin(f*x+e)^4,x, algorithm="giac")
 

Output:

integrate(sqrt(d*csc(f*x + e))*sin(f*x + e)^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {d \csc (e+f x)} \sin ^4(e+f x) \, dx=\int {\sin \left (e+f\,x\right )}^4\,\sqrt {\frac {d}{\sin \left (e+f\,x\right )}} \,d x \] Input:

int(sin(e + f*x)^4*(d/sin(e + f*x))^(1/2),x)
 

Output:

int(sin(e + f*x)^4*(d/sin(e + f*x))^(1/2), x)
 

Reduce [F]

\[ \int \sqrt {d \csc (e+f x)} \sin ^4(e+f x) \, dx=\sqrt {d}\, \left (\int \sqrt {\csc \left (f x +e \right )}\, \sin \left (f x +e \right )^{4}d x \right ) \] Input:

int((d*csc(f*x+e))^(1/2)*sin(f*x+e)^4,x)
 

Output:

sqrt(d)*int(sqrt(csc(e + f*x))*sin(e + f*x)**4,x)