Integrand size = 21, antiderivative size = 77 \[ \int (d \csc (e+f x))^{3/2} \sin ^4(e+f x) \, dx=-\frac {2 d^3 \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}}+\frac {6 d^2 E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right )}{5 f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \] Output:
-2/5*d^3*cos(f*x+e)/f/(d*csc(f*x+e))^(3/2)-6/5*d^2*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)
Time = 0.42 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.81 \[ \int (d \csc (e+f x))^{3/2} \sin ^4(e+f x) \, dx=-\frac {2 (d \csc (e+f x))^{3/2} \left (3 E\left (\left .\frac {1}{4} (-2 e+\pi -2 f x)\right |2\right ) \sin ^{\frac {3}{2}}(e+f x)+\cos (e+f x) \sin ^3(e+f x)\right )}{5 f} \] Input:
Integrate[(d*Csc[e + f*x])^(3/2)*Sin[e + f*x]^4,x]
Output:
(-2*(d*Csc[e + f*x])^(3/2)*(3*EllipticE[(-2*e + Pi - 2*f*x)/4, 2]*Sin[e + f*x]^(3/2) + Cos[e + f*x]*Sin[e + f*x]^3))/(5*f)
Time = 0.61 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.05, 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.
\(\displaystyle \int \sin ^4(e+f x) (d \csc (e+f x))^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(d \csc (e+f x))^{3/2}}{\csc (e+f x)^4}dx\) |
\(\Big \downarrow \) 2030 |
\(\displaystyle d^4 \int \frac {1}{(d \csc (e+f x))^{5/2}}dx\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle d^4 \left (\frac {3 \int \frac {1}{\sqrt {d \csc (e+f x)}}dx}{5 d^2}-\frac {2 \cos (e+f x)}{5 d f (d \csc (e+f x))^{3/2}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle d^4 \left (\frac {3 \int \frac {1}{\sqrt {d \csc (e+f x)}}dx}{5 d^2}-\frac {2 \cos (e+f x)}{5 d f (d \csc (e+f x))^{3/2}}\right )\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle d^4 \left (\frac {3 \int \sqrt {\sin (e+f x)}dx}{5 d^2 \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}-\frac {2 \cos (e+f x)}{5 d f (d \csc (e+f x))^{3/2}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle d^4 \left (\frac {3 \int \sqrt {\sin (e+f x)}dx}{5 d^2 \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}-\frac {2 \cos (e+f x)}{5 d f (d \csc (e+f x))^{3/2}}\right )\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle d^4 \left (\frac {6 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{5 d^2 f \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}-\frac {2 \cos (e+f x)}{5 d f (d \csc (e+f x))^{3/2}}\right )\) |
Input:
Int[(d*Csc[e + f*x])^(3/2)*Sin[e + f*x]^4,x]
Output:
d^4*((-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]]))
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]
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[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]
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 = 0.96 (sec) , antiderivative size = 249, normalized size of antiderivative = 3.23
method | result | size |
default | \(\frac {\sqrt {2}\, \left (\left (-6 \cos \left (f x +e \right )-6\right ) \sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}\, \sqrt {-i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {1-i \cot \left (f x +e \right )+i \csc \left (f x +e \right )}+\left (3 \cos \left (f x +e \right )+3\right ) \sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}\, \sqrt {-i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}\, \sqrt {1-i \cot \left (f x +e \right )+i \csc \left (f x +e \right )}\, \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (f x +e \right )-i \csc \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )+\left (\cos \left (f x +e \right )^{3}-4 \cos \left (f x +e \right )+3\right ) \sqrt {2}\right ) d \sqrt {d \csc \left (f x +e \right )}}{5 f}\) | \(249\) |
Input:
int((d*csc(f*x+e))^(3/2)*sin(f*x+e)^4,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*( -csc(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* cot(f*x+e)-I*csc(f*x+e))^(1/2)*(-I*(-csc(f*x+e)+cot(f*x+e)))^(1/2)*(1-I*co t(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*(d*csc(f*x+e))^(1/2 )
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.27 \[ \int (d \csc (e+f x))^{3/2} \sin ^4(e+f x) \, dx=\frac {3 \, \sqrt {2 i \, d} d {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 3 \, \sqrt {-2 i \, d} d {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (d \cos \left (f x + e\right )^{3} - d \cos \left (f x + e\right )\right )} \sqrt {\frac {d}{\sin \left (f x + e\right )}}}{5 \, f} \] Input:
integrate((d*csc(f*x+e))^(3/2)*sin(f*x+e)^4,x, algorithm="fricas")
Output:
1/5*(3*sqrt(2*I*d)*d*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(f *x + e) + I*sin(f*x + e))) + 3*sqrt(-2*I*d)*d*weierstrassZeta(4, 0, weiers trassPInverse(4, 0, cos(f*x + e) - I*sin(f*x + e))) + 2*(d*cos(f*x + e)^3 - d*cos(f*x + e))*sqrt(d/sin(f*x + e)))/f
Timed out. \[ \int (d \csc (e+f x))^{3/2} \sin ^4(e+f x) \, dx=\text {Timed out} \] Input:
integrate((d*csc(f*x+e))**(3/2)*sin(f*x+e)**4,x)
Output:
Timed out
\[ \int (d \csc (e+f x))^{3/2} \sin ^4(e+f x) \, dx=\int { \left (d \csc \left (f x + e\right )\right )^{\frac {3}{2}} \sin \left (f x + e\right )^{4} \,d x } \] Input:
integrate((d*csc(f*x+e))^(3/2)*sin(f*x+e)^4,x, algorithm="maxima")
Output:
integrate((d*csc(f*x + e))^(3/2)*sin(f*x + e)^4, x)
\[ \int (d \csc (e+f x))^{3/2} \sin ^4(e+f x) \, dx=\int { \left (d \csc \left (f x + e\right )\right )^{\frac {3}{2}} \sin \left (f x + e\right )^{4} \,d x } \] Input:
integrate((d*csc(f*x+e))^(3/2)*sin(f*x+e)^4,x, algorithm="giac")
Output:
integrate((d*csc(f*x + e))^(3/2)*sin(f*x + e)^4, x)
Timed out. \[ \int (d \csc (e+f x))^{3/2} \sin ^4(e+f x) \, dx=\int {\sin \left (e+f\,x\right )}^4\,{\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^{3/2} \,d x \] Input:
int(sin(e + f*x)^4*(d/sin(e + f*x))^(3/2),x)
Output:
int(sin(e + f*x)^4*(d/sin(e + f*x))^(3/2), x)
\[ \int (d \csc (e+f x))^{3/2} \sin ^4(e+f x) \, dx=\sqrt {d}\, \left (\int \sqrt {\csc \left (f x +e \right )}\, \csc \left (f x +e \right ) \sin \left (f x +e \right )^{4}d x \right ) d \] Input:
int((d*csc(f*x+e))^(3/2)*sin(f*x+e)^4,x)
Output:
sqrt(d)*int(sqrt(csc(e + f*x))*csc(e + f*x)*sin(e + f*x)**4,x)*d