\(\int \csc ^4(e+f x) (b \sec (e+f x))^{3/2} \, dx\) [396]

Optimal result

Integrand size = 21, antiderivative size = 124 \[ \int \csc ^4(e+f x) (b \sec (e+f x))^{3/2} \, dx=-\frac {7 b^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{2 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {7 b \csc (e+f x) \sqrt {b \sec (e+f x)}}{6 f}-\frac {b \csc ^3(e+f x) \sqrt {b \sec (e+f x)}}{3 f}+\frac {7 b \sqrt {b \sec (e+f x)} \sin (e+f x)}{2 f} \]

[Out]

-7/2*b^2*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticE(sin(1/2*f*x+1/2*e),2^(1/2))/f/cos(f*x+e)^(1
/2)/(b*sec(f*x+e))^(1/2)-7/6*b*csc(f*x+e)*(b*sec(f*x+e))^(1/2)/f-1/3*b*csc(f*x+e)^3*(b*sec(f*x+e))^(1/2)/f+7/2
*b*sin(f*x+e)*(b*sec(f*x+e))^(1/2)/f

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2705, 3853, 3856, 2719} \[ \int \csc ^4(e+f x) (b \sec (e+f x))^{3/2} \, dx=-\frac {7 b^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{2 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {b \csc ^3(e+f x) \sqrt {b \sec (e+f x)}}{3 f}-\frac {7 b \csc (e+f x) \sqrt {b \sec (e+f x)}}{6 f}+\frac {7 b \sin (e+f x) \sqrt {b \sec (e+f x)}}{2 f} \]

[In]

Int[Csc[e + f*x]^4*(b*Sec[e + f*x])^(3/2),x]

[Out]

(-7*b^2*EllipticE[(e + f*x)/2, 2])/(2*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) - (7*b*Csc[e + f*x]*Sqrt[b*Se
c[e + f*x]])/(6*f) - (b*Csc[e + f*x]^3*Sqrt[b*Sec[e + f*x]])/(3*f) + (7*b*Sqrt[b*Sec[e + f*x]]*Sin[e + f*x])/(
2*f)

Rule 2705

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[(-a)*b*(a*Cs
c[e + f*x])^(m - 1)*((b*Sec[e + f*x])^(n - 1)/(f*(m - 1))), x] + Dist[a^2*((m + n - 2)/(m - 1)), Int[(a*Csc[e
+ f*x])^(m - 2)*(b*Sec[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && IntegersQ[2*m, 2*n] &&
!GtQ[n, m]

Rule 2719

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 3853

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] + Dist[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]

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(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]

Rubi steps \begin{align*} \text {integral}& = -\frac {b \csc ^3(e+f x) \sqrt {b \sec (e+f x)}}{3 f}+\frac {7}{6} \int \csc ^2(e+f x) (b \sec (e+f x))^{3/2} \, dx \\ & = -\frac {7 b \csc (e+f x) \sqrt {b \sec (e+f x)}}{6 f}-\frac {b \csc ^3(e+f x) \sqrt {b \sec (e+f x)}}{3 f}+\frac {7}{4} \int (b \sec (e+f x))^{3/2} \, dx \\ & = -\frac {7 b \csc (e+f x) \sqrt {b \sec (e+f x)}}{6 f}-\frac {b \csc ^3(e+f x) \sqrt {b \sec (e+f x)}}{3 f}+\frac {7 b \sqrt {b \sec (e+f x)} \sin (e+f x)}{2 f}-\frac {1}{4} \left (7 b^2\right ) \int \frac {1}{\sqrt {b \sec (e+f x)}} \, dx \\ & = -\frac {7 b \csc (e+f x) \sqrt {b \sec (e+f x)}}{6 f}-\frac {b \csc ^3(e+f x) \sqrt {b \sec (e+f x)}}{3 f}+\frac {7 b \sqrt {b \sec (e+f x)} \sin (e+f x)}{2 f}-\frac {\left (7 b^2\right ) \int \sqrt {\cos (e+f x)} \, dx}{4 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}} \\ & = -\frac {7 b^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{2 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {7 b \csc (e+f x) \sqrt {b \sec (e+f x)}}{6 f}-\frac {b \csc ^3(e+f x) \sqrt {b \sec (e+f x)}}{3 f}+\frac {7 b \sqrt {b \sec (e+f x)} \sin (e+f x)}{2 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.62 \[ \int \csc ^4(e+f x) (b \sec (e+f x))^{3/2} \, dx=-\frac {b \left (-21+7 \csc ^2(e+f x)+2 \csc ^4(e+f x)+21 \sqrt {\cos (e+f x)} \csc (e+f x) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )\right ) \sqrt {b \sec (e+f x)} \sin (e+f x)}{6 f} \]

[In]

Integrate[Csc[e + f*x]^4*(b*Sec[e + f*x])^(3/2),x]

[Out]

-1/6*(b*(-21 + 7*Csc[e + f*x]^2 + 2*Csc[e + f*x]^4 + 21*Sqrt[Cos[e + f*x]]*Csc[e + f*x]*EllipticE[(e + f*x)/2,
2])*Sqrt[b*Sec[e + f*x]]*Sin[e + f*x])/f

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.93 (sec) , antiderivative size = 276, normalized size of antiderivative = 2.23

method	result	size
default	\(-\frac {i b \sqrt {b \sec \left (f x +e \right )}\, \left (21 \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, E\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right ) \cos \left (f x +e \right )-21 \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right ) \cos \left (f x +e \right )+21 \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, E\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right )-21 \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right )-21 i \cot \left (f x +e \right )+14 i \csc \left (f x +e \right )-2 i \left (\csc ^{3}\left (f x +e \right )\right )\right )}{6 f}\)	\(276\)

[In]

int(csc(f*x+e)^4*(b*sec(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/6*I/f*b*(b*sec(f*x+e))^(1/2)*(21*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(-c
ot(f*x+e)+csc(f*x+e)),I)*cos(f*x+e)-21*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*
(-cot(f*x+e)+csc(f*x+e)),I)*cos(f*x+e)+21*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE
(I*(-cot(f*x+e)+csc(f*x+e)),I)-21*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(-cot
(f*x+e)+csc(f*x+e)),I)-21*I*cot(f*x+e)+14*I*csc(f*x+e)-2*I*csc(f*x+e)^3)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.13 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.35 \[ \int \csc ^4(e+f x) (b \sec (e+f x))^{3/2} \, dx=-\frac {21 \, \sqrt {2} {\left (i \, b \cos \left (f x + e\right )^{2} - i \, b\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 21 \, \sqrt {2} {\left (-i \, b \cos \left (f x + e\right )^{2} + i \, b\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (21 \, b \cos \left (f x + e\right )^{4} - 35 \, b \cos \left (f x + e\right )^{2} + 12 \, b\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{12 \, {\left (f \cos \left (f x + e\right )^{2} - f\right )} \sin \left (f x + e\right )} \]

[In]

integrate(csc(f*x+e)^4*(b*sec(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

-1/12*(21*sqrt(2)*(I*b*cos(f*x + e)^2 - I*b)*sqrt(b)*sin(f*x + e)*weierstrassZeta(-4, 0, weierstrassPInverse(-
4, 0, cos(f*x + e) + I*sin(f*x + e))) + 21*sqrt(2)*(-I*b*cos(f*x + e)^2 + I*b)*sqrt(b)*sin(f*x + e)*weierstras
sZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))) + 2*(21*b*cos(f*x + e)^4 - 35*b*cos(f*
x + e)^2 + 12*b)*sqrt(b/cos(f*x + e)))/((f*cos(f*x + e)^2 - f)*sin(f*x + e))

Sympy [F(-1)]

Timed out. \[ \int \csc ^4(e+f x) (b \sec (e+f x))^{3/2} \, dx=\text {Timed out} \]

[In]

integrate(csc(f*x+e)**4*(b*sec(f*x+e))**(3/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \csc ^4(e+f x) (b \sec (e+f x))^{3/2} \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}} \csc \left (f x + e\right )^{4} \,d x } \]

[In]

integrate(csc(f*x+e)^4*(b*sec(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*sec(f*x + e))^(3/2)*csc(f*x + e)^4, x)

Giac [F]

\[ \int \csc ^4(e+f x) (b \sec (e+f x))^{3/2} \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}} \csc \left (f x + e\right )^{4} \,d x } \]

[In]

integrate(csc(f*x+e)^4*(b*sec(f*x+e))^(3/2),x, algorithm="giac")

[Out]

integrate((b*sec(f*x + e))^(3/2)*csc(f*x + e)^4, x)

Mupad [F(-1)]

Timed out. \[ \int \csc ^4(e+f x) (b \sec (e+f x))^{3/2} \, dx=\int \frac {{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{{\sin \left (e+f\,x\right )}^4} \,d x \]

[In]

int((b/cos(e + f*x))^(3/2)/sin(e + f*x)^4,x)

[Out]

int((b/cos(e + f*x))^(3/2)/sin(e + f*x)^4, x)