\(\int (a \csc ^2(x))^{3/2} \, dx\) [49]

Optimal result

Integrand size = 10, antiderivative size = 46 \[ \int \left (a \csc ^2(x)\right )^{3/2} \, dx=-\frac {1}{2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc ^2(x)}}\right )-\frac {1}{2} a \cot (x) \sqrt {a \csc ^2(x)} \]

[Out]

-1/2*a^(3/2)*arctanh(cot(x)*a^(1/2)/(a*csc(x)^2)^(1/2))-1/2*a*cot(x)*(a*csc(x)^2)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4207, 201, 223, 212} \[ \int \left (a \csc ^2(x)\right )^{3/2} \, dx=-\frac {1}{2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc ^2(x)}}\right )-\frac {1}{2} a \cot (x) \sqrt {a \csc ^2(x)} \]

[In]

Int[(a*Csc[x]^2)^(3/2),x]

[Out]

-1/2*(a^(3/2)*ArcTanh[(Sqrt[a]*Cot[x])/Sqrt[a*Csc[x]^2]]) - (a*Cot[x]*Sqrt[a*Csc[x]^2])/2

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 4207

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[b*(ff/
f), Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rubi steps \begin{align*} \text {integral}& = -\left (a \text {Subst}\left (\int \sqrt {a+a x^2} \, dx,x,\cot (x)\right )\right ) \\ & = -\frac {1}{2} a \cot (x) \sqrt {a \csc ^2(x)}-\frac {1}{2} a^2 \text {Subst}\left (\int \frac {1}{\sqrt {a+a x^2}} \, dx,x,\cot (x)\right ) \\ & = -\frac {1}{2} a \cot (x) \sqrt {a \csc ^2(x)}-\frac {1}{2} a^2 \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\cot (x)}{\sqrt {a \csc ^2(x)}}\right ) \\ & = -\frac {1}{2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc ^2(x)}}\right )-\frac {1}{2} a \cot (x) \sqrt {a \csc ^2(x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85 \[ \int \left (a \csc ^2(x)\right )^{3/2} \, dx=-\frac {1}{2} a \sqrt {a \csc ^2(x)} \left (\cot (x) \csc (x)+\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right ) \sin (x) \]

[In]

Integrate[(a*Csc[x]^2)^(3/2),x]

[Out]

-1/2*(a*Sqrt[a*Csc[x]^2]*(Cot[x]*Csc[x] + Log[Cos[x/2]] - Log[Sin[x/2]])*Sin[x])

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.65

method	result	size
default	\(-\frac {a \sqrt {a \csc \left (x \right )^{2}}\, \left (-\sin \left (x \right ) \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+\cot \left (x \right )\right ) \sqrt {4}}{4}\)	\(30\)
risch	\(-\frac {i a \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}+1\right )}{{\mathrm e}^{2 i x}-1}-a \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+1\right ) \sin \left (x \right )+a \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-1\right ) \sin \left (x \right )\)	\(104\)

[In]

int((a*csc(x)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/4*a*(a*csc(x)^2)^(1/2)*(-sin(x)*ln(csc(x)-cot(x))+cot(x))*4^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.07 \[ \int \left (a \csc ^2(x)\right )^{3/2} \, dx=-\frac {{\left (2 \, a \cos \left (x\right ) + {\left (a \cos \left (x\right )^{2} - a\right )} \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right )\right )} \sqrt {-\frac {a}{\cos \left (x\right )^{2} - 1}}}{4 \, \sin \left (x\right )} \]

[In]

integrate((a*csc(x)^2)^(3/2),x, algorithm="fricas")

[Out]

-1/4*(2*a*cos(x) + (a*cos(x)^2 - a)*log(-(cos(x) - 1)/(cos(x) + 1)))*sqrt(-a/(cos(x)^2 - 1))/sin(x)

Sympy [F]

\[ \int \left (a \csc ^2(x)\right )^{3/2} \, dx=\int \left (a \csc ^{2}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \]

[In]

integrate((a*csc(x)**2)**(3/2),x)

[Out]

Integral((a*csc(x)**2)**(3/2), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (34) = 68\).

Time = 0.32 (sec) , antiderivative size = 318, normalized size of antiderivative = 6.91 \[ \int \left (a \csc ^2(x)\right )^{3/2} \, dx=-\frac {{\left (4 \, a \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a \cos \left (x\right ) \sin \left (2 \, x\right ) - 4 \, a \cos \left (2 \, x\right ) \sin \left (x\right ) - {\left (a \cos \left (4 \, x\right )^{2} + 4 \, a \cos \left (2 \, x\right )^{2} + a \sin \left (4 \, x\right )^{2} - 4 \, a \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a \sin \left (2 \, x\right )^{2} - 2 \, {\left (2 \, a \cos \left (2 \, x\right ) - a\right )} \cos \left (4 \, x\right ) - 4 \, a \cos \left (2 \, x\right ) + a\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) + {\left (a \cos \left (4 \, x\right )^{2} + 4 \, a \cos \left (2 \, x\right )^{2} + a \sin \left (4 \, x\right )^{2} - 4 \, a \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a \sin \left (2 \, x\right )^{2} - 2 \, {\left (2 \, a \cos \left (2 \, x\right ) - a\right )} \cos \left (4 \, x\right ) - 4 \, a \cos \left (2 \, x\right ) + a\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right ) + 2 \, {\left (a \sin \left (3 \, x\right ) + a \sin \left (x\right )\right )} \cos \left (4 \, x\right ) - 2 \, {\left (a \cos \left (3 \, x\right ) + a \cos \left (x\right )\right )} \sin \left (4 \, x\right ) - 2 \, {\left (2 \, a \cos \left (2 \, x\right ) - a\right )} \sin \left (3 \, x\right ) + 2 \, a \sin \left (x\right )\right )} \sqrt {-a}}{2 \, {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )}} \]

[In]

integrate((a*csc(x)^2)^(3/2),x, algorithm="maxima")

[Out]

-1/2*(4*a*cos(3*x)*sin(2*x) + 4*a*cos(x)*sin(2*x) - 4*a*cos(2*x)*sin(x) - (a*cos(4*x)^2 + 4*a*cos(2*x)^2 + a*s
in(4*x)^2 - 4*a*sin(4*x)*sin(2*x) + 4*a*sin(2*x)^2 - 2*(2*a*cos(2*x) - a)*cos(4*x) - 4*a*cos(2*x) + a)*arctan2
(sin(x), cos(x) + 1) + (a*cos(4*x)^2 + 4*a*cos(2*x)^2 + a*sin(4*x)^2 - 4*a*sin(4*x)*sin(2*x) + 4*a*sin(2*x)^2
- 2*(2*a*cos(2*x) - a)*cos(4*x) - 4*a*cos(2*x) + a)*arctan2(sin(x), cos(x) - 1) + 2*(a*sin(3*x) + a*sin(x))*co
s(4*x) - 2*(a*cos(3*x) + a*cos(x))*sin(4*x) - 2*(2*a*cos(2*x) - a)*sin(3*x) + 2*a*sin(x))*sqrt(-a)/(2*(2*cos(2
*x) - 1)*cos(4*x) - cos(4*x)^2 - 4*cos(2*x)^2 - sin(4*x)^2 + 4*sin(4*x)*sin(2*x) - 4*sin(2*x)^2 + 4*cos(2*x) -
1)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (34) = 68\).

Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.57 \[ \int \left (a \csc ^2(x)\right )^{3/2} \, dx=\frac {1}{8} \, {\left (2 \, \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \mathrm {sgn}\left (\sin \left (x\right )\right ) - \frac {{\left (\frac {2 \, {\left (\cos \left (x\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{\cos \left (x\right ) + 1} - \mathrm {sgn}\left (\sin \left (x\right )\right )\right )} {\left (\cos \left (x\right ) + 1\right )}}{\cos \left (x\right ) - 1} - \frac {{\left (\cos \left (x\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{\cos \left (x\right ) + 1}\right )} a^{\frac {3}{2}} \]

[In]

integrate((a*csc(x)^2)^(3/2),x, algorithm="giac")

[Out]

1/8*(2*log(-(cos(x) - 1)/(cos(x) + 1))*sgn(sin(x)) - (2*(cos(x) - 1)*sgn(sin(x))/(cos(x) + 1) - sgn(sin(x)))*(
cos(x) + 1)/(cos(x) - 1) - (cos(x) - 1)*sgn(sin(x))/(cos(x) + 1))*a^(3/2)

Mupad [F(-1)]

Timed out. \[ \int \left (a \csc ^2(x)\right )^{3/2} \, dx=\int {\left (\frac {a}{{\sin \left (x\right )}^2}\right )}^{3/2} \,d x \]

[In]

int((a/sin(x)^2)^(3/2),x)

[Out]

int((a/sin(x)^2)^(3/2), x)