\(\int (a+a \csc (x))^{3/2} \, dx\) [14]

Optimal result

Integrand size = 10, antiderivative size = 44 \[ \int (a+a \csc (x))^{3/2} \, dx=-2 a^{3/2} \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {a+a \csc (x)}}\right )-\frac {2 a^2 \cot (x)}{\sqrt {a+a \csc (x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 44, 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 = {3860, 21, 3859, 209} \[ \int (a+a \csc (x))^{3/2} \, dx=-2 a^{3/2} \arctan \left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc (x)+a}}\right )-\frac {2 a^2 \cot (x)}{\sqrt {a \csc (x)+a}} \]

[In]

Int[(a + a*Csc[x])^(3/2),x]

[Out]

-2*a^(3/2)*ArcTan[(Sqrt[a]*Cot[x])/Sqrt[a + a*Csc[x]]] - (2*a^2*Cot[x])/Sqrt[a + a*Csc[x]]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 209

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

Rule 3859

Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[-2*(b/d), Subst[Int[1/(a + x^2), x], x, b*(C
ot[c + d*x]/Sqrt[a + b*Csc[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 3860

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[(-b^2)*Cot[c + d*x]*((a + b*Csc[c + d*x])^(
n - 2)/(d*(n - 1))), x] + Dist[a/(n - 1), Int[(a + b*Csc[c + d*x])^(n - 2)*(a*(n - 1) + b*(3*n - 4)*Csc[c + d*
x]), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.57 \[ \int (a+a \csc (x))^{3/2} \, dx=-\frac {2 a \left (\arctan \left (\sqrt {-1+\csc (x)}\right )+\sqrt {-1+\csc (x)}\right ) \sqrt {a (1+\csc (x))} \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )}{\sqrt {-1+\csc (x)} \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )} \]

[In]

Integrate[(a + a*Csc[x])^(3/2),x]

[Out]

(-2*a*(ArcTan[Sqrt[-1 + Csc[x]]] + Sqrt[-1 + Csc[x]])*Sqrt[a*(1 + Csc[x])]*(Cos[x/2] - Sin[x/2]))/(Sqrt[-1 + C
sc[x]]*(Cos[x/2] + Sin[x/2]))

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(244\) vs. \(2(36)=72\).

Time = 0.55 (sec) , antiderivative size = 245, normalized size of antiderivative = 5.57

method	result	size
default	\(\frac {\csc \left (x \right ) {\left (\frac {a \left (\csc \left (x \right ) \left (1-\cos \left (x \right )\right )^{2}+2-2 \cos \left (x \right )+\sin \left (x \right )\right )}{1-\cos \left (x \right )}\right )}^{\frac {3}{2}} \left (1-\cos \left (x \right )\right ) \left (\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}\, \ln \left (-\frac {\csc \left (x \right )-\cot \left (x \right )+\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}+1}{\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}-\csc \left (x \right )+\cot \left (x \right )-1}\right )+4 \sqrt {2}\, \sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \arctan \left (\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}+1\right )+4 \sqrt {2}\, \sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \arctan \left (\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}-1\right )+\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}\, \ln \left (-\frac {\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}-\csc \left (x \right )+\cot \left (x \right )-1}{\csc \left (x \right )-\cot \left (x \right )+\sqrt {\csc \left (x \right )-\cot \left (x \right )}\, \sqrt {2}+1}\right )+4 \csc \left (x \right )-4 \cot \left (x \right )-4\right ) \sqrt {2}}{4 \left (\csc \left (x \right )-\cot \left (x \right )+1\right )^{3}}\)	\(245\)

[In]

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

[Out]

1/4*csc(x)*(a/(1-cos(x))*(csc(x)*(1-cos(x))^2+2-2*cos(x)+sin(x)))^(3/2)/(csc(x)-cot(x)+1)^3*(1-cos(x))*((csc(x
)-cot(x))^(1/2)*2^(1/2)*ln(-(csc(x)-cot(x)+(csc(x)-cot(x))^(1/2)*2^(1/2)+1)/((csc(x)-cot(x))^(1/2)*2^(1/2)-csc
(x)+cot(x)-1))+4*2^(1/2)*(csc(x)-cot(x))^(1/2)*arctan((csc(x)-cot(x))^(1/2)*2^(1/2)+1)+4*2^(1/2)*(csc(x)-cot(x
))^(1/2)*arctan((csc(x)-cot(x))^(1/2)*2^(1/2)-1)+(csc(x)-cot(x))^(1/2)*2^(1/2)*ln(-((csc(x)-cot(x))^(1/2)*2^(1
/2)-csc(x)+cot(x)-1)/(csc(x)-cot(x)+(csc(x)-cot(x))^(1/2)*2^(1/2)+1))+4*csc(x)-4*cot(x)-4)*2^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (36) = 72\).

Time = 0.26 (sec) , antiderivative size = 212, normalized size of antiderivative = 4.82 \[ \int (a+a \csc (x))^{3/2} \, dx=\left [\frac {{\left (a \cos \left (x\right ) + a \sin \left (x\right ) + a\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (x\right )^{2} - 2 \, {\left (\cos \left (x\right )^{2} + {\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 1\right )} \sqrt {-a} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}} + a \cos \left (x\right ) - {\left (2 \, a \cos \left (x\right ) + a\right )} \sin \left (x\right ) - a}{\cos \left (x\right ) + \sin \left (x\right ) + 1}\right ) - 2 \, {\left (a \cos \left (x\right ) - a \sin \left (x\right ) + a\right )} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}}}{\cos \left (x\right ) + \sin \left (x\right ) + 1}, \frac {2 \, {\left ({\left (a \cos \left (x\right ) + a \sin \left (x\right ) + a\right )} \sqrt {a} \arctan \left (-\frac {\sqrt {a} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}} {\left (\cos \left (x\right ) - \sin \left (x\right ) + 1\right )}}{a \cos \left (x\right ) + a \sin \left (x\right ) + a}\right ) - {\left (a \cos \left (x\right ) - a \sin \left (x\right ) + a\right )} \sqrt {\frac {a \sin \left (x\right ) + a}{\sin \left (x\right )}}\right )}}{\cos \left (x\right ) + \sin \left (x\right ) + 1}\right ] \]

[In]

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

[Out]

[((a*cos(x) + a*sin(x) + a)*sqrt(-a)*log((2*a*cos(x)^2 - 2*(cos(x)^2 + (cos(x) + 1)*sin(x) - 1)*sqrt(-a)*sqrt(
(a*sin(x) + a)/sin(x)) + a*cos(x) - (2*a*cos(x) + a)*sin(x) - a)/(cos(x) + sin(x) + 1)) - 2*(a*cos(x) - a*sin(
x) + a)*sqrt((a*sin(x) + a)/sin(x)))/(cos(x) + sin(x) + 1), 2*((a*cos(x) + a*sin(x) + a)*sqrt(a)*arctan(-sqrt(
a)*sqrt((a*sin(x) + a)/sin(x))*(cos(x) - sin(x) + 1)/(a*cos(x) + a*sin(x) + a)) - (a*cos(x) - a*sin(x) + a)*sq
rt((a*sin(x) + a)/sin(x)))/(cos(x) + sin(x) + 1)]

Sympy [F]

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

[In]

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

[Out]

Integral((a*csc(x) + a)**(3/2), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (36) = 72\).

Time = 0.33 (sec) , antiderivative size = 200, normalized size of antiderivative = 4.55 \[ \int (a+a \csc (x))^{3/2} \, dx=\sqrt {2} {\left (\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}}\right )}\right ) + \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}}\right )}\right )\right )} a^{\frac {3}{2}} - \frac {1}{5} \, \sqrt {2} {\left (a^{\frac {3}{2}} \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )^{\frac {5}{2}} + 5 \, a^{\frac {3}{2}} \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )^{\frac {3}{2}} + 10 \, a^{\frac {3}{2}} \sqrt {\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}}\right )} - \frac {\frac {5 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {15 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {5 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {\sqrt {2} a^{\frac {3}{2}} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}}}{5 \, \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )^{\frac {3}{2}}} \]

[In]

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

[Out]

sqrt(2)*(sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(sin(x)/(cos(x) + 1)))) + sqrt(2)*arctan(-1/2*sqrt(2)*(sq
rt(2) - 2*sqrt(sin(x)/(cos(x) + 1)))))*a^(3/2) - 1/5*sqrt(2)*(a^(3/2)*(sin(x)/(cos(x) + 1))^(5/2) + 5*a^(3/2)*
(sin(x)/(cos(x) + 1))^(3/2) + 10*a^(3/2)*sqrt(sin(x)/(cos(x) + 1))) - 1/5*(5*sqrt(2)*a^(3/2)*sin(x)/(cos(x) +
1) - 15*sqrt(2)*a^(3/2)*sin(x)^2/(cos(x) + 1)^2 - 5*sqrt(2)*a^(3/2)*sin(x)^3/(cos(x) + 1)^3 - sqrt(2)*a^(3/2)*
sin(x)^4/(cos(x) + 1)^4)/(sin(x)/(cos(x) + 1))^(3/2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (36) = 72\).

Time = 0.42 (sec) , antiderivative size = 195, normalized size of antiderivative = 4.43 \[ \int (a+a \csc (x))^{3/2} \, dx=\sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, x\right )} a - \frac {\sqrt {2} a^{2}}{\sqrt {a \tan \left (\frac {1}{2} \, x\right )}} + {\left (a \sqrt {{\left | a \right |}} + {\left | a \right |}^{\frac {3}{2}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | a \right |}} + 2 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )}\right )}}{2 \, \sqrt {{\left | a \right |}}}\right ) + {\left (a \sqrt {{\left | a \right |}} + {\left | a \right |}^{\frac {3}{2}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | a \right |}} - 2 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )}\right )}}{2 \, \sqrt {{\left | a \right |}}}\right ) + \frac {1}{2} \, {\left (a \sqrt {{\left | a \right |}} - {\left | a \right |}^{\frac {3}{2}}\right )} \log \left (a \tan \left (\frac {1}{2} \, x\right ) + \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, x\right )} \sqrt {{\left | a \right |}} + {\left | a \right |}\right ) - \frac {1}{2} \, {\left (a \sqrt {{\left | a \right |}} - {\left | a \right |}^{\frac {3}{2}}\right )} \log \left (a \tan \left (\frac {1}{2} \, x\right ) - \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, x\right )} \sqrt {{\left | a \right |}} + {\left | a \right |}\right ) \]

[In]

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

[Out]

sqrt(2)*sqrt(a*tan(1/2*x))*a - sqrt(2)*a^2/sqrt(a*tan(1/2*x)) + (a*sqrt(abs(a)) + abs(a)^(3/2))*arctan(1/2*sqr
t(2)*(sqrt(2)*sqrt(abs(a)) + 2*sqrt(a*tan(1/2*x)))/sqrt(abs(a))) + (a*sqrt(abs(a)) + abs(a)^(3/2))*arctan(-1/2
*sqrt(2)*(sqrt(2)*sqrt(abs(a)) - 2*sqrt(a*tan(1/2*x)))/sqrt(abs(a))) + 1/2*(a*sqrt(abs(a)) - abs(a)^(3/2))*log
(a*tan(1/2*x) + sqrt(2)*sqrt(a*tan(1/2*x))*sqrt(abs(a)) + abs(a)) - 1/2*(a*sqrt(abs(a)) - abs(a)^(3/2))*log(a*
tan(1/2*x) - sqrt(2)*sqrt(a*tan(1/2*x))*sqrt(abs(a)) + abs(a))

Mupad [F(-1)]

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

[In]

int((a + a/sin(x))^(3/2),x)

[Out]

int((a + a/sin(x))^(3/2), x)