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

Optimal. Leaf size=44 \[ -\frac{2 a^2 \cot (x)}{\sqrt{a \csc (x)+a}}-2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc (x)+a}}\right ) \]

[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]]

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Rubi [A]  time = 0.0304638, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3775, 21, 3774, 203} \[ -\frac{2 a^2 \cot (x)}{\sqrt{a \csc (x)+a}}-2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc (x)+a}}\right ) \]

Antiderivative was successfully verified.

[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 3775

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]

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 3774

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 203

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

Rubi steps

\begin{align*} \int (a+a \csc (x))^{3/2} \, dx &=-\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 ) \operatorname{Subst}\left (\int \frac{1}{a+x^2} \, dx,x,\frac{a \cot (x)}{\sqrt{a+a \csc (x)}}\right )\\ &=-2 a^{3/2} \tan ^{-1}\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]  time = 0.0853796, size = 69, normalized size = 1.57 \[ -\frac{2 a \sqrt{a (\csc (x)+1)} \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right ) \left (\sqrt{\csc (x)-1}+\tan ^{-1}\left (\sqrt{\csc (x)-1}\right )\right )}{\sqrt{\csc (x)-1} \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )} \]

Antiderivative was successfully verified.

[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]))

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Maple [B]  time = 0.16, size = 273, normalized size = 6.2 \begin{align*} -{\frac{\sin \left ( x \right ) \sqrt{2}}{2\,\cos \left ( x \right ) \sin \left ( x \right ) +2\, \left ( \cos \left ( x \right ) \right ) ^{2}-4\,\sin \left ( x \right ) +2\,\cos \left ( x \right ) -4} \left ( \sin \left ( x \right ) \sqrt{-{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}\ln \left ( -{ \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}\sin \left ( x \right ) +\sin \left ( x \right ) -\cos \left ( x \right ) +1 \right ) \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}\sin \left ( x \right ) -\sin \left ( x \right ) +\cos \left ( x \right ) -1 \right ) ^{-1}} \right ) +4\,\sin \left ( x \right ) \sqrt{-{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}\arctan \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}+1 \right ) +4\,\sin \left ( x \right ) \sqrt{-{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}\arctan \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}-1 \right ) +\sin \left ( x \right ) \sqrt{-{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}\ln \left ( -{ \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}\sin \left ( x \right ) -\sin \left ( x \right ) +\cos \left ( x \right ) -1 \right ) \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }}}\sin \left ( x \right ) +\sin \left ( x \right ) -\cos \left ( x \right ) +1 \right ) ^{-1}} \right ) -2\,\sin \left ( x \right ) \sqrt{2}-2\,\cos \left ( x \right ) \sqrt{2}+2\,\sqrt{2} \right ) \left ({\frac{a \left ( \sin \left ( x \right ) +1 \right ) }{\sin \left ( x \right ) }} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.54284, size = 270, normalized size = 6.14 \begin{align*} \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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 0.501578, size = 666, normalized size = 15.14 \begin{align*} \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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \csc{\left (x \right )} + a\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*csc(x) + a)**(3/2), x)

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Giac [B]  time = 2.07439, size = 263, normalized size = 5.98 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))