3.13 \(\int (a+a \csc (x))^{5/2} \, dx\)

Optimal. Leaf size=65 \[ -\frac{14 a^3 \cot (x)}{3 \sqrt{a \csc (x)+a}}-\frac{2}{3} a^2 \cot (x) \sqrt{a \csc (x)+a}-2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc (x)+a}}\right ) \]

[Out]

-2*a^(5/2)*ArcTan[(Sqrt[a]*Cot[x])/Sqrt[a + a*Csc[x]]] - (14*a^3*Cot[x])/(3*Sqrt[a + a*Csc[x]]) - (2*a^2*Cot[x
]*Sqrt[a + a*Csc[x]])/3

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Rubi [A]  time = 0.0913834, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3775, 3915, 3774, 203, 3792} \[ -\frac{14 a^3 \cot (x)}{3 \sqrt{a \csc (x)+a}}-\frac{2}{3} a^2 \cot (x) \sqrt{a \csc (x)+a}-2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc (x)+a}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Csc[x])^(5/2),x]

[Out]

-2*a^(5/2)*ArcTan[(Sqrt[a]*Cot[x])/Sqrt[a + a*Csc[x]]] - (14*a^3*Cot[x])/(3*Sqrt[a + a*Csc[x]]) - (2*a^2*Cot[x
]*Sqrt[a + a*Csc[x]])/3

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 3915

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Dist[c, In
t[Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[d, Int[Sqrt[a + b*Csc[e + f*x]]*Csc[e + f*x], x], x] /; FreeQ[{a, b,
c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]

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])

Rule 3792

Int[csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(-2*b*Cot[e + f*x])/
(f*Sqrt[a + b*Csc[e + f*x]]), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int (a+a \csc (x))^{5/2} \, dx &=-\frac{2}{3} a^2 \cot (x) \sqrt{a+a \csc (x)}+\frac{1}{3} (2 a) \int \sqrt{a+a \csc (x)} \left (\frac{3 a}{2}+\frac{7}{2} a \csc (x)\right ) \, dx\\ &=-\frac{2}{3} a^2 \cot (x) \sqrt{a+a \csc (x)}+a^2 \int \sqrt{a+a \csc (x)} \, dx+\frac{1}{3} \left (7 a^2\right ) \int \csc (x) \sqrt{a+a \csc (x)} \, dx\\ &=-\frac{14 a^3 \cot (x)}{3 \sqrt{a+a \csc (x)}}-\frac{2}{3} a^2 \cot (x) \sqrt{a+a \csc (x)}-\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{a+x^2} \, dx,x,\frac{a \cot (x)}{\sqrt{a+a \csc (x)}}\right )\\ &=-2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{a+a \csc (x)}}\right )-\frac{14 a^3 \cot (x)}{3 \sqrt{a+a \csc (x)}}-\frac{2}{3} a^2 \cot (x) \sqrt{a+a \csc (x)}\\ \end{align*}

Mathematica [A]  time = 1.62999, size = 80, normalized size = 1.23 \[ -\frac{2 a^2 \sqrt{a (\csc (x)+1)} \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right ) \left (\sqrt{\csc (x)-1} (\csc (x)+8)+3 \tan ^{-1}\left (\sqrt{\csc (x)-1}\right )\right )}{3 \sqrt{\csc (x)-1} \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Csc[x])^(5/2),x]

[Out]

(-2*a^2*Sqrt[a*(1 + Csc[x])]*(3*ArcTan[Sqrt[-1 + Csc[x]]] + Sqrt[-1 + Csc[x]]*(8 + Csc[x]))*(Cos[x/2] - Sin[x/
2]))/(3*Sqrt[-1 + Csc[x]]*(Cos[x/2] + Sin[x/2]))

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Maple [B]  time = 0.21, size = 535, normalized size = 8.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*csc(x))^(5/2),x)

[Out]

1/6*2^(1/2)*(-12*sin(x)*cos(x)*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)-1)*(-(-1+cos(x))/sin(x))^(3/2)-3*sin
(x)*cos(x)*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))*(-(-1+cos(x))/sin(x))^(3/2)-3*sin(x)*cos(x)*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))*(-(-1+cos(x))/sin(x))
^(3/2)-12*sin(x)*cos(x)*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)+1)*(-(-1+cos(x))/sin(x))^(3/2)-12*sin(x)*ar
ctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)-1)*(-(-1+cos(x))/sin(x))^(3/2)-3*sin(x)*ln(-(2^(1/2)*(-(-1+cos(x))/si
n(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))*(-(-1+cos(x)
)/sin(x))^(3/2)-3*sin(x)*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))*(-(-1+cos(x))/sin(x))^(3/2)-12*sin(x)*arctan(2^(1/2)*(-(-1+cos(x))/s
in(x))^(1/2)+1)*(-(-1+cos(x))/sin(x))^(3/2)+16*sin(x)*cos(x)*2^(1/2)-16*cos(x)^2*2^(1/2)-14*sin(x)*2^(1/2)+2*c
os(x)*2^(1/2)+14*2^(1/2))*sin(x)*(a*(sin(x)+1)/sin(x))^(5/2)/(cos(x)^2*sin(x)-cos(x)^3+2*cos(x)*sin(x)+3*cos(x
)^2-4*sin(x)+2*cos(x)-4)

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Maxima [B]  time = 1.58343, size = 563, normalized size = 8.66 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*csc(x))^(5/2),x, algorithm="maxima")

[Out]

1/22*sqrt(2)*a^(5/2)*(sin(x)/(cos(x) + 1))^(11/2) + 5/18*sqrt(2)*a^(5/2)*(sin(x)/(cos(x) + 1))^(9/2) + 9/14*sq
rt(2)*a^(5/2)*(sin(x)/(cos(x) + 1))^(7/2) + 1/2*sqrt(2)*a^(5/2)*(sin(x)/(cos(x) + 1))^(5/2) - 2/3*sqrt(2)*a^(5
/2)*(sin(x)/(cos(x) + 1))^(3/2) + 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)*(sqrt(2) - 2*sqrt(sin(x)/(cos(x) + 1)))))*a^(5/2) - 2*sqrt(2)*a^(5/2)*sqrt(sin(
x)/(cos(x) + 1)) - 1/1386*(693*sqrt(2)*a^(5/2)*sin(x)/(cos(x) + 1) + 1155*sqrt(2)*a^(5/2)*sin(x)^2/(cos(x) + 1
)^2 + 1386*sqrt(2)*a^(5/2)*sin(x)^3/(cos(x) + 1)^3 + 990*sqrt(2)*a^(5/2)*sin(x)^4/(cos(x) + 1)^4 + 385*sqrt(2)
*a^(5/2)*sin(x)^5/(cos(x) + 1)^5 + 63*sqrt(2)*a^(5/2)*sin(x)^6/(cos(x) + 1)^6)/sqrt(sin(x)/(cos(x) + 1)) - 1/4
2*(7*sqrt(2)*a^(5/2)*sin(x)/(cos(x) + 1) + 105*sqrt(2)*a^(5/2)*sin(x)^2/(cos(x) + 1)^2 - 210*sqrt(2)*a^(5/2)*s
in(x)^3/(cos(x) + 1)^3 - 70*sqrt(2)*a^(5/2)*sin(x)^4/(cos(x) + 1)^4 - 21*sqrt(2)*a^(5/2)*sin(x)^5/(cos(x) + 1)
^5 - 3*sqrt(2)*a^(5/2)*sin(x)^6/(cos(x) + 1)^6)/(sin(x)/(cos(x) + 1))^(5/2)

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Fricas [B]  time = 0.514679, size = 900, normalized size = 13.85 \begin{align*} \left [\frac{3 \,{\left (a^{2} \cos \left (x\right )^{2} - a^{2} -{\left (a^{2} \cos \left (x\right ) + a^{2}\right )} \sin \left (x\right )\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 (8 \, a^{2} \cos \left (x\right )^{2} + a^{2} \cos \left (x\right ) - 7 \, a^{2} +{\left (8 \, a^{2} \cos \left (x\right ) + 7 \, a^{2}\right )} \sin \left (x\right )\right )} \sqrt{\frac{a \sin \left (x\right ) + a}{\sin \left (x\right )}}}{3 \,{\left (\cos \left (x\right )^{2} -{\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 1\right )}}, \frac{2 \,{\left (3 \,{\left (a^{2} \cos \left (x\right )^{2} - a^{2} -{\left (a^{2} \cos \left (x\right ) + a^{2}\right )} \sin \left (x\right )\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 (8 \, a^{2} \cos \left (x\right )^{2} + a^{2} \cos \left (x\right ) - 7 \, a^{2} +{\left (8 \, a^{2} \cos \left (x\right ) + 7 \, a^{2}\right )} \sin \left (x\right )\right )} \sqrt{\frac{a \sin \left (x\right ) + a}{\sin \left (x\right )}}\right )}}{3 \,{\left (\cos \left (x\right )^{2} -{\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 1\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*csc(x))^(5/2),x, algorithm="fricas")

[Out]

[1/3*(3*(a^2*cos(x)^2 - a^2 - (a^2*cos(x) + a^2)*sin(x))*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*(8*a^2*cos(x)^2 + a^2*cos(x) - 7*a^2 + (8*a^2*cos(x) + 7*a^2)*sin(x))*sqrt((a*sin(x) + a)/sin(x)))/
(cos(x)^2 - (cos(x) + 1)*sin(x) - 1), 2/3*(3*(a^2*cos(x)^2 - a^2 - (a^2*cos(x) + a^2)*sin(x))*sqrt(a)*arctan(-
sqrt(a)*sqrt((a*sin(x) + a)/sin(x))*(cos(x) - sin(x) + 1)/(a*cos(x) + a*sin(x) + a)) + (8*a^2*cos(x)^2 + a^2*c
os(x) - 7*a^2 + (8*a^2*cos(x) + 7*a^2)*sin(x))*sqrt((a*sin(x) + a)/sin(x)))/(cos(x)^2 - (cos(x) + 1)*sin(x) -
1)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*csc(x))**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 2.24225, size = 339, normalized size = 5.22 \begin{align*} \frac{1}{6} \, \sqrt{2} \sqrt{a \tan \left (\frac{1}{2} \, x\right )} a^{2} \tan \left (\frac{1}{2} \, x\right ) + \frac{5}{2} \, \sqrt{2} \sqrt{a \tan \left (\frac{1}{2} \, x\right )} a^{2} +{\left (a^{2} \sqrt{{\left | a \right |}} + a{\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^{2} \sqrt{{\left | a \right |}} + a{\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^{2} \sqrt{{\left | a \right |}} - a{\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^{2} \sqrt{{\left | a \right |}} - a{\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{\sqrt{2}{\left (15 \, a^{4} \tan \left (\frac{1}{2} \, x\right ) + a^{4}\right )}}{6 \, \sqrt{a \tan \left (\frac{1}{2} \, x\right )} a \tan \left (\frac{1}{2} \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*csc(x))^(5/2),x, algorithm="giac")

[Out]

1/6*sqrt(2)*sqrt(a*tan(1/2*x))*a^2*tan(1/2*x) + 5/2*sqrt(2)*sqrt(a*tan(1/2*x))*a^2 + (a^2*sqrt(abs(a)) + 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))) + (a^2*sqrt(abs(a))
+ 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^2*s
qrt(abs(a)) - 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^2*
sqrt(abs(a)) - a*abs(a)^(3/2))*log(a*tan(1/2*x) - sqrt(2)*sqrt(a*tan(1/2*x))*sqrt(abs(a)) + abs(a)) - 1/6*sqrt
(2)*(15*a^4*tan(1/2*x) + a^4)/(sqrt(a*tan(1/2*x))*a*tan(1/2*x))