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

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

[Out]

-2*a^(5/2)*arctan(cot(x)*a^(1/2)/(a+a*csc(x))^(1/2))-14/3*a^3*cot(x)/(a+a*csc(x))^(1/2)-2/3*a^2*cot(x)*(a+a*cs
c(x))^(1/2)

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Rubi [A]  time = 0.09, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 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 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 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 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]

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]

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*}

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Mathematica [A]  time = 1.90, 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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fricas [B]  time = 0.56, size = 318, normalized size = 4.89 \[ \left [\frac {3 \, {\left (a^{2} \cos \relax (x)^{2} - a^{2} - {\left (a^{2} \cos \relax (x) + a^{2}\right )} \sin \relax (x)\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \relax (x)^{2} - 2 \, {\left (\cos \relax (x)^{2} + {\left (\cos \relax (x) + 1\right )} \sin \relax (x) - 1\right )} \sqrt {-a} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}} + a \cos \relax (x) - {\left (2 \, a \cos \relax (x) + a\right )} \sin \relax (x) - a}{\cos \relax (x) + \sin \relax (x) + 1}\right ) + 2 \, {\left (8 \, a^{2} \cos \relax (x)^{2} + a^{2} \cos \relax (x) - 7 \, a^{2} + {\left (8 \, a^{2} \cos \relax (x) + 7 \, a^{2}\right )} \sin \relax (x)\right )} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}}}{3 \, {\left (\cos \relax (x)^{2} - {\left (\cos \relax (x) + 1\right )} \sin \relax (x) - 1\right )}}, \frac {2 \, {\left (3 \, {\left (a^{2} \cos \relax (x)^{2} - a^{2} - {\left (a^{2} \cos \relax (x) + a^{2}\right )} \sin \relax (x)\right )} \sqrt {a} \arctan \left (-\frac {\sqrt {a} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}} {\left (\cos \relax (x) - \sin \relax (x) + 1\right )}}{a \cos \relax (x) + a \sin \relax (x) + a}\right ) + {\left (8 \, a^{2} \cos \relax (x)^{2} + a^{2} \cos \relax (x) - 7 \, a^{2} + {\left (8 \, a^{2} \cos \relax (x) + 7 \, a^{2}\right )} \sin \relax (x)\right )} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}}\right )}}{3 \, {\left (\cos \relax (x)^{2} - {\left (\cos \relax (x) + 1\right )} \sin \relax (x) - 1\right )}}\right ] \]

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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giac [B]  time = 1.01, size = 250, normalized size = 3.85 \[ {\left (a^{2} \sqrt {{\left | a \right |}} + a {\left | a \right |}^{\frac {3}{2}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | a \right |}} + 2 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )}\right )}}{2 \, \sqrt {{\left | a \right |}}}\right ) + {\left (a^{2} \sqrt {{\left | a \right |}} + a {\left | a \right |}^{\frac {3}{2}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | a \right |}} - 2 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )}\right )}}{2 \, \sqrt {{\left | a \right |}}}\right ) + \frac {1}{2} \, {\left (a^{2} \sqrt {{\left | a \right |}} - a {\left | a \right |}^{\frac {3}{2}}\right )} \log \left (a \tan \left (\frac {1}{2} \, x\right ) + \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, x\right )} \sqrt {{\left | a \right |}} + {\left | a \right |}\right ) - \frac {1}{2} \, {\left (a^{2} \sqrt {{\left | a \right |}} - a {\left | a \right |}^{\frac {3}{2}}\right )} \log \left (a \tan \left (\frac {1}{2} \, x\right ) - \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, x\right )} \sqrt {{\left | a \right |}} + {\left | a \right |}\right ) + \frac {1}{6} \, \sqrt {2} {\left (\sqrt {a \tan \left (\frac {1}{2} \, x\right )} a^{2} \tan \left (\frac {1}{2} \, x\right ) + 15 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )} a^{2}\right )} - \frac {\sqrt {2} {\left (15 \, a^{4} \tan \left (\frac {1}{2} \, x\right ) + a^{4}\right )}}{6 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )} a \tan \left (\frac {1}{2} \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.92, size = 535, normalized size = 8.23 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(-3*sin(x)*cos(x)*(-(-1+cos(x))/sin(x))^(3/2)*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))-12*sin(x)*cos(x)*(-(-1+cos(x))/sin(x))^(3/2
)*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)+1)-12*sin(x)*cos(x)*(-(-1+cos(x))/sin(x))^(3/2)*arctan(2^(1/2)*(-
(-1+cos(x))/sin(x))^(1/2)-1)-3*sin(x)*cos(x)*(-(-1+cos(x))/sin(x))^(3/2)*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))-3*sin(x)*(-(-1+cos(x
))/sin(x))^(3/2)*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))-12*sin(x)*(-(-1+cos(x))/sin(x))^(3/2)*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(
1/2)+1)-12*sin(x)*(-(-1+cos(x))/sin(x))^(3/2)*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)-1)-3*sin(x)*(-(-1+cos
(x))/sin(x))^(3/2)*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))+16*sin(x)*cos(x)*2^(1/2)-16*cos(x)^2*2^(1/2)-14*sin(x)*2^(1/2)+2*cos(x)*2^
(1/2)+14*2^(1/2))*sin(x)*(a*(1+sin(x))/sin(x))^(5/2)/(cos(x)^2*sin(x)-cos(x)^3+2*cos(x)*sin(x)+3*cos(x)^2-4*si
n(x)+2*cos(x)-4)*2^(1/2)

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maxima [B]  time = 0.47, size = 417, normalized size = 6.42 \[ \frac {1}{22} \, \sqrt {2} a^{\frac {5}{2}} \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )^{\frac {11}{2}} + \frac {5}{18} \, \sqrt {2} a^{\frac {5}{2}} \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )^{\frac {9}{2}} + \frac {9}{14} \, \sqrt {2} a^{\frac {5}{2}} \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )^{\frac {7}{2}} + \frac {1}{2} \, \sqrt {2} a^{\frac {5}{2}} \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )^{\frac {5}{2}} - \frac {2}{3} \, \sqrt {2} a^{\frac {5}{2}} \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )^{\frac {3}{2}} + \sqrt {2} {\left (\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\frac {\sin \relax (x)}{\cos \relax (x) + 1}}\right )}\right ) + \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\frac {\sin \relax (x)}{\cos \relax (x) + 1}}\right )}\right )\right )} a^{\frac {5}{2}} - 2 \, \sqrt {2} a^{\frac {5}{2}} \sqrt {\frac {\sin \relax (x)}{\cos \relax (x) + 1}} - \frac {\frac {693 \, \sqrt {2} a^{\frac {5}{2}} \sin \relax (x)}{\cos \relax (x) + 1} + \frac {1155 \, \sqrt {2} a^{\frac {5}{2}} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {1386 \, \sqrt {2} a^{\frac {5}{2}} \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {990 \, \sqrt {2} a^{\frac {5}{2}} \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {385 \, \sqrt {2} a^{\frac {5}{2}} \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}} + \frac {63 \, \sqrt {2} a^{\frac {5}{2}} \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}}}{1386 \, \sqrt {\frac {\sin \relax (x)}{\cos \relax (x) + 1}}} - \frac {\frac {7 \, \sqrt {2} a^{\frac {5}{2}} \sin \relax (x)}{\cos \relax (x) + 1} + \frac {105 \, \sqrt {2} a^{\frac {5}{2}} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} - \frac {210 \, \sqrt {2} a^{\frac {5}{2}} \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} - \frac {70 \, \sqrt {2} a^{\frac {5}{2}} \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} - \frac {21 \, \sqrt {2} a^{\frac {5}{2}} \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}} - \frac {3 \, \sqrt {2} a^{\frac {5}{2}} \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}}}{42 \, \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )^{\frac {5}{2}}} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (a+\frac {a}{\sin \relax (x)}\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a/sin(x))^(5/2),x)

[Out]

int((a + a/sin(x))^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \csc {\relax (x )} + a\right )^{\frac {5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*csc(x) + a)**(5/2), x)

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