3.702 \(\int (-1+x)^{5/2} \csc ^{-1}(x) \, dx\)

Optimal. Leaf size=82 \[ \frac{4 x \sqrt{x^2-1} \left (3 x^2-19 x+83\right )}{105 \sqrt{x^2} \sqrt{x-1}}+\frac{4 x \tanh ^{-1}\left (\frac{\sqrt{x^2-1}}{\sqrt{x-1}}\right )}{7 \sqrt{x^2}}+\frac{2}{7} (x-1)^{7/2} \csc ^{-1}(x) \]

[Out]

(4*x*Sqrt[-1 + x^2]*(83 - 19*x + 3*x^2))/(105*Sqrt[-1 + x]*Sqrt[x^2]) + (2*(-1 +
x)^(7/2)*ArcCsc[x])/7 + (4*x*ArcTanh[Sqrt[-1 + x^2]/Sqrt[-1 + x]])/(7*Sqrt[x^2]
)

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Rubi [A]  time = 0.21222, antiderivative size = 140, normalized size of antiderivative = 1.71, number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6 \[ \frac{4 (x+1)^3 \sqrt{x-1}}{35 \sqrt{1-\frac{1}{x^2}} x}-\frac{20 (x+1)^2 \sqrt{x-1}}{21 \sqrt{1-\frac{1}{x^2}} x}+\frac{4 (x+1) \sqrt{x-1}}{\sqrt{1-\frac{1}{x^2}} x}+\frac{4 \sqrt{x+1} \sqrt{x-1} \tanh ^{-1}\left (\sqrt{x+1}\right )}{7 \sqrt{1-\frac{1}{x^2}} x}+\frac{2}{7} (x-1)^{7/2} \csc ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(-1 + x)^(5/2)*ArcCsc[x],x]

[Out]

(4*Sqrt[-1 + x]*(1 + x))/(Sqrt[1 - x^(-2)]*x) - (20*Sqrt[-1 + x]*(1 + x)^2)/(21*
Sqrt[1 - x^(-2)]*x) + (4*Sqrt[-1 + x]*(1 + x)^3)/(35*Sqrt[1 - x^(-2)]*x) + (2*(-
1 + x)^(7/2)*ArcCsc[x])/7 + (4*Sqrt[-1 + x]*Sqrt[1 + x]*ArcTanh[Sqrt[1 + x]])/(7
*Sqrt[1 - x^(-2)]*x)

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Rubi in Sympy [A]  time = 5.36338, size = 143, normalized size = 1.74 \[ \frac{4 x \left (x + 1\right )^{2} \sqrt{x^{2} - 1}}{35 \sqrt{x - 1} \sqrt{x^{2}}} - \frac{20 x \left (x + 1\right ) \sqrt{x^{2} - 1}}{21 \sqrt{x - 1} \sqrt{x^{2}}} + \frac{4 x \sqrt{x^{2} - 1}}{\sqrt{x - 1} \sqrt{x^{2}}} + \frac{4 x \sqrt{x^{2} - 1} \operatorname{atanh}{\left (\sqrt{x + 1} \right )}}{7 \sqrt{x - 1} \sqrt{x + 1} \sqrt{x^{2}}} + \frac{2 \left (x - 1\right )^{\frac{7}{2}} \operatorname{acsc}{\left (x \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-1+x)**(5/2)*acsc(x),x)

[Out]

4*x*(x + 1)**2*sqrt(x**2 - 1)/(35*sqrt(x - 1)*sqrt(x**2)) - 20*x*(x + 1)*sqrt(x*
*2 - 1)/(21*sqrt(x - 1)*sqrt(x**2)) + 4*x*sqrt(x**2 - 1)/(sqrt(x - 1)*sqrt(x**2)
) + 4*x*sqrt(x**2 - 1)*atanh(sqrt(x + 1))/(7*sqrt(x - 1)*sqrt(x + 1)*sqrt(x**2))
+ 2*(x - 1)**(7/2)*acsc(x)/7

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Mathematica [A]  time = 0.105234, size = 72, normalized size = 0.88 \[ \frac{4 \sqrt{1-\frac{1}{x^2}} x \left (3 x^2-19 x+83\right )}{105 \sqrt{x-1}}+\frac{4}{7} \tanh ^{-1}\left (\frac{\sqrt{1-\frac{1}{x^2}} x}{\sqrt{x-1}}\right )+\frac{2}{7} (x-1)^{7/2} \csc ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Integrate[(-1 + x)^(5/2)*ArcCsc[x],x]

[Out]

(4*Sqrt[1 - x^(-2)]*x*(83 - 19*x + 3*x^2))/(105*Sqrt[-1 + x]) + (2*(-1 + x)^(7/2
)*ArcCsc[x])/7 + (4*ArcTanh[(Sqrt[1 - x^(-2)]*x)/Sqrt[-1 + x]])/7

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Maple [A]  time = 0.024, size = 76, normalized size = 0.9 \[{\frac{2\,{\rm arccsc} \left (x\right )}{7} \left ( -1+x \right ) ^{{\frac{7}{2}}}}+{\frac{4}{105\,x}\sqrt{-1+x}\sqrt{1+x} \left ( 3\, \left ( -1+x \right ) ^{2}\sqrt{1+x}-13\, \left ( -1+x \right ) \sqrt{1+x}+15\,{\it Artanh} \left ( \sqrt{1+x} \right ) +67\,\sqrt{1+x} \right ){\frac{1}{\sqrt{{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) }{{x}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-1+x)^(5/2)*arccsc(x),x)

[Out]

2/7*(-1+x)^(7/2)*arccsc(x)+4/105*(-1+x)^(1/2)*(1+x)^(1/2)*(3*(-1+x)^2*(1+x)^(1/2
)-13*(-1+x)*(1+x)^(1/2)+15*arctanh((1+x)^(1/2))+67*(1+x)^(1/2))/((-1+x)*(1+x)/x^
2)^(1/2)/x

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Maxima [A]  time = 1.59334, size = 69, normalized size = 0.84 \[ \frac{2}{7} \,{\left (x - 1\right )}^{\frac{7}{2}} \operatorname{arccsc}\left (x\right ) + \frac{4}{35} \,{\left (x + 1\right )}^{\frac{5}{2}} - \frac{20}{21} \,{\left (x + 1\right )}^{\frac{3}{2}} + 4 \, \sqrt{x + 1} + \frac{2}{7} \, \log \left (\sqrt{x + 1} + 1\right ) - \frac{2}{7} \, \log \left (\sqrt{x + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x - 1)^(5/2)*arccsc(x),x, algorithm="maxima")

[Out]

2/7*(x - 1)^(7/2)*arccsc(x) + 4/35*(x + 1)^(5/2) - 20/21*(x + 1)^(3/2) + 4*sqrt(
x + 1) + 2/7*log(sqrt(x + 1) + 1) - 2/7*log(sqrt(x + 1) - 1)

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Fricas [A]  time = 0.234169, size = 198, normalized size = 2.41 \[ \frac{2 \,{\left (6 \, x^{4} - 38 \, x^{3} + 15 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )} \sqrt{x^{2} - 1} \operatorname{arccsc}\left (x\right ) + 160 \, x^{2} + 15 \, \sqrt{x^{2} - 1} \sqrt{x - 1} \log \left (\frac{x^{2} + \sqrt{x^{2} - 1} \sqrt{x - 1} - 1}{x^{2} - 1}\right ) - 15 \, \sqrt{x^{2} - 1} \sqrt{x - 1} \log \left (-\frac{x^{2} - \sqrt{x^{2} - 1} \sqrt{x - 1} - 1}{x^{2} - 1}\right ) + 38 \, x - 166\right )}}{105 \, \sqrt{x^{2} - 1} \sqrt{x - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x - 1)^(5/2)*arccsc(x),x, algorithm="fricas")

[Out]

2/105*(6*x^4 - 38*x^3 + 15*(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)*sqrt(x^2 - 1)*arccsc(
x) + 160*x^2 + 15*sqrt(x^2 - 1)*sqrt(x - 1)*log((x^2 + sqrt(x^2 - 1)*sqrt(x - 1)
- 1)/(x^2 - 1)) - 15*sqrt(x^2 - 1)*sqrt(x - 1)*log(-(x^2 - sqrt(x^2 - 1)*sqrt(x
- 1) - 1)/(x^2 - 1)) + 38*x - 166)/(sqrt(x^2 - 1)*sqrt(x - 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-1+x)**(5/2)*acsc(x),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.308175, size = 266, normalized size = 3.24 \[ \frac{2}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} \arcsin \left (\frac{1}{x}\right ) + \frac{2}{105} \,{\left (15 \,{\left (x - 1\right )}^{\frac{7}{2}} + 42 \,{\left (x - 1\right )}^{\frac{5}{2}} + 35 \,{\left (x - 1\right )}^{\frac{3}{2}}\right )} \arcsin \left (\frac{1}{x}\right ) - \frac{4}{15} \,{\left (3 \,{\left (x - 1\right )}^{\frac{5}{2}} + 5 \,{\left (x - 1\right )}^{\frac{3}{2}}\right )} \arcsin \left (\frac{1}{x}\right ) + \frac{4 \,{\left (3 \,{\left (x + 1\right )}^{\frac{5}{2}} - 11 \,{\left (x + 1\right )}^{\frac{3}{2}} + 14 \, \sqrt{x + 1}\right )}}{105 \,{\rm sign}\left ({\left (x - 1\right )}^{\frac{3}{2}} + \sqrt{x - 1}\right )} - \frac{8 \,{\left ({\left (x + 1\right )}^{\frac{3}{2}} - 4 \, \sqrt{x + 1}\right )}}{15 \,{\rm sign}\left ({\left (x - 1\right )}^{\frac{3}{2}} + \sqrt{x - 1}\right )} + \frac{2 \,{\rm ln}\left (\sqrt{x + 1} + 1\right )}{7 \,{\rm sign}\left ({\left (x - 1\right )}^{\frac{3}{2}} + \sqrt{x - 1}\right )} - \frac{2 \,{\rm ln}\left (\sqrt{x + 1} - 1\right )}{7 \,{\rm sign}\left ({\left (x - 1\right )}^{\frac{3}{2}} + \sqrt{x - 1}\right )} + \frac{4 \, \sqrt{x + 1}}{3 \,{\rm sign}\left ({\left (x - 1\right )}^{\frac{3}{2}} + \sqrt{x - 1}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x - 1)^(5/2)*arccsc(x),x, algorithm="giac")

[Out]

2/3*(x - 1)^(3/2)*arcsin(1/x) + 2/105*(15*(x - 1)^(7/2) + 42*(x - 1)^(5/2) + 35*
(x - 1)^(3/2))*arcsin(1/x) - 4/15*(3*(x - 1)^(5/2) + 5*(x - 1)^(3/2))*arcsin(1/x
) + 4/105*(3*(x + 1)^(5/2) - 11*(x + 1)^(3/2) + 14*sqrt(x + 1))/sign((x - 1)^(3/
2) + sqrt(x - 1)) - 8/15*((x + 1)^(3/2) - 4*sqrt(x + 1))/sign((x - 1)^(3/2) + sq
rt(x - 1)) + 2/7*ln(sqrt(x + 1) + 1)/sign((x - 1)^(3/2) + sqrt(x - 1)) - 2/7*ln(
sqrt(x + 1) - 1)/sign((x - 1)^(3/2) + sqrt(x - 1)) + 4/3*sqrt(x + 1)/sign((x - 1
)^(3/2) + sqrt(x - 1))