3.441 \(\int \frac{\tan (x)}{\left (1+5 \tan ^2(x)\right )^{5/2}} \, dx\)

Optimal. Leaf size=54 \[ \frac{1}{16 \sqrt{5 \tan ^2(x)+1}}-\frac{1}{12 \left (5 \tan ^2(x)+1\right )^{3/2}}+\frac{1}{32} \tan ^{-1}\left (\frac{1}{2} \sqrt{5 \tan ^2(x)+1}\right ) \]

[Out]

ArcTan[Sqrt[1 + 5*Tan[x]^2]/2]/32 - 1/(12*(1 + 5*Tan[x]^2)^(3/2)) + 1/(16*Sqrt[1
+ 5*Tan[x]^2])

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Rubi [A]  time = 0.119809, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{1}{16 \sqrt{5 \tan ^2(x)+1}}-\frac{1}{12 \left (5 \tan ^2(x)+1\right )^{3/2}}+\frac{1}{32} \tan ^{-1}\left (\frac{1}{2} \sqrt{5 \tan ^2(x)+1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Tan[x]/(1 + 5*Tan[x]^2)^(5/2),x]

[Out]

ArcTan[Sqrt[1 + 5*Tan[x]^2]/2]/32 - 1/(12*(1 + 5*Tan[x]^2)^(3/2)) + 1/(16*Sqrt[1
+ 5*Tan[x]^2])

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Rubi in Sympy [A]  time = 6.85549, size = 46, normalized size = 0.85 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{5 \tan ^{2}{\left (x \right )} + 1}}{2} \right )}}{32} + \frac{1}{16 \sqrt{5 \tan ^{2}{\left (x \right )} + 1}} - \frac{1}{12 \left (5 \tan ^{2}{\left (x \right )} + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(sqrt(5*tan(x)**2 + 1)/2)/32 + 1/(16*sqrt(5*tan(x)**2 + 1)) - 1/(12*(5*tan(x
)**2 + 1)**(3/2))

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Mathematica [A]  time = 0.536195, size = 71, normalized size = 1.31 \[ \frac{(2 \cos (2 x)-3) \sec ^5(x) \left (-6 \cos (x)+8 \cos (3 x)-3 (2 \cos (2 x)-3)^{3/2} \log \left (2 \cos (x)+\sqrt{2 \cos (2 x)-3}\right )\right )}{96 \left (5 \tan ^2(x)+1\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((-3 + 2*Cos[2*x])*(-6*Cos[x] + 8*Cos[3*x] - 3*(-3 + 2*Cos[2*x])^(3/2)*Log[2*Cos
[x] + Sqrt[-3 + 2*Cos[2*x]]])*Sec[x]^5)/(96*(1 + 5*Tan[x]^2)^(5/2))

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Maple [A]  time = 0.026, size = 41, normalized size = 0.8 \[{\frac{1}{32}\arctan \left ({\frac{1}{2}\sqrt{1+5\, \left ( \tan \left ( x \right ) \right ) ^{2}}} \right ) }+{\frac{1}{16}{\frac{1}{\sqrt{1+5\, \left ( \tan \left ( x \right ) \right ) ^{2}}}}}-{\frac{1}{12} \left ( 1+5\, \left ( \tan \left ( x \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(tan(x)/(1+5*tan(x)^2)^(5/2),x)

[Out]

1/32*arctan(1/2*(1+5*tan(x)^2)^(1/2))+1/16/(1+5*tan(x)^2)^(1/2)-1/12/(1+5*tan(x)
^2)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\tan \left (x\right )}{{\left (5 \, \tan \left (x\right )^{2} + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(tan(x)/(5*tan(x)^2 + 1)^(5/2), x)

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Fricas [A]  time = 0.250353, size = 103, normalized size = 1.91 \[ \frac{3 \,{\left (25 \, \tan \left (x\right )^{4} + 10 \, \tan \left (x\right )^{2} + 1\right )} \arctan \left (\frac{5 \, \tan \left (x\right )^{2} - 3}{4 \, \sqrt{5 \, \tan \left (x\right )^{2} + 1}}\right ) + 4 \,{\left (15 \, \tan \left (x\right )^{2} - 1\right )} \sqrt{5 \, \tan \left (x\right )^{2} + 1}}{192 \,{\left (25 \, \tan \left (x\right )^{4} + 10 \, \tan \left (x\right )^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*(3*(25*tan(x)^4 + 10*tan(x)^2 + 1)*arctan(1/4*(5*tan(x)^2 - 3)/sqrt(5*tan(
x)^2 + 1)) + 4*(15*tan(x)^2 - 1)*sqrt(5*tan(x)^2 + 1))/(25*tan(x)^4 + 10*tan(x)^
2 + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\tan{\left (x \right )}}{\left (5 \tan ^{2}{\left (x \right )} + 1\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(tan(x)/(5*tan(x)**2 + 1)**(5/2), x)

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GIAC/XCAS [A]  time = 0.203296, size = 49, normalized size = 0.91 \[ \frac{15 \, \tan \left (x\right )^{2} - 1}{48 \,{\left (5 \, \tan \left (x\right )^{2} + 1\right )}^{\frac{3}{2}}} + \frac{1}{32} \, \arctan \left (\frac{1}{2} \, \sqrt{5 \, \tan \left (x\right )^{2} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(15*tan(x)^2 - 1)/(5*tan(x)^2 + 1)^(3/2) + 1/32*arctan(1/2*sqrt(5*tan(x)^2
+ 1))