3.452 \(\int \frac{\csc (x) \sec (x) \left (1+\sqrt [3]{1-8 \tan ^2(x)}\right )}{\left (1-8 \tan ^2(x)\right )^{2/3}} \, dx\)

Optimal. Leaf size=27 \[ \frac{3}{2} \log \left (1-\sqrt [3]{1-8 \tan ^2(x)}\right )-\log (\tan (x)) \]

[Out]

-Log[Tan[x]] + (3*Log[1 - (1 - 8*Tan[x]^2)^(1/3)])/2

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Rubi [A]  time = 1.63165, antiderivative size = 35, normalized size of antiderivative = 1.3, number of steps used = 15, number of rules used = 9, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.29 \[ \frac{3}{2} \log \left (1-\sqrt [3]{9-8 \sec ^2(x)}\right )-\frac{1}{2} \log \left (1-\sec ^2(x)\right ) \]

Antiderivative was successfully verified.

[In]  Int[(Csc[x]*Sec[x]*(1 + (1 - 8*Tan[x]^2)^(1/3)))/(1 - 8*Tan[x]^2)^(2/3),x]

[Out]

-Log[1 - Sec[x]^2]/2 + (3*Log[1 - (9 - 8*Sec[x]^2)^(1/3)])/2

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int ^{\tan{\left (x \right )}} \frac{\sqrt [3]{- 8 x^{2} + 1} + 1}{x \left (- 8 x^{2} + 1\right )^{\frac{2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cot(x)*(1+(1-8*tan(x)**2)**(1/3))/cos(x)**2/(1-8*tan(x)**2)**(2/3),x)

[Out]

Integral(((-8*x**2 + 1)**(1/3) + 1)/(x*(-8*x**2 + 1)**(2/3)), (x, tan(x)))

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Mathematica [C]  time = 4.67344, size = 93, normalized size = 3.44 \[ -\frac{3 \sqrt [3]{8-\cot ^2(x)} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{\cot ^2(x)}{8}\right )}{4 \sqrt [3]{1-8 \tan ^2(x)}}-\frac{3 \left (8-\cot ^2(x)\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{\cot ^2(x)}{8}\right )}{16 \left (1-8 \tan ^2(x)\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Csc[x]*Sec[x]*(1 + (1 - 8*Tan[x]^2)^(1/3)))/(1 - 8*Tan[x]^2)^(2/3),x]

[Out]

(-3*(8 - Cot[x]^2)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, Cot[x]^2/8])/(16*(1 -
8*Tan[x]^2)^(2/3)) - (3*(8 - Cot[x]^2)^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, Co
t[x]^2/8])/(4*(1 - 8*Tan[x]^2)^(1/3))

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Maple [F]  time = 0.471, size = 0, normalized size = 0. \[ \int{\frac{\cot \left ( x \right ) }{ \left ( \cos \left ( x \right ) \right ) ^{2}} \left ( 1+\sqrt [3]{1-8\, \left ( \tan \left ( x \right ) \right ) ^{2}} \right ) \left ( 1-8\, \left ( \tan \left ( x \right ) \right ) ^{2} \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cot(x)*(1+(1-8*tan(x)^2)^(1/3))/cos(x)^2/(1-8*tan(x)^2)^(2/3),x)

[Out]

int(cot(x)*(1+(1-8*tan(x)^2)^(1/3))/cos(x)^2/(1-8*tan(x)^2)^(2/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((-8*tan(x)^2 + 1)^(1/3) + 1)*cot(x)/((-8*tan(x)^2 + 1)^(2/3)*cos(x)^2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 1.29129, size = 126, normalized size = 4.67 \[ -\frac{1}{2} \, \log \left (\frac{16 \,{\left (145 \, \cos \left (x\right )^{4} - 200 \, \cos \left (x\right )^{2} + 3 \,{\left (11 \, \cos \left (x\right )^{4} - 8 \, \cos \left (x\right )^{2}\right )} \left (\frac{9 \, \cos \left (x\right )^{2} - 8}{\cos \left (x\right )^{2}}\right )^{\frac{2}{3}} + 3 \,{\left (19 \, \cos \left (x\right )^{4} - 16 \, \cos \left (x\right )^{2}\right )} \left (\frac{9 \, \cos \left (x\right )^{2} - 8}{\cos \left (x\right )^{2}}\right )^{\frac{1}{3}} + 64\right )}}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((-8*tan(x)^2 + 1)^(1/3) + 1)*cot(x)/((-8*tan(x)^2 + 1)^(2/3)*cos(x)^2),x, algorithm="fricas")

[Out]

-1/2*log(16*(145*cos(x)^4 - 200*cos(x)^2 + 3*(11*cos(x)^4 - 8*cos(x)^2)*((9*cos(
x)^2 - 8)/cos(x)^2)^(2/3) + 3*(19*cos(x)^4 - 16*cos(x)^2)*((9*cos(x)^2 - 8)/cos(
x)^2)^(1/3) + 64)/(cos(x)^4 - 2*cos(x)^2 + 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(cot(x)*(1+(1-8*tan(x)**2)**(1/3))/cos(x)**2/(1-8*tan(x)**2)**(2/3),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.229704, size = 55, normalized size = 2.04 \[ -\frac{1}{2} \,{\rm ln}\left ({\left (-8 \, \tan \left (x\right )^{2} + 1\right )}^{\frac{2}{3}} +{\left (-8 \, \tan \left (x\right )^{2} + 1\right )}^{\frac{1}{3}} + 1\right ) +{\rm ln}\left (-{\left (-8 \, \tan \left (x\right )^{2} + 1\right )}^{\frac{1}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((-8*tan(x)^2 + 1)^(1/3) + 1)*cot(x)/((-8*tan(x)^2 + 1)^(2/3)*cos(x)^2),x, algorithm="giac")

[Out]

-1/2*ln((-8*tan(x)^2 + 1)^(2/3) + (-8*tan(x)^2 + 1)^(1/3) + 1) + ln(-(-8*tan(x)^
2 + 1)^(1/3) + 1)