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