3.106 \(\int \tan ^3(x) \, dx\)

Optimal. Leaf size=12 \[ \frac {\tan ^2(x)}{2}+\log (\cos (x)) \]

[Out]

ln(cos(x))+1/2*tan(x)^2

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3473, 3475} \[ \frac {\tan ^2(x)}{2}+\log (\cos (x)) \]

Antiderivative was successfully verified.

[In]

Int[Tan[x]^3,x]

[Out]

Log[Cos[x]] + Tan[x]^2/2

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \tan ^3(x) \, dx &=\frac {\tan ^2(x)}{2}-\int \tan (x) \, dx\\ &=\log (\cos (x))+\frac {\tan ^2(x)}{2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 1.00 \[ \frac {\sec ^2(x)}{2}+\log (\cos (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[x]^3,x]

[Out]

Log[Cos[x]] + Sec[x]^2/2

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fricas [A]  time = 0.42, size = 18, normalized size = 1.50 \[ \frac {1}{2} \, \tan \relax (x)^{2} + \frac {1}{2} \, \log \left (\frac {1}{\tan \relax (x)^{2} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)^3,x, algorithm="fricas")

[Out]

1/2*tan(x)^2 + 1/2*log(1/(tan(x)^2 + 1))

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giac [A]  time = 0.99, size = 16, normalized size = 1.33 \[ \frac {1}{2} \, \tan \relax (x)^{2} - \frac {1}{2} \, \log \left (\tan \relax (x)^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)^3,x, algorithm="giac")

[Out]

1/2*tan(x)^2 - 1/2*log(tan(x)^2 + 1)

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maple [A]  time = 0.00, size = 17, normalized size = 1.42 \[ \frac {\left (\tan ^{2}\relax (x )\right )}{2}-\frac {\ln \left (\tan ^{2}\relax (x )+1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)^3,x)

[Out]

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

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maxima [A]  time = 0.44, size = 20, normalized size = 1.67 \[ -\frac {1}{2 \, {\left (\sin \relax (x)^{2} - 1\right )}} + \frac {1}{2} \, \log \left (\sin \relax (x)^{2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)^3,x, algorithm="maxima")

[Out]

-1/2/(sin(x)^2 - 1) + 1/2*log(sin(x)^2 - 1)

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mupad [B]  time = 0.02, size = 16, normalized size = 1.33 \[ \ln \left (\cos \relax (x)\right )-\frac {{\cos \relax (x)}^2-1}{2\,{\cos \relax (x)}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)^3,x)

[Out]

log(cos(x)) - (cos(x)^2 - 1)/(2*cos(x)^2)

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sympy [A]  time = 0.09, size = 12, normalized size = 1.00 \[ \log {\left (\cos {\relax (x )} \right )} + \frac {1}{2 \cos ^{2}{\relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)**3,x)

[Out]

log(cos(x)) + 1/(2*cos(x)**2)

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