3.1.86 \(\int \sec ^2(x) \tan ^4(x) \, dx\) [86]

Optimal. Leaf size=8 \[ \frac {\tan ^5(x)}{5} \]

[Out]

1/5*tan(x)^5

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Rubi [A]
time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2687, 30} \begin {gather*} \frac {\tan ^5(x)}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[x]^2*Tan[x]^4,x]

[Out]

Tan[x]^5/5

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2687

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rubi steps

\begin {align*} \int \sec ^2(x) \tan ^4(x) \, dx &=\text {Subst}\left (\int x^4 \, dx,x,\tan (x)\right )\\ &=\frac {\tan ^5(x)}{5}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} \frac {\tan ^5(x)}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]^2*Tan[x]^4,x]

[Out]

Tan[x]^5/5

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Mathics [A]
time = 1.82, size = 6, normalized size = 0.75 \begin {gather*} \frac {\text {Tan}\left [x\right ]^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Tan[x]^4*Sec[x]^2,x]')

[Out]

Tan[x] ^ 5 / 5

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Maple [A]
time = 0.03, size = 11, normalized size = 1.38

method	result	size
default	\(\frac {\sin ^{5}\left (x \right )}{5 \cos \left (x \right )^{5}}\)	\(11\)
risch	\(\frac {2 i \left (5 \,{\mathrm e}^{8 i x}+10 \,{\mathrm e}^{4 i x}+1\right )}{5 \left ({\mathrm e}^{2 i x}+1\right )^{5}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^2*tan(x)^4,x,method=_RETURNVERBOSE)

[Out]

1/5*sin(x)^5/cos(x)^5

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Maxima [A]
time = 0.28, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{5} \, \tan \left (x\right )^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2*tan(x)^4,x, algorithm="maxima")

[Out]

1/5*tan(x)^5

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 20 vs. \(2 (6) = 12\).
time = 0.34, size = 20, normalized size = 2.50 \begin {gather*} \frac {{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right )}{5 \, \cos \left (x\right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2*tan(x)^4,x, algorithm="fricas")

[Out]

1/5*(cos(x)^4 - 2*cos(x)^2 + 1)*sin(x)/cos(x)^5

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (5) = 10\)
time = 0.03, size = 29, normalized size = 3.62 \begin {gather*} \frac {\sin {\left (x \right )}}{5 \cos {\left (x \right )}} - \frac {2 \sin {\left (x \right )}}{5 \cos ^{3}{\left (x \right )}} + \frac {\sin {\left (x \right )}}{5 \cos ^{5}{\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)**2*tan(x)**4,x)

[Out]

sin(x)/(5*cos(x)) - 2*sin(x)/(5*cos(x)**3) + sin(x)/(5*cos(x)**5)

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Giac [A]
time = 0.00, size = 7, normalized size = 0.88 \begin {gather*} \frac {\tan ^{5}x}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2*tan(x)^4,x)

[Out]

1/5*tan(x)^5

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Mupad [B]
time = 0.03, size = 6, normalized size = 0.75 \begin {gather*} \frac {{\mathrm {tan}\left (x\right )}^5}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)^4/cos(x)^2,x)

[Out]

tan(x)^5/5

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