3.1.94 \(\int \sec ^6(x) \tan (x) \, dx\) [94]

Optimal. Leaf size=8 \[ \frac {\sec ^6(x)}{6} \]

[Out]

1/6*sec(x)^6

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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 = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2686, 30} \begin {gather*} \frac {\sec ^6(x)}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[x]^6*Tan[x],x]

[Out]

Sec[x]^6/6

Rule 30

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

Rule 2686

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

Rubi steps

\begin {align*} \int \sec ^6(x) \tan (x) \, dx &=\text {Subst}\left (\int x^5 \, dx,x,\sec (x)\right )\\ &=\frac {\sec ^6(x)}{6}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} \frac {\sec ^6(x)}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]^6*Tan[x],x]

[Out]

Sec[x]^6/6

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Mathics [A]
time = 1.62, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{6 \text {Cos}\left [x\right ]^6} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Tan[x]*Sec[x]^6,x]')

[Out]

1 / (6 Cos[x] ^ 6)

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Maple [A]
time = 0.02, size = 7, normalized size = 0.88

method	result	size
derivativedivides	\(\frac {\left (\sec ^{6}\left (x \right )\right )}{6}\)	\(7\)
default	\(\frac {\left (\sec ^{6}\left (x \right )\right )}{6}\)	\(7\)
risch	\(\frac {32 \,{\mathrm e}^{6 i x}}{3 \left ({\mathrm e}^{2 i x}+1\right )^{6}}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^6*tan(x),x,method=_RETURNVERBOSE)

[Out]

1/6*sec(x)^6

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Maxima [A]
time = 0.27, size = 10, normalized size = 1.25 \begin {gather*} -\frac {1}{6 \, {\left (\sin \left (x\right )^{2} - 1\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^6*tan(x),x, algorithm="maxima")

[Out]

-1/6/(sin(x)^2 - 1)^3

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Fricas [A]
time = 0.34, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{6 \, \cos \left (x\right )^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^6*tan(x),x, algorithm="fricas")

[Out]

1/6/cos(x)^6

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Sympy [A]
time = 0.03, size = 7, normalized size = 0.88 \begin {gather*} \frac {1}{6 \cos ^{6}{\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)**6*tan(x),x)

[Out]

1/(6*cos(x)**6)

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Giac [A]
time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} \frac {1}{6 \cos ^{6}x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^6*tan(x),x)

[Out]

1/6/cos(x)^6

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Mupad [B]
time = 0.17, size = 18, normalized size = 2.25 \begin {gather*} \frac {{\mathrm {tan}\left (x\right )}^2\,\left ({\mathrm {tan}\left (x\right )}^4+3\,{\mathrm {tan}\left (x\right )}^2+3\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)/cos(x)^6,x)

[Out]

(tan(x)^2*(3*tan(x)^2 + tan(x)^4 + 3))/6

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