3.1.95 \(\int \sec ^6(x) \tan ^3(x) \, dx\) [95]

Optimal. Leaf size=17 \[ -\frac {1}{6} \sec ^6(x)+\frac {\sec ^8(x)}{8} \]

[Out]

-1/6*sec(x)^6+1/8*sec(x)^8

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Rubi [A]
time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2686, 14} \begin {gather*} \frac {\sec ^8(x)}{8}-\frac {\sec ^6(x)}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[x]^6*Tan[x]^3,x]

[Out]

-1/6*Sec[x]^6 + Sec[x]^8/8

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 ^3(x) \, dx &=\text {Subst}\left (\int x^5 \left (-1+x^2\right ) \, dx,x,\sec (x)\right )\\ &=\text {Subst}\left (\int \left (-x^5+x^7\right ) \, dx,x,\sec (x)\right )\\ &=-\frac {1}{6} \sec ^6(x)+\frac {\sec ^8(x)}{8}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} -\frac {1}{6} \sec ^6(x)+\frac {\sec ^8(x)}{8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]^6*Tan[x]^3,x]

[Out]

-1/6*Sec[x]^6 + Sec[x]^8/8

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Mathics [A]
time = 1.75, size = 14, normalized size = 0.82 \begin {gather*} \frac {3-4 \text {Cos}\left [x\right ]^2}{24 \text {Cos}\left [x\right ]^8} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Tan[x]^3*Sec[x]^6,x]')

[Out]

(3 - 4 Cos[x] ^ 2) / (24 Cos[x] ^ 8)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(31\) vs. \(2(13)=26\).
time = 0.04, size = 32, normalized size = 1.88

method	result	size
risch	\(-\frac {32 \left ({\mathrm e}^{10 i x}-{\mathrm e}^{8 i x}+{\mathrm e}^{6 i x}\right )}{3 \left ({\mathrm e}^{2 i x}+1\right )^{8}}\)	\(30\)
default	\(\frac {\sin ^{4}\left (x \right )}{8 \cos \left (x \right )^{8}}+\frac {\sin ^{4}\left (x \right )}{12 \cos \left (x \right )^{6}}+\frac {\sin ^{4}\left (x \right )}{24 \cos \left (x \right )^{4}}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^6*tan(x)^3,x,method=_RETURNVERBOSE)

[Out]

1/8*sin(x)^4/cos(x)^8+1/12*sin(x)^4/cos(x)^6+1/24*sin(x)^4/cos(x)^4

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (13) = 26\).
time = 0.29, size = 36, normalized size = 2.12 \begin {gather*} \frac {4 \, \sin \left (x\right )^{2} - 1}{24 \, {\left (\sin \left (x\right )^{8} - 4 \, \sin \left (x\right )^{6} + 6 \, \sin \left (x\right )^{4} - 4 \, \sin \left (x\right )^{2} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^6*tan(x)^3,x, algorithm="maxima")

[Out]

1/24*(4*sin(x)^2 - 1)/(sin(x)^8 - 4*sin(x)^6 + 6*sin(x)^4 - 4*sin(x)^2 + 1)

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Fricas [A]
time = 0.33, size = 14, normalized size = 0.82 \begin {gather*} -\frac {4 \, \cos \left (x\right )^{2} - 3}{24 \, \cos \left (x\right )^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^6*tan(x)^3,x, algorithm="fricas")

[Out]

-1/24*(4*cos(x)^2 - 3)/cos(x)^8

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Sympy [A]
time = 0.06, size = 14, normalized size = 0.82 \begin {gather*} \frac {3 - 4 \cos ^{2}{\left (x \right )}}{24 \cos ^{8}{\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)**6*tan(x)**3,x)

[Out]

(3 - 4*cos(x)**2)/(24*cos(x)**8)

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Giac [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {-4 \cos ^{2}x+3}{24 \cos ^{8}x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^6*tan(x)^3,x)

[Out]

-1/24*(4*cos(x)^2 - 3)/cos(x)^8

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Mupad [B]
time = 0.17, size = 20, normalized size = 1.18 \begin {gather*} \frac {{\mathrm {tan}\left (x\right )}^4\,\left (3\,{\mathrm {tan}\left (x\right )}^4+8\,{\mathrm {tan}\left (x\right )}^2+6\right )}{24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)^3/cos(x)^6,x)

[Out]

(tan(x)^4*(8*tan(x)^2 + 3*tan(x)^4 + 6))/24

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