3.1.89 \(\int \sec ^3(x) \tan ^3(x) \, dx\) [89]

Optimal. Leaf size=17 \[ -\frac {1}{3} \sec ^3(x)+\frac {\sec ^5(x)}{5} \]

[Out]

-1/3*sec(x)^3+1/5*sec(x)^5

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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 ^5(x)}{5}-\frac {\sec ^3(x)}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*Sec[x]^3 + Sec[x]^5/5

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 ^3(x) \tan ^3(x) \, dx &=\text {Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\sec (x)\right )\\ &=\text {Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,\sec (x)\right )\\ &=-\frac {1}{3} \sec ^3(x)+\frac {\sec ^5(x)}{5}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 17, normalized size = 1.00 \begin {gather*} -\frac {1}{3} \sec ^3(x)+\frac {\sec ^5(x)}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*Sec[x]^3 + Sec[x]^5/5

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(41\) vs. \(2(13)=26\).
time = 0.04, size = 42, normalized size = 2.47

method	result	size
risch	\(-\frac {8 \left (5 \,{\mathrm e}^{7 i x}-2 \,{\mathrm e}^{5 i x}+5 \,{\mathrm e}^{3 i x}\right )}{15 \left ({\mathrm e}^{2 i x}+1\right )^{5}}\)	\(34\)
default	\(\frac {\sin ^{4}\left (x \right )}{5 \cos \left (x \right )^{5}}+\frac {\sin ^{4}\left (x \right )}{15 \cos \left (x \right )^{3}}-\frac {\sin ^{4}\left (x \right )}{15 \cos \left (x \right )}-\frac {\left (2+\sin ^{2}\left (x \right )\right ) \cos \left (x \right )}{15}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*sin(x)^4/cos(x)^5+1/15*sin(x)^4/cos(x)^3-1/15*sin(x)^4/cos(x)-1/15*(2+sin(x)^2)*cos(x)

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Maxima [A]
time = 1.61, size = 14, normalized size = 0.82 \begin {gather*} -\frac {5 \, \cos \left (x\right )^{2} - 3}{15 \, \cos \left (x\right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(5*cos(x)^2 - 3)/cos(x)^5

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Fricas [A]
time = 0.68, size = 14, normalized size = 0.82 \begin {gather*} -\frac {5 \, \cos \left (x\right )^{2} - 3}{15 \, \cos \left (x\right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(5*cos(x)^2 - 3)/cos(x)^5

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Sympy [A]
time = 0.03, size = 14, normalized size = 0.82 \begin {gather*} \frac {3 - 5 \cos ^{2}{\left (x \right )}}{15 \cos ^{5}{\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3 - 5*cos(x)**2)/(15*cos(x)**5)

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Giac [A]
time = 0.94, size = 14, normalized size = 0.82 \begin {gather*} -\frac {5 \, \cos \left (x\right )^{2} - 3}{15 \, \cos \left (x\right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(5*cos(x)^2 - 3)/cos(x)^5

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Mupad [B]
time = 0.38, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{5\,{\cos \left (x\right )}^5}-\frac {1}{3\,{\cos \left (x\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(5*cos(x)^5) - 1/(3*cos(x)^3)

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