3.3.71 \(\int \sec ^4(x) \tan ^3(x) \, dx\) [271]

Optimal. Leaf size=17 \[ -\frac {1}{4} \sec ^4(x)+\frac {\sec ^6(x)}{6} \]

[Out]

-1/4*sec(x)^4+1/6*sec(x)^6

________________________________________________________________________________________

Rubi [A]
time = 0.01, 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 ^6(x)}{6}-\frac {\sec ^4(x)}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[x]^4*Tan[x]^3,x]

[Out]

-1/4*Sec[x]^4 + Sec[x]^6/6

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 ^4(x) \tan ^3(x) \, dx &=\text {Subst}\left (\int x^3 \left (-1+x^2\right ) \, dx,x,\sec (x)\right )\\ &=\text {Subst}\left (\int \left (-x^3+x^5\right ) \, dx,x,\sec (x)\right )\\ &=-\frac {1}{4} \sec ^4(x)+\frac {\sec ^6(x)}{6}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \sec ^4(x)+\frac {\sec ^6(x)}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]^4*Tan[x]^3,x]

[Out]

-1/4*Sec[x]^4 + Sec[x]^6/6

________________________________________________________________________________________

Mathics [A]
time = 1.74, size = 14, normalized size = 0.82 \begin {gather*} \frac {2-3 \text {Cos}\left [x\right ]^2}{12 \text {Cos}\left [x\right ]^6} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Sec[x]^4*Tan[x]^3,x]')

[Out]

(2 - 3 Cos[x] ^ 2) / (12 Cos[x] ^ 6)

________________________________________________________________________________________

Maple [A]
time = 0.03, size = 22, normalized size = 1.29

method	result	size
default	\(\frac {\sin ^{4}\left (x \right )}{6 \cos \left (x \right )^{6}}+\frac {\sin ^{4}\left (x \right )}{12 \cos \left (x \right )^{4}}\)	\(22\)
risch	\(-\frac {4 \left (3 \,{\mathrm e}^{8 i x}-2 \,{\mathrm e}^{6 i x}+3 \,{\mathrm e}^{4 i x}\right )}{3 \left ({\mathrm e}^{2 i x}+1\right )^{6}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^4*tan(x)^3,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (13) = 26\).
time = 0.26, size = 30, normalized size = 1.76 \begin {gather*} -\frac {3 \, \sin \left (x\right )^{2} - 1}{12 \, {\left (\sin \left (x\right )^{6} - 3 \, \sin \left (x\right )^{4} + 3 \, \sin \left (x\right )^{2} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^4*tan(x)^3,x, algorithm="maxima")

[Out]

-1/12*(3*sin(x)^2 - 1)/(sin(x)^6 - 3*sin(x)^4 + 3*sin(x)^2 - 1)

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 14, normalized size = 0.82 \begin {gather*} -\frac {3 \, \cos \left (x\right )^{2} - 2}{12 \, \cos \left (x\right )^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^4*tan(x)^3,x, algorithm="fricas")

[Out]

-1/12*(3*cos(x)^2 - 2)/cos(x)^6

________________________________________________________________________________________

Sympy [A]
time = 0.05, size = 14, normalized size = 0.82 \begin {gather*} \frac {2 - 3 \cos ^{2}{\left (x \right )}}{12 \cos ^{6}{\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)**4*tan(x)**3,x)

[Out]

(2 - 3*cos(x)**2)/(12*cos(x)**6)

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {-3 \cos ^{2}x+2}{12 \cos ^{6}x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^4*tan(x)^3,x)

[Out]

-1/12*(3*cos(x)^2 - 2)/cos(x)^6

________________________________________________________________________________________

Mupad [B]
time = 0.19, size = 14, normalized size = 0.82 \begin {gather*} \frac {{\mathrm {tan}\left (x\right )}^4\,\left (2\,{\mathrm {tan}\left (x\right )}^2+3\right )}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)^3/cos(x)^4,x)

[Out]

(tan(x)^4*(2*tan(x)^2 + 3))/12

________________________________________________________________________________________