3.1.92 \(\int \sec (x) \tan ^5(x) \, dx\) [92]

Optimal. Leaf size=19 \[ \sec (x)-\frac {2 \sec ^3(x)}{3}+\frac {\sec ^5(x)}{5} \]

[Out]

sec(x)-2/3*sec(x)^3+1/5*sec(x)^5

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Rubi [A]
time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2686, 200} \begin {gather*} \frac {\sec ^5(x)}{5}-\frac {2 \sec ^3(x)}{3}+\sec (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[x]*Tan[x]^5,x]

[Out]

Sec[x] - (2*Sec[x]^3)/3 + Sec[x]^5/5

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

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

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Mathematica [A]
time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} \sec (x)-\frac {2 \sec ^3(x)}{3}+\frac {\sec ^5(x)}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]*Tan[x]^5,x]

[Out]

Sec[x] - (2*Sec[x]^3)/3 + Sec[x]^5/5

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(47\) vs. \(2(15)=30\).
time = 0.04, size = 48, normalized size = 2.53

method	result	size
default	\(\frac {\sin ^{6}\left (x \right )}{5 \cos \left (x \right )^{5}}-\frac {\sin ^{6}\left (x \right )}{15 \cos \left (x \right )^{3}}+\frac {\sin ^{6}\left (x \right )}{5 \cos \left (x \right )}+\frac {\left (\frac {8}{3}+\sin ^{4}\left (x \right )+\frac {4 \left (\sin ^{2}\left (x \right )\right )}{3}\right ) \cos \left (x \right )}{5}\)	\(48\)
risch	\(\frac {2 \,{\mathrm e}^{9 i x}+\frac {8 \,{\mathrm e}^{7 i x}}{3}+\frac {116 \,{\mathrm e}^{5 i x}}{15}+\frac {8 \,{\mathrm e}^{3 i x}}{3}+2 \,{\mathrm e}^{i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{5}}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)*tan(x)^5,x,method=_RETURNVERBOSE)

[Out]

1/5*sin(x)^6/cos(x)^5-1/15*sin(x)^6/cos(x)^3+1/5*sin(x)^6/cos(x)+1/5*(8/3+sin(x)^4+4/3*sin(x)^2)*cos(x)

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Maxima [A]
time = 1.77, size = 20, normalized size = 1.05 \begin {gather*} \frac {15 \, \cos \left (x\right )^{4} - 10 \, \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)*tan(x)^5,x, algorithm="maxima")

[Out]

1/15*(15*cos(x)^4 - 10*cos(x)^2 + 3)/cos(x)^5

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Fricas [A]
time = 0.72, size = 20, normalized size = 1.05 \begin {gather*} \frac {15 \, \cos \left (x\right )^{4} - 10 \, \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)*tan(x)^5,x, algorithm="fricas")

[Out]

1/15*(15*cos(x)^4 - 10*cos(x)^2 + 3)/cos(x)^5

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Sympy [A]
time = 0.04, size = 22, normalized size = 1.16 \begin {gather*} - \frac {- 15 \cos ^{4}{\left (x \right )} + 10 \cos ^{2}{\left (x \right )} - 3}{15 \cos ^{5}{\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)*tan(x)**5,x)

[Out]

-(-15*cos(x)**4 + 10*cos(x)**2 - 3)/(15*cos(x)**5)

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Giac [A]
time = 0.76, size = 20, normalized size = 1.05 \begin {gather*} \frac {15 \, \cos \left (x\right )^{4} - 10 \, \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)*tan(x)^5,x, algorithm="giac")

[Out]

1/15*(15*cos(x)^4 - 10*cos(x)^2 + 3)/cos(x)^5

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Mupad [B]
time = 0.27, size = 17, normalized size = 0.89 \begin {gather*} \frac {{\cos \left (x\right )}^4-\frac {2\,{\cos \left (x\right )}^2}{3}+\frac {1}{5}}{{\cos \left (x\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)^5/cos(x),x)

[Out]

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

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