### 3.93 $$\int \sec ^3(x) \tan ^5(x) \, dx$$

Optimal. Leaf size=25 $\frac{\sec ^7(x)}{7}-\frac{2 \sec ^5(x)}{5}+\frac{\sec ^3(x)}{3}$

[Out]

Sec[x]^3/3 - (2*Sec[x]^5)/5 + Sec[x]^7/7

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Rubi [A]  time = 0.0291992, antiderivative size = 25, normalized size of antiderivative = 1., 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 = {2606, 270} $\frac{\sec ^7(x)}{7}-\frac{2 \sec ^5(x)}{5}+\frac{\sec ^3(x)}{3}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sec[x]^3*Tan[x]^5,x]

[Out]

Sec[x]^3/3 - (2*Sec[x]^5)/5 + Sec[x]^7/7

Rule 2606

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])

Rule 270

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

Rubi steps

\begin{align*} \int \sec ^3(x) \tan ^5(x) \, dx &=\operatorname{Subst}\left (\int x^2 \left (-1+x^2\right )^2 \, dx,x,\sec (x)\right )\\ &=\operatorname{Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\sec (x)\right )\\ &=\frac{\sec ^3(x)}{3}-\frac{2 \sec ^5(x)}{5}+\frac{\sec ^7(x)}{7}\\ \end{align*}

Mathematica [A]  time = 0.011756, size = 25, normalized size = 1. $\frac{\sec ^7(x)}{7}-\frac{2 \sec ^5(x)}{5}+\frac{\sec ^3(x)}{3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sec[x]^3*Tan[x]^5,x]

[Out]

Sec[x]^3/3 - (2*Sec[x]^5)/5 + Sec[x]^7/7

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Maple [B]  time = 0.012, size = 58, normalized size = 2.3 \begin{align*}{\frac{ \left ( \sin \left ( x \right ) \right ) ^{6}}{7\, \left ( \cos \left ( x \right ) \right ) ^{7}}}+{\frac{ \left ( \sin \left ( x \right ) \right ) ^{6}}{35\, \left ( \cos \left ( x \right ) \right ) ^{5}}}-{\frac{ \left ( \sin \left ( x \right ) \right ) ^{6}}{105\, \left ( \cos \left ( x \right ) \right ) ^{3}}}+{\frac{ \left ( \sin \left ( x \right ) \right ) ^{6}}{35\,\cos \left ( x \right ) }}+{\frac{\cos \left ( x \right ) }{35} \left ({\frac{8}{3}}+ \left ( \sin \left ( x \right ) \right ) ^{4}+{\frac{4\, \left ( \sin \left ( x \right ) \right ) ^{2}}{3}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^3*tan(x)^5,x)

[Out]

1/7*sin(x)^6/cos(x)^7+1/35*sin(x)^6/cos(x)^5-1/105*sin(x)^6/cos(x)^3+1/35*sin(x)^6/cos(x)+1/35*(8/3+sin(x)^4+4
/3*sin(x)^2)*cos(x)

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Maxima [A]  time = 0.926329, size = 27, normalized size = 1.08 \begin{align*} \frac{35 \, \cos \left (x\right )^{4} - 42 \, \cos \left (x\right )^{2} + 15}{105 \, \cos \left (x\right )^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^3*tan(x)^5,x, algorithm="maxima")

[Out]

1/105*(35*cos(x)^4 - 42*cos(x)^2 + 15)/cos(x)^7

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Fricas [A]  time = 2.05767, size = 66, normalized size = 2.64 \begin{align*} \frac{35 \, \cos \left (x\right )^{4} - 42 \, \cos \left (x\right )^{2} + 15}{105 \, \cos \left (x\right )^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^3*tan(x)^5,x, algorithm="fricas")

[Out]

1/105*(35*cos(x)^4 - 42*cos(x)^2 + 15)/cos(x)^7

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Sympy [A]  time = 0.114735, size = 20, normalized size = 0.8 \begin{align*} \frac{35 \cos ^{4}{\left (x \right )} - 42 \cos ^{2}{\left (x \right )} + 15}{105 \cos ^{7}{\left (x \right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)**3*tan(x)**5,x)

[Out]

(35*cos(x)**4 - 42*cos(x)**2 + 15)/(105*cos(x)**7)

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Giac [A]  time = 1.06472, size = 27, normalized size = 1.08 \begin{align*} \frac{35 \, \cos \left (x\right )^{4} - 42 \, \cos \left (x\right )^{2} + 15}{105 \, \cos \left (x\right )^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^3*tan(x)^5,x, algorithm="giac")

[Out]

1/105*(35*cos(x)^4 - 42*cos(x)^2 + 15)/cos(x)^7