3.705 \(\int \sec ^2(x) \tan ^6(x) (1+\tan ^2(x))^3 \, dx\)

Optimal. Leaf size=33 \[ \frac{\tan ^{13}(x)}{13}+\frac{3 \tan ^{11}(x)}{11}+\frac{\tan ^9(x)}{3}+\frac{\tan ^7(x)}{7} \]

[Out]

Tan[x]^7/7 + Tan[x]^9/3 + (3*Tan[x]^11)/11 + Tan[x]^13/13

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Rubi [A]  time = 0.0920761, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {3657, 2607, 270} \[ \frac{\tan ^{13}(x)}{13}+\frac{3 \tan ^{11}(x)}{11}+\frac{\tan ^9(x)}{3}+\frac{\tan ^7(x)}{7} \]

Antiderivative was successfully verified.

[In]

Int[Sec[x]^2*Tan[x]^6*(1 + Tan[x]^2)^3,x]

[Out]

Tan[x]^7/7 + Tan[x]^9/3 + (3*Tan[x]^11)/11 + Tan[x]^13/13

Rule 3657

Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*sec[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a, b]

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 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 ^2(x) \tan ^6(x) \left (1+\tan ^2(x)\right )^3 \, dx &=\int \sec ^8(x) \tan ^6(x) \, dx\\ &=\operatorname{Subst}\left (\int x^6 \left (1+x^2\right )^3 \, dx,x,\tan (x)\right )\\ &=\operatorname{Subst}\left (\int \left (x^6+3 x^8+3 x^{10}+x^{12}\right ) \, dx,x,\tan (x)\right )\\ &=\frac{\tan ^7(x)}{7}+\frac{\tan ^9(x)}{3}+\frac{3 \tan ^{11}(x)}{11}+\frac{\tan ^{13}(x)}{13}\\ \end{align*}

Mathematica [B]  time = 0.0263524, size = 67, normalized size = 2.03 \[ -\frac{16 \tan (x)}{3003}+\frac{1}{13} \tan (x) \sec ^{12}(x)-\frac{27}{143} \tan (x) \sec ^{10}(x)+\frac{53}{429} \tan (x) \sec ^8(x)-\frac{5 \tan (x) \sec ^6(x)}{3003}-\frac{2 \tan (x) \sec ^4(x)}{1001}-\frac{8 \tan (x) \sec ^2(x)}{3003} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]^2*Tan[x]^6*(1 + Tan[x]^2)^3,x]

[Out]

(-16*Tan[x])/3003 - (8*Sec[x]^2*Tan[x])/3003 - (2*Sec[x]^4*Tan[x])/1001 - (5*Sec[x]^6*Tan[x])/3003 + (53*Sec[x
]^8*Tan[x])/429 - (27*Sec[x]^10*Tan[x])/143 + (Sec[x]^12*Tan[x])/13

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Maple [A]  time = 0.022, size = 42, normalized size = 1.3 \begin{align*}{\frac{ \left ( \sin \left ( x \right ) \right ) ^{7}}{7\, \left ( \cos \left ( x \right ) \right ) ^{7}}}+{\frac{ \left ( \sin \left ( x \right ) \right ) ^{9}}{3\, \left ( \cos \left ( x \right ) \right ) ^{9}}}+{\frac{3\, \left ( \sin \left ( x \right ) \right ) ^{11}}{11\, \left ( \cos \left ( x \right ) \right ) ^{11}}}+{\frac{ \left ( \sin \left ( x \right ) \right ) ^{13}}{13\, \left ( \cos \left ( x \right ) \right ) ^{13}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^2*tan(x)^6*(1+tan(x)^2)^3,x)

[Out]

1/7*sin(x)^7/cos(x)^7+1/3*sin(x)^9/cos(x)^9+3/11*sin(x)^11/cos(x)^11+1/13*sin(x)^13/cos(x)^13

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Maxima [A]  time = 0.959681, size = 34, normalized size = 1.03 \begin{align*} \frac{1}{13} \, \tan \left (x\right )^{13} + \frac{3}{11} \, \tan \left (x\right )^{11} + \frac{1}{3} \, \tan \left (x\right )^{9} + \frac{1}{7} \, \tan \left (x\right )^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2*tan(x)^6*(1+tan(x)^2)^3,x, algorithm="maxima")

[Out]

1/13*tan(x)^13 + 3/11*tan(x)^11 + 1/3*tan(x)^9 + 1/7*tan(x)^7

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Fricas [A]  time = 2.2282, size = 158, normalized size = 4.79 \begin{align*} -\frac{{\left (16 \, \cos \left (x\right )^{12} + 8 \, \cos \left (x\right )^{10} + 6 \, \cos \left (x\right )^{8} + 5 \, \cos \left (x\right )^{6} - 371 \, \cos \left (x\right )^{4} + 567 \, \cos \left (x\right )^{2} - 231\right )} \sin \left (x\right )}{3003 \, \cos \left (x\right )^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2*tan(x)^6*(1+tan(x)^2)^3,x, algorithm="fricas")

[Out]

-1/3003*(16*cos(x)^12 + 8*cos(x)^10 + 6*cos(x)^8 + 5*cos(x)^6 - 371*cos(x)^4 + 567*cos(x)^2 - 231)*sin(x)/cos(
x)^13

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Sympy [A]  time = 92.6382, size = 27, normalized size = 0.82 \begin{align*} \frac{\tan ^{13}{\left (x \right )}}{13} + \frac{3 \tan ^{11}{\left (x \right )}}{11} + \frac{\tan ^{9}{\left (x \right )}}{3} + \frac{\tan ^{7}{\left (x \right )}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)**2*tan(x)**6*(1+tan(x)**2)**3,x)

[Out]

tan(x)**13/13 + 3*tan(x)**11/11 + tan(x)**9/3 + tan(x)**7/7

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Giac [A]  time = 1.10194, size = 34, normalized size = 1.03 \begin{align*} \frac{1}{13} \, \tan \left (x\right )^{13} + \frac{3}{11} \, \tan \left (x\right )^{11} + \frac{1}{3} \, \tan \left (x\right )^{9} + \frac{1}{7} \, \tan \left (x\right )^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2*tan(x)^6*(1+tan(x)^2)^3,x, algorithm="giac")

[Out]

1/13*tan(x)^13 + 3/11*tan(x)^11 + 1/3*tan(x)^9 + 1/7*tan(x)^7