### 3.175 $$\int \log (a \sec ^n(x)) \, dx$$

Optimal. Leaf size=51 $-\frac{1}{2} i n \text{PolyLog}\left (2,-e^{2 i x}\right )+x \log \left (a \sec ^n(x)\right )-\frac{1}{2} i n x^2+n x \log \left (1+e^{2 i x}\right )$

[Out]

(-I/2)*n*x^2 + n*x*Log[1 + E^((2*I)*x)] + x*Log[a*Sec[x]^n] - (I/2)*n*PolyLog[2, -E^((2*I)*x)]

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Rubi [A]  time = 0.0538001, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.857, Rules used = {2548, 12, 3719, 2190, 2279, 2391} $-\frac{1}{2} i n \text{PolyLog}\left (2,-e^{2 i x}\right )+x \log \left (a \sec ^n(x)\right )-\frac{1}{2} i n x^2+n x \log \left (1+e^{2 i x}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[Log[a*Sec[x]^n],x]

[Out]

(-I/2)*n*x^2 + n*x*Log[1 + E^((2*I)*x)] + x*Log[a*Sec[x]^n] - (I/2)*n*PolyLog[2, -E^((2*I)*x)]

Rule 2548

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/u, x], x] /; InverseFunctionFr
eeQ[u, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3719

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(I*(c + d*x)^(m + 1))/(d*(m + 1)), x
] - Dist[2*I, Int[((c + d*x)^m*E^(2*I*(e + f*x)))/(1 + E^(2*I*(e + f*x))), x], x] /; FreeQ[{c, d, e, f}, x] &&
IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \log \left (a \sec ^n(x)\right ) \, dx &=x \log \left (a \sec ^n(x)\right )-\int n x \tan (x) \, dx\\ &=x \log \left (a \sec ^n(x)\right )-n \int x \tan (x) \, dx\\ &=-\frac{1}{2} i n x^2+x \log \left (a \sec ^n(x)\right )+(2 i n) \int \frac{e^{2 i x} x}{1+e^{2 i x}} \, dx\\ &=-\frac{1}{2} i n x^2+n x \log \left (1+e^{2 i x}\right )+x \log \left (a \sec ^n(x)\right )-n \int \log \left (1+e^{2 i x}\right ) \, dx\\ &=-\frac{1}{2} i n x^2+n x \log \left (1+e^{2 i x}\right )+x \log \left (a \sec ^n(x)\right )+\frac{1}{2} (i n) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i x}\right )\\ &=-\frac{1}{2} i n x^2+n x \log \left (1+e^{2 i x}\right )+x \log \left (a \sec ^n(x)\right )-\frac{1}{2} i n \text{Li}_2\left (-e^{2 i x}\right )\\ \end{align*}

Mathematica [A]  time = 0.022417, size = 51, normalized size = 1. $-\frac{1}{2} i n \text{PolyLog}\left (2,-e^{2 i x}\right )+x \log \left (a \sec ^n(x)\right )-\frac{1}{2} i n x^2+n x \log \left (1+e^{2 i x}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Log[a*Sec[x]^n],x]

[Out]

(-I/2)*n*x^2 + n*x*Log[1 + E^((2*I)*x)] + x*Log[a*Sec[x]^n] - (I/2)*n*PolyLog[2, -E^((2*I)*x)]

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Maple [F]  time = 0.118, size = 0, normalized size = 0. \begin{align*} \int \ln \left ( a \left ( \sec \left ( x \right ) \right ) ^{n} \right ) \, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(ln(a*sec(x)^n),x)

[Out]

int(ln(a*sec(x)^n),x)

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Maxima [A]  time = 2.42739, size = 88, normalized size = 1.73 \begin{align*} \frac{1}{2} \,{\left (-i \, x^{2} + 2 i \, x \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right ) + 1\right ) + x \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right ) - i \,{\rm Li}_2\left (-e^{\left (2 i \, x\right )}\right )\right )} n + x \log \left (a \sec \left (x\right )^{n}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(a*sec(x)^n),x, algorithm="maxima")

[Out]

1/2*(-I*x^2 + 2*I*x*arctan2(sin(2*x), cos(2*x) + 1) + x*log(cos(2*x)^2 + sin(2*x)^2 + 2*cos(2*x) + 1) - I*dilo
g(-e^(2*I*x)))*n + x*log(a*sec(x)^n)

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Fricas [B]  time = 2.15378, size = 435, normalized size = 8.53 \begin{align*} n x \log \left (\frac{1}{\cos \left (x\right )}\right ) + \frac{1}{2} \, n x \log \left (i \, \cos \left (x\right ) + \sin \left (x\right ) + 1\right ) + \frac{1}{2} \, n x \log \left (i \, \cos \left (x\right ) - \sin \left (x\right ) + 1\right ) + \frac{1}{2} \, n x \log \left (-i \, \cos \left (x\right ) + \sin \left (x\right ) + 1\right ) + \frac{1}{2} \, n x \log \left (-i \, \cos \left (x\right ) - \sin \left (x\right ) + 1\right ) + \frac{1}{2} i \, n{\rm Li}_2\left (i \, \cos \left (x\right ) + \sin \left (x\right )\right ) - \frac{1}{2} i \, n{\rm Li}_2\left (i \, \cos \left (x\right ) - \sin \left (x\right )\right ) - \frac{1}{2} i \, n{\rm Li}_2\left (-i \, \cos \left (x\right ) + \sin \left (x\right )\right ) + \frac{1}{2} i \, n{\rm Li}_2\left (-i \, \cos \left (x\right ) - \sin \left (x\right )\right ) + x \log \left (a\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(a*sec(x)^n),x, algorithm="fricas")

[Out]

n*x*log(1/cos(x)) + 1/2*n*x*log(I*cos(x) + sin(x) + 1) + 1/2*n*x*log(I*cos(x) - sin(x) + 1) + 1/2*n*x*log(-I*c
os(x) + sin(x) + 1) + 1/2*n*x*log(-I*cos(x) - sin(x) + 1) + 1/2*I*n*dilog(I*cos(x) + sin(x)) - 1/2*I*n*dilog(I
*cos(x) - sin(x)) - 1/2*I*n*dilog(-I*cos(x) + sin(x)) + 1/2*I*n*dilog(-I*cos(x) - sin(x)) + x*log(a)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \log{\left (a \sec ^{n}{\left (x \right )} \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(a*sec(x)**n),x)

[Out]

Integral(log(a*sec(x)**n), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \log \left (a \sec \left (x\right )^{n}\right )\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(a*sec(x)^n),x, algorithm="giac")

[Out]

integrate(log(a*sec(x)^n), x)