3.2.61 \(\int \tan (d (a+b \log (c x^n))) \, dx\) [161]

Optimal. Leaf size=67 \[ -i x+2 i x \, _2F_1\left (1,-\frac {i}{2 b d n};1-\frac {i}{2 b d n};-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \]

[Out]

-I*x+2*I*x*hypergeom([1, -1/2*I/b/d/n],[1-1/2*I/b/d/n],-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {4589, 4591, 470, 371} \begin {gather*} 2 i x \, _2F_1\left (1,-\frac {i}{2 b d n};1-\frac {i}{2 b d n};-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )-i x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tan[d*(a + b*Log[c*x^n])],x]

[Out]

(-I)*x + (2*I)*x*Hypergeometric2F1[1, (-1/2*I)/(b*d*n), 1 - (I/2)/(b*d*n), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)
)]

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

Rule 4589

Int[Tan[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x
^(1/n - 1)*Tan[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n,
1])

Rule 4591

Int[((e_.)*(x_))^(m_.)*Tan[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Int[(e*x)^m*((I - I*E^(2*I*a*d)*
x^(2*I*b*d))/(1 + E^(2*I*a*d)*x^(2*I*b*d)))^p, x] /; FreeQ[{a, b, d, e, m, p}, x]

Rubi steps

\begin {align*} \int \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(151\) vs. \(2(67)=134\).
time = 11.37, size = 151, normalized size = 2.25 \begin {gather*} \frac {x \left (-e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1-\frac {i}{2 b d n};2-\frac {i}{2 b d n};-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(1+2 i b d n) \, _2F_1\left (1,-\frac {i}{2 b d n};1-\frac {i}{2 b d n};-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{-i+2 b d n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tan[d*(a + b*Log[c*x^n])],x]

[Out]

(x*(-(E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - (I/2)/(b*d*n), 2 - (I/2)/(b*d*n), -E^((2*I)*d*(a
+ b*Log[c*x^n]))]) + (1 + (2*I)*b*d*n)*Hypergeometric2F1[1, (-1/2*I)/(b*d*n), 1 - (I/2)/(b*d*n), -E^((2*I)*d*
(a + b*Log[c*x^n]))]))/(-I + 2*b*d*n)

________________________________________________________________________________________

Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \tan \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*(a+b*ln(c*x^n))),x)

[Out]

int(tan(d*(a+b*ln(c*x^n))),x)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*(a+b*log(c*x^n))),x, algorithm="maxima")

[Out]

integrate(tan((b*log(c*x^n) + a)*d), x)

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*(a+b*log(c*x^n))),x, algorithm="fricas")

[Out]

integral(tan(b*d*log(c*x^n) + a*d), x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \tan {\left (d \left (a + b \log {\left (c x^{n} \right )}\right ) \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*(a+b*ln(c*x**n))),x)

[Out]

Integral(tan(d*(a + b*log(c*x**n))), x)

________________________________________________________________________________________

Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*(a+b*log(c*x^n))),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \mathrm {tan}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*(a + b*log(c*x^n))),x)

[Out]

int(tan(d*(a + b*log(c*x^n))), x)

________________________________________________________________________________________