3.2.77 \(\int \frac {\tanh (d (a+b \log (c x^n)))}{x^2} \, dx\) [177]

Optimal. Leaf size=59 \[ -\frac {1}{x}+\frac {2 \, _2F_1\left (1,-\frac {1}{2 b d n};1-\frac {1}{2 b d n};-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{x} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {5658, 5656, 470, 371} \begin {gather*} \frac {2 \, _2F_1\left (1,-\frac {1}{2 b d n};1-\frac {1}{2 b d n};-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{x}-\frac {1}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tanh[d*(a + b*Log[c*x^n])]/x^2,x]

[Out]

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

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 5656

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

Rule 5658

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

Rubi steps

\begin {align*} \int \frac {\tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx &=\int \frac {\tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(126\) vs. \(2(59)=118\).
time = 2.29, size = 126, normalized size = 2.14 \begin {gather*} \frac {\frac {e^{2 d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1-\frac {1}{2 b d n};2-\frac {1}{2 b d n};-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )}{-1+2 b d n}+\, _2F_1\left (1,-\frac {1}{2 b d n};1-\frac {1}{2 b d n};-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tanh[d*(a + b*Log[c*x^n])]/x^2,x]

[Out]

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

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Maple [F]
time = 0.48, size = 0, normalized size = 0.00 \[\int \frac {\tanh \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(d*(a+b*ln(c*x^n)))/x^2,x)

[Out]

int(tanh(d*(a+b*ln(c*x^n)))/x^2,x)

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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(tanh(d*(a+b*log(c*x^n)))/x^2,x, algorithm="maxima")

[Out]

-1/x - 2*integrate(1/(c^(2*b*d)*x^2*e^(2*b*d*log(x^n) + 2*a*d) + x^2), x)

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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(tanh(d*(a+b*log(c*x^n)))/x^2,x, algorithm="fricas")

[Out]

integral(tanh(b*d*log(c*x^n) + a*d)/x^2, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh {\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*(a+b*ln(c*x**n)))/x**2,x)

[Out]

Integral(tanh(a*d + b*d*log(c*x**n))/x**2, x)

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Giac [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(tanh(d*(a+b*log(c*x^n)))/x^2,x, algorithm="giac")

[Out]

integrate(tanh((b*log(c*x^n) + a)*d)/x^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\mathrm {tanh}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(d*(a + b*log(c*x^n)))/x^2,x)

[Out]

int(tanh(d*(a + b*log(c*x^n)))/x^2, x)

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