\(\int x^3 \tanh (d (a+b \log (c x^n))) \, dx\) [172]

Optimal result

Integrand size = 17, antiderivative size = 59 \[ \int x^3 \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {x^4}{4}-\frac {1}{2} x^4 \operatorname {Hypergeometric2F1}\left (1,\frac {2}{b d n},1+\frac {2}{b d n},-e^{2 a d} \left (c x^n\right )^{2 b d}\right ) \]

[Out]

1/4*x^4-1/2*x^4*hypergeom([1, 2/b/d/n],[1+2/b/d/n],-exp(2*a*d)*(c*x^n)^(2*b*d))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {5658, 5656, 470, 371} \[ \int x^3 \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {x^4}{4}-\frac {1}{2} x^4 \operatorname {Hypergeometric2F1}\left (1,\frac {2}{b d n},1+\frac {2}{b d n},-e^{2 a d} \left (c x^n\right )^{2 b d}\right ) \]

[In]

Int[x^3*Tanh[d*(a + b*Log[c*x^n])],x]

[Out]

x^4/4 - (x^4*Hypergeometric2F1[1, 2/(b*d*n), 1 + 2/(b*d*n), -(E^(2*a*d)*(c*x^n)^(2*b*d))])/2

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*} \text {integral}& = \frac {\left (x^4 \left (c x^n\right )^{-4/n}\right ) \text {Subst}\left (\int x^{-1+\frac {4}{n}} \tanh (d (a+b \log (x))) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x^4 \left (c x^n\right )^{-4/n}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {4}{n}} \left (-1+e^{2 a d} x^{2 b d}\right )}{1+e^{2 a d} x^{2 b d}} \, dx,x,c x^n\right )}{n} \\ & = \frac {x^4}{4}-\frac {\left (2 x^4 \left (c x^n\right )^{-4/n}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {4}{n}}}{1+e^{2 a d} x^{2 b d}} \, dx,x,c x^n\right )}{n} \\ & = \frac {x^4}{4}-\frac {1}{2} x^4 \operatorname {Hypergeometric2F1}\left (1,\frac {2}{b d n},1+\frac {2}{b d n},-e^{2 a d} \left (c x^n\right )^{2 b d}\right ) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(127\) vs. \(2(59)=118\).

Time = 8.17 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.15 \[ \int x^3 \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {x^4 \left (2 e^{2 d \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,1+\frac {2}{b d n},2+\frac {2}{b d n},-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )-(2+b d n) \operatorname {Hypergeometric2F1}\left (1,\frac {2}{b d n},1+\frac {2}{b d n},-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{8+4 b d n} \]

[In]

Integrate[x^3*Tanh[d*(a + b*Log[c*x^n])],x]

[Out]

(x^4*(2*E^(2*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 + 2/(b*d*n), 2 + 2/(b*d*n), -E^(2*d*(a + b*Log[c*x^n
]))] - (2 + b*d*n)*Hypergeometric2F1[1, 2/(b*d*n), 1 + 2/(b*d*n), -E^(2*d*(a + b*Log[c*x^n]))]))/(8 + 4*b*d*n)

Maple [F]

\[\int x^{3} \tanh \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]

[In]

int(x^3*tanh(d*(a+b*ln(c*x^n))),x)

[Out]

int(x^3*tanh(d*(a+b*ln(c*x^n))),x)

Fricas [F]

\[ \int x^3 \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{3} \tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]

[In]

integrate(x^3*tanh(d*(a+b*log(c*x^n))),x, algorithm="fricas")

[Out]

integral(x^3*tanh(b*d*log(c*x^n) + a*d), x)

Sympy [F]

\[ \int x^3 \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x^{3} \tanh {\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]

[In]

integrate(x**3*tanh(d*(a+b*ln(c*x**n))),x)

[Out]

Integral(x**3*tanh(a*d + b*d*log(c*x**n)), x)

Maxima [F]

\[ \int x^3 \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{3} \tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]

[In]

integrate(x^3*tanh(d*(a+b*log(c*x^n))),x, algorithm="maxima")

[Out]

1/4*x^4 - 2*integrate(x^3/(c^(2*b*d)*e^(2*b*d*log(x^n) + 2*a*d) + 1), x)

Giac [F]

\[ \int x^3 \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{3} \tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]

[In]

integrate(x^3*tanh(d*(a+b*log(c*x^n))),x, algorithm="giac")

[Out]

integrate(x^3*tanh((b*log(c*x^n) + a)*d), x)

Mupad [F(-1)]

Timed out. \[ \int x^3 \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x^3\,\mathrm {tanh}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \]

[In]

int(x^3*tanh(d*(a + b*log(c*x^n))),x)

[Out]

int(x^3*tanh(d*(a + b*log(c*x^n))), x)