\(\int (e x)^m \tanh ^3(a+2 \log (x)) \, dx\) [162]

Optimal result

Integrand size = 15, antiderivative size = 176 \[ \int (e x)^m \tanh ^3(a+2 \log (x)) \, dx=\frac {(3+m) (5+m) (e x)^{1+m}}{8 e (1+m)}-\frac {(e x)^{1+m} \left (1-e^{2 a} x^4\right )^2}{4 e \left (1+e^{2 a} x^4\right )^2}-\frac {e^{-2 a} (e x)^{1+m} \left (e^{2 a} (3-m)+e^{4 a} (5+m) x^4\right )}{8 e \left (1+e^{2 a} x^4\right )}-\frac {\left (9+2 m+m^2\right ) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{4},\frac {5+m}{4},-e^{2 a} x^4\right )}{4 e (1+m)} \]

[Out]

1/8*(3+m)*(5+m)*(e*x)^(1+m)/e/(1+m)-1/4*(e*x)^(1+m)*(1-exp(2*a)*x^4)^2/e/(1+exp(2*a)*x^4)^2-1/8*(e*x)^(1+m)*(e
xp(2*a)*(3-m)+exp(4*a)*(5+m)*x^4)/e/exp(2*a)/(1+exp(2*a)*x^4)-1/4*(m^2+2*m+9)*(e*x)^(1+m)*hypergeom([1, 1/4+1/
4*m],[5/4+1/4*m],-exp(2*a)*x^4)/e/(1+m)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5656, 479, 591, 470, 371} \[ \int (e x)^m \tanh ^3(a+2 \log (x)) \, dx=-\frac {\left (m^2+2 m+9\right ) (e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{4},\frac {m+5}{4},-e^{2 a} x^4\right )}{4 e (m+1)}-\frac {e^{-2 a} \left (e^{4 a} (m+5) x^4+e^{2 a} (3-m)\right ) (e x)^{m+1}}{8 e \left (e^{2 a} x^4+1\right )}-\frac {\left (1-e^{2 a} x^4\right )^2 (e x)^{m+1}}{4 e \left (e^{2 a} x^4+1\right )^2}+\frac {(m+3) (m+5) (e x)^{m+1}}{8 e (m+1)} \]

[In]

Int[(e*x)^m*Tanh[a + 2*Log[x]]^3,x]

[Out]

((3 + m)*(5 + m)*(e*x)^(1 + m))/(8*e*(1 + m)) - ((e*x)^(1 + m)*(1 - E^(2*a)*x^4)^2)/(4*e*(1 + E^(2*a)*x^4)^2)
- ((e*x)^(1 + m)*(E^(2*a)*(3 - m) + E^(4*a)*(5 + m)*x^4))/(8*e*E^(2*a)*(1 + E^(2*a)*x^4)) - ((9 + 2*m + m^2)*(
e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, -(E^(2*a)*x^4)])/(4*e*(1 + m))

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 479

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-(c*b -
a*d))*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(a*b*e*n*(p + 1))), x] + Dist[1/(a*b*n*(p + 1)),
Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c*b - a*d)*(m + 1)) + d*(c*b*n*(
p + 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 591

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*b*g*n*(p + 1))), x] + Dis
t[1/(a*b*n*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(b*e*n*(p + 1) + (b*e - a*f)*(
m + 1)) + d*(b*e*n*(p + 1) + (b*e - a*f)*(m + n*q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 0] &&  !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])

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]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(e x)^m \left (-1+e^{2 a} x^4\right )^3}{\left (1+e^{2 a} x^4\right )^3} \, dx \\ & = -\frac {(e x)^{1+m} \left (1-e^{2 a} x^4\right )^2}{4 e \left (1+e^{2 a} x^4\right )^2}-\frac {1}{8} e^{-2 a} \int \frac {(e x)^m \left (-1+e^{2 a} x^4\right ) \left (-2 e^{2 a} (3-m)-2 e^{4 a} (5+m) x^4\right )}{\left (1+e^{2 a} x^4\right )^2} \, dx \\ & = -\frac {(e x)^{1+m} \left (1-e^{2 a} x^4\right )^2}{4 e \left (1+e^{2 a} x^4\right )^2}-\frac {e^{-2 a} (e x)^{1+m} \left (e^{2 a} (3-m)+e^{4 a} (5+m) x^4\right )}{8 e \left (1+e^{2 a} x^4\right )}+\frac {1}{32} e^{-4 a} \int \frac {(e x)^m \left (-4 e^{4 a} (1-m) (3-m)+4 e^{6 a} (3+m) (5+m) x^4\right )}{1+e^{2 a} x^4} \, dx \\ & = \frac {(3+m) (5+m) (e x)^{1+m}}{8 e (1+m)}-\frac {(e x)^{1+m} \left (1-e^{2 a} x^4\right )^2}{4 e \left (1+e^{2 a} x^4\right )^2}-\frac {e^{-2 a} (e x)^{1+m} \left (e^{2 a} (3-m)+e^{4 a} (5+m) x^4\right )}{8 e \left (1+e^{2 a} x^4\right )}+\frac {1}{4} \left (-9-2 m-m^2\right ) \int \frac {(e x)^m}{1+e^{2 a} x^4} \, dx \\ & = \frac {(3+m) (5+m) (e x)^{1+m}}{8 e (1+m)}-\frac {(e x)^{1+m} \left (1-e^{2 a} x^4\right )^2}{4 e \left (1+e^{2 a} x^4\right )^2}-\frac {e^{-2 a} (e x)^{1+m} \left (e^{2 a} (3-m)+e^{4 a} (5+m) x^4\right )}{8 e \left (1+e^{2 a} x^4\right )}-\frac {\left (9+2 m+m^2\right ) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{4},\frac {5+m}{4},-e^{2 a} x^4\right )}{4 e (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.63 \[ \int (e x)^m \tanh ^3(a+2 \log (x)) \, dx=-\frac {x (e x)^m \left (-1+6 \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{4},\frac {5+m}{4},-x^4 (\cosh (2 a)+\sinh (2 a))\right )-12 \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{4},\frac {5+m}{4},-x^4 (\cosh (2 a)+\sinh (2 a))\right )+8 \operatorname {Hypergeometric2F1}\left (3,\frac {1+m}{4},\frac {5+m}{4},-x^4 (\cosh (2 a)+\sinh (2 a))\right )\right )}{1+m} \]

[In]

Integrate[(e*x)^m*Tanh[a + 2*Log[x]]^3,x]

[Out]

-((x*(e*x)^m*(-1 + 6*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, -(x^4*(Cosh[2*a] + Sinh[2*a]))] - 12*Hypergeom
etric2F1[2, (1 + m)/4, (5 + m)/4, -(x^4*(Cosh[2*a] + Sinh[2*a]))] + 8*Hypergeometric2F1[3, (1 + m)/4, (5 + m)/
4, -(x^4*(Cosh[2*a] + Sinh[2*a]))]))/(1 + m))

Maple [F]

\[\int \left (e x \right )^{m} \tanh \left (a +2 \ln \left (x \right )\right )^{3}d x\]

[In]

int((e*x)^m*tanh(a+2*ln(x))^3,x)

[Out]

int((e*x)^m*tanh(a+2*ln(x))^3,x)

Fricas [F]

\[ \int (e x)^m \tanh ^3(a+2 \log (x)) \, dx=\int { \left (e x\right )^{m} \tanh \left (a + 2 \, \log \left (x\right )\right )^{3} \,d x } \]

[In]

integrate((e*x)^m*tanh(a+2*log(x))^3,x, algorithm="fricas")

[Out]

integral((e*x)^m*tanh(a + 2*log(x))^3, x)

Sympy [F]

\[ \int (e x)^m \tanh ^3(a+2 \log (x)) \, dx=\int \left (e x\right )^{m} \tanh ^{3}{\left (a + 2 \log {\left (x \right )} \right )}\, dx \]

[In]

integrate((e*x)**m*tanh(a+2*ln(x))**3,x)

[Out]

Integral((e*x)**m*tanh(a + 2*log(x))**3, x)

Maxima [F]

\[ \int (e x)^m \tanh ^3(a+2 \log (x)) \, dx=\int { \left (e x\right )^{m} \tanh \left (a + 2 \, \log \left (x\right )\right )^{3} \,d x } \]

[In]

integrate((e*x)^m*tanh(a+2*log(x))^3,x, algorithm="maxima")

[Out]

integrate((e*x)^m*tanh(a + 2*log(x))^3, x)

Giac [F]

\[ \int (e x)^m \tanh ^3(a+2 \log (x)) \, dx=\int { \left (e x\right )^{m} \tanh \left (a + 2 \, \log \left (x\right )\right )^{3} \,d x } \]

[In]

integrate((e*x)^m*tanh(a+2*log(x))^3,x, algorithm="giac")

[Out]

integrate((e*x)^m*tanh(a + 2*log(x))^3, x)

Mupad [F(-1)]

Timed out. \[ \int (e x)^m \tanh ^3(a+2 \log (x)) \, dx=\int {\mathrm {tanh}\left (a+2\,\ln \left (x\right )\right )}^3\,{\left (e\,x\right )}^m \,d x \]

[In]

int(tanh(a + 2*log(x))^3*(e*x)^m,x)

[Out]

int(tanh(a + 2*log(x))^3*(e*x)^m, x)