Integrand size = 19, antiderivative size = 88 \[ \int (e x)^m \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^{1+m}}{e (1+m)}-\frac {2 (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2 b d n},1+\frac {1+m}{2 b d n},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{e (1+m)} \]
(e*x)^(1+m)/e/(1+m)-2*(e*x)^(1+m)*hypergeom([1, 1/2*(1+m)/b/d/n],[1+1/2*(1 +m)/b/d/n],-exp(2*a*d)*(c*x^n)^(2*b*d))/e/(1+m)
Time = 13.66 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.82 \[ \int (e x)^m \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {x (e x)^m \left (-\operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2 b d n},1+\frac {1+m}{2 b d n},-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+\frac {e^{2 a d} (1+m) \left (c x^n\right )^{2 b d} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m+2 b d n}{2 b d n},\frac {1+m+4 b d n}{2 b d n},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{1+m+2 b d n}\right )}{1+m} \]
(x*(e*x)^m*(-Hypergeometric2F1[1, (1 + m)/(2*b*d*n), 1 + (1 + m)/(2*b*d*n) , -E^(2*d*(a + b*Log[c*x^n]))] + (E^(2*a*d)*(1 + m)*(c*x^n)^(2*b*d)*Hyperg eometric2F1[1, (1 + m + 2*b*d*n)/(2*b*d*n), (1 + m + 4*b*d*n)/(2*b*d*n), - (E^(2*a*d)*(c*x^n)^(2*b*d))])/(1 + m + 2*b*d*n)))/(1 + m)
Time = 0.37 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.41, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6073, 6071, 959, 888}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (e x)^m \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\) |
\(\Big \downarrow \) 6073 |
\(\displaystyle \frac {(e x)^{m+1} \left (c x^n\right )^{-\frac {m+1}{n}} \int \left (c x^n\right )^{\frac {m+1}{n}-1} \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right )d\left (c x^n\right )}{e n}\) |
\(\Big \downarrow \) 6071 |
\(\displaystyle \frac {(e x)^{m+1} \left (c x^n\right )^{-\frac {m+1}{n}} \int \frac {\left (c x^n\right )^{\frac {m+1}{n}-1} \left (e^{2 a d} \left (c x^n\right )^{2 b d}-1\right )}{e^{2 a d} \left (c x^n\right )^{2 b d}+1}d\left (c x^n\right )}{e n}\) |
\(\Big \downarrow \) 959 |
\(\displaystyle \frac {(e x)^{m+1} \left (c x^n\right )^{-\frac {m+1}{n}} \left (\frac {n \left (c x^n\right )^{\frac {m+1}{n}}}{m+1}-2 \int \frac {\left (c x^n\right )^{\frac {m+1}{n}-1}}{e^{2 a d} \left (c x^n\right )^{2 b d}+1}d\left (c x^n\right )\right )}{e n}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {(e x)^{m+1} \left (c x^n\right )^{-\frac {m+1}{n}} \left (\frac {n \left (c x^n\right )^{\frac {m+1}{n}}}{m+1}-\frac {2 n \left (c x^n\right )^{\frac {m+1}{n}} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2 b d n},\frac {m+1}{2 b d n}+1,-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{m+1}\right )}{e n}\) |
((e*x)^(1 + m)*((n*(c*x^n)^((1 + m)/n))/(1 + m) - (2*n*(c*x^n)^((1 + m)/n) *Hypergeometric2F1[1, (1 + m)/(2*b*d*n), 1 + (1 + m)/(2*b*d*n), -(E^(2*a*d )*(c*x^n)^(2*b*d))])/(1 + m)))/(e*n*(c*x^n)^((1 + m)/n))
3.2.89.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
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] - Simp[(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]
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]
Int[((e_.)*(x_))^(m_.)*Tanh[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p _.), x_Symbol] :> Simp[(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])
\[\int \left (e x \right )^{m} \tanh \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]
\[ \int (e x)^m \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
\[ \int (e x)^m \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \left (e x\right )^{m} \tanh {\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
\[ \int (e x)^m \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
\[ \int (e x)^m \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
Timed out. \[ \int (e x)^m \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \mathrm {tanh}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )\,{\left (e\,x\right )}^m \,d x \]