Integrand size = 21, antiderivative size = 135 \[ \int (e x)^m \tanh ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^{1+m} \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )^{-p} \left (-1+e^{2 a d} \left (c x^n\right )^{2 b d}\right )^p \operatorname {AppellF1}\left (\frac {1+m}{2 b d n},-p,p,1+\frac {1+m}{2 b d n},e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{e (1+m)} \]
(e*x)^(1+m)*(-1+exp(2*a*d)*(c*x^n)^(2*b*d))^p*AppellF1(1/2*(1+m)/b/d/n,-p, p,1+1/2*(1+m)/b/d/n,exp(2*a*d)*(c*x^n)^(2*b*d),-exp(2*a*d)*(c*x^n)^(2*b*d) )/e/(1+m)/((1-exp(2*a*d)*(c*x^n)^(2*b*d))^p)
Time = 1.28 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.29 \[ \int (e x)^m \tanh ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {x (e x)^m \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )^{-p} \left (\frac {-1+e^{2 a d} \left (c x^n\right )^{2 b d}}{1+e^{2 a d} \left (c x^n\right )^{2 b d}}\right )^p \left (1+e^{2 a d} \left (c x^n\right )^{2 b d}\right )^p \operatorname {AppellF1}\left (\frac {1+m}{2 b d n},-p,p,1+\frac {1+m}{2 b d n},e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{1+m} \]
(x*(e*x)^m*((-1 + E^(2*a*d)*(c*x^n)^(2*b*d))/(1 + E^(2*a*d)*(c*x^n)^(2*b*d )))^p*(1 + E^(2*a*d)*(c*x^n)^(2*b*d))^p*AppellF1[(1 + m)/(2*b*d*n), -p, p, 1 + (1 + m)/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d), -(E^(2*a*d)*(c*x^n)^(2* b*d))])/((1 + m)*(1 - E^(2*a*d)*(c*x^n)^(2*b*d))^p)
Time = 0.44 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6073, 6071, 1013, 1012}
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.
\(\displaystyle \int (e x)^m \tanh ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\) |
\(\Big \downarrow \) 6073 |
\(\displaystyle \frac {(e x)^{m+1} \left (c x^n\right )^{-\frac {m+1}{n}} \int \left (c x^n\right )^{\frac {m+1}{n}-1} \tanh ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right )d\left (c x^n\right )}{e n}\) |
\(\Big \downarrow \) 6071 |
\(\displaystyle \frac {(e x)^{m+1} \left (c x^n\right )^{-\frac {m+1}{n}} \int \left (c x^n\right )^{\frac {m+1}{n}-1} \left (e^{2 a d} \left (c x^n\right )^{2 b d}-1\right )^p \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )^{-p}d\left (c x^n\right )}{e n}\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle \frac {(e x)^{m+1} \left (c x^n\right )^{-\frac {m+1}{n}} \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )^{-p} \left (e^{2 a d} \left (c x^n\right )^{2 b d}-1\right )^p \int \left (c x^n\right )^{\frac {m+1}{n}-1} \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )^p \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )^{-p}d\left (c x^n\right )}{e n}\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle \frac {(e x)^{m+1} \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )^{-p} \left (e^{2 a d} \left (c x^n\right )^{2 b d}-1\right )^p \operatorname {AppellF1}\left (\frac {m+1}{2 b d n},-p,p,\frac {m+1}{2 b d n}+1,e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{e (m+1)}\) |
((e*x)^(1 + m)*(-1 + E^(2*a*d)*(c*x^n)^(2*b*d))^p*AppellF1[(1 + m)/(2*b*d* n), -p, p, 1 + (1 + m)/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d), -(E^(2*a*d)*( c*x^n)^(2*b*d))])/(e*(1 + m)*(1 - E^(2*a*d)*(c*x^n)^(2*b*d))^p)
3.2.93.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^ n/a))^FracPart[p]) Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] & & NeQ[m, n - 1] && !(IntegerQ[p] || GtQ[a, 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])
\[\int \left (e x \right )^{m} {\tanh \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}^{p}d x\]
\[ \int (e x)^m \tanh ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{p} \,d x } \]
Timed out. \[ \int (e x)^m \tanh ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\text {Timed out} \]
\[ \int (e x)^m \tanh ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{p} \,d x } \]
\[ \int (e x)^m \tanh ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{p} \,d x } \]
Timed out. \[ \int (e x)^m \tanh ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int {\mathrm {tanh}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^p\,{\left (e\,x\right )}^m \,d x \]