Integrand size = 14, antiderivative size = 100 \[ \int (c+d x) \tanh ^3(e+f x) \, dx=\frac {d x}{2 f}-\frac {(c+d x)^2}{2 d}+\frac {(c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {d \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}-\frac {d \tanh (e+f x)}{2 f^2}-\frac {(c+d x) \tanh ^2(e+f x)}{2 f} \]
1/2*d*x/f-1/2*(d*x+c)^2/d+(d*x+c)*ln(1+exp(2*f*x+2*e))/f+1/2*d*polylog(2,- exp(2*f*x+2*e))/f^2-1/2*d*tanh(f*x+e)/f^2-1/2*(d*x+c)*tanh(f*x+e)^2/f
Time = 1.81 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.97 \[ \int (c+d x) \tanh ^3(e+f x) \, dx=\frac {-d \operatorname {PolyLog}\left (2,-e^{-2 (e+f x)}\right )+d f x \text {sech}^2(e+f x)-d \text {sech}(e) \text {sech}(e+f x) \sinh (f x)+f \left (d f x^2+2 d x \log \left (1+e^{-2 (e+f x)}\right )+2 c \log (\cosh (e+f x))-c \tanh ^2(e+f x)\right )}{2 f^2} \]
(-(d*PolyLog[2, -E^(-2*(e + f*x))]) + d*f*x*Sech[e + f*x]^2 - d*Sech[e]*Se ch[e + f*x]*Sinh[f*x] + f*(d*f*x^2 + 2*d*x*Log[1 + E^(-2*(e + f*x))] + 2*c *Log[Cosh[e + f*x]] - c*Tanh[e + f*x]^2))/(2*f^2)
Result contains complex when optimal does not.
Time = 0.56 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.17, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 26, 4203, 25, 26, 3042, 25, 26, 3954, 24, 4201, 2620, 2715, 2838}
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.
| \(\displaystyle \int (c+d x) \tanh ^3(e+f x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int i (c+d x) \tan (i e+i f x)^3dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int (c+d x) \tan (i e+i f x)^3dx\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle i \left (-\int i (c+d x) \tanh (e+f x)dx+\frac {i d \int -\tanh ^2(e+f x)dx}{2 f}+\frac {i (c+d x) \tanh ^2(e+f x)}{2 f}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (-\int i (c+d x) \tanh (e+f x)dx-\frac {i d \int \tanh ^2(e+f x)dx}{2 f}+\frac {i (c+d x) \tanh ^2(e+f x)}{2 f}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-i \int (c+d x) \tanh (e+f x)dx-\frac {i d \int \tanh ^2(e+f x)dx}{2 f}+\frac {i (c+d x) \tanh ^2(e+f x)}{2 f}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-i \int -i (c+d x) \tan (i e+i f x)dx-\frac {i d \int -\tan (i e+i f x)^2dx}{2 f}+\frac {i (c+d x) \tanh ^2(e+f x)}{2 f}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (-i \int -i (c+d x) \tan (i e+i f x)dx+\frac {i d \int \tan (i e+i f x)^2dx}{2 f}+\frac {i (c+d x) \tanh ^2(e+f x)}{2 f}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\int (c+d x) \tan (i e+i f x)dx+\frac {i d \int \tan (i e+i f x)^2dx}{2 f}+\frac {i (c+d x) \tanh ^2(e+f x)}{2 f}\right )\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle i \left (-\int (c+d x) \tan (i e+i f x)dx+\frac {i d \left (\frac {\tanh (e+f x)}{f}-\int 1dx\right )}{2 f}+\frac {i (c+d x) \tanh ^2(e+f x)}{2 f}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle i \left (-\int (c+d x) \tan (i e+i f x)dx+\frac {i (c+d x) \tanh ^2(e+f x)}{2 f}+\frac {i d \left (\frac {\tanh (e+f x)}{f}-x\right )}{2 f}\right )\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle i \left (-2 i \int \frac {e^{2 (e+f x)} (c+d x)}{1+e^{2 (e+f x)}}dx+\frac {i (c+d x) \tanh ^2(e+f x)}{2 f}+\frac {i (c+d x)^2}{2 d}+\frac {i d \left (\frac {\tanh (e+f x)}{f}-x\right )}{2 f}\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle i \left (-2 i \left (\frac {(c+d x) \log \left (e^{2 (e+f x)}+1\right )}{2 f}-\frac {d \int \log \left (1+e^{2 (e+f x)}\right )dx}{2 f}\right )+\frac {i (c+d x) \tanh ^2(e+f x)}{2 f}+\frac {i (c+d x)^2}{2 d}+\frac {i d \left (\frac {\tanh (e+f x)}{f}-x\right )}{2 f}\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle i \left (-2 i \left (\frac {(c+d x) \log \left (e^{2 (e+f x)}+1\right )}{2 f}-\frac {d \int e^{-2 (e+f x)} \log \left (1+e^{2 (e+f x)}\right )de^{2 (e+f x)}}{4 f^2}\right )+\frac {i (c+d x) \tanh ^2(e+f x)}{2 f}+\frac {i (c+d x)^2}{2 d}+\frac {i d \left (\frac {\tanh (e+f x)}{f}-x\right )}{2 f}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle i \left (-2 i \left (\frac {(c+d x) \log \left (e^{2 (e+f x)}+1\right )}{2 f}+\frac {d \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{4 f^2}\right )+\frac {i (c+d x) \tanh ^2(e+f x)}{2 f}+\frac {i (c+d x)^2}{2 d}+\frac {i d \left (\frac {\tanh (e+f x)}{f}-x\right )}{2 f}\right )\) |
I*(((I/2)*(c + d*x)^2)/d - (2*I)*(((c + d*x)*Log[1 + E^(2*(e + f*x))])/(2* f) + (d*PolyLog[2, -E^(2*(e + f*x))])/(4*f^2)) + ((I/2)*(c + d*x)*Tanh[e + f*x]^2)/f + ((I/2)*d*(-x + Tanh[e + f*x]/f))/f)
3.1.13.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symb ol] :> Simp[b*(c + d*x)^m*((b*Tan[e + f*x])^(n - 1)/(f*(n - 1))), x] + (-Si mp[b*d*(m/(f*(n - 1))) Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1), x] , x] - Simp[b^2 Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; Free Q[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 0]
Time = 0.06 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.66
| method | result | size |
| risch | \(-\frac {d \,x^{2}}{2}+c x +\frac {2 d f x \,{\mathrm e}^{2 f x +2 e}+2 c f \,{\mathrm e}^{2 f x +2 e}+{\mathrm e}^{2 f x +2 e} d +d}{f^{2} \left (1+{\mathrm e}^{2 f x +2 e}\right )^{2}}+\frac {c \ln \left (1+{\mathrm e}^{2 f x +2 e}\right )}{f}-\frac {2 c \ln \left ({\mathrm e}^{f x +e}\right )}{f}-\frac {2 d e x}{f}-\frac {d \,e^{2}}{f^{2}}+\frac {d \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x}{f}+\frac {d \operatorname {polylog}\left (2, -{\mathrm e}^{2 f x +2 e}\right )}{2 f^{2}}+\frac {2 e d \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}\) | \(166\) |
-1/2*d*x^2+c*x+(2*d*f*x*exp(2*f*x+2*e)+2*c*f*exp(2*f*x+2*e)+exp(2*f*x+2*e) *d+d)/f^2/(1+exp(2*f*x+2*e))^2+1/f*c*ln(1+exp(2*f*x+2*e))-2/f*c*ln(exp(f*x +e))-2/f*d*e*x-1/f^2*d*e^2+1/f*d*ln(1+exp(2*f*x+2*e))*x+1/2*d*polylog(2,-e xp(2*f*x+2*e))/f^2+2/f^2*e*d*ln(exp(f*x+e))
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 1462, normalized size of antiderivative = 14.62 \[ \int (c+d x) \tanh ^3(e+f x) \, dx=\text {Too large to display} \]
-1/2*(d*f^2*x^2 + (d*f^2*x^2 + 2*c*f^2*x - 2*d*e^2 + 4*c*e*f)*cosh(f*x + e )^4 + 4*(d*f^2*x^2 + 2*c*f^2*x - 2*d*e^2 + 4*c*e*f)*cosh(f*x + e)*sinh(f*x + e)^3 + (d*f^2*x^2 + 2*c*f^2*x - 2*d*e^2 + 4*c*e*f)*sinh(f*x + e)^4 + 2* c*f^2*x - 2*d*e^2 + 4*c*e*f + 2*(d*f^2*x^2 - 2*d*e^2 + 2*(2*c*e - c)*f + 2 *(c*f^2 - d*f)*x - d)*cosh(f*x + e)^2 + 2*(d*f^2*x^2 - 2*d*e^2 + 3*(d*f^2* x^2 + 2*c*f^2*x - 2*d*e^2 + 4*c*e*f)*cosh(f*x + e)^2 + 2*(2*c*e - c)*f + 2 *(c*f^2 - d*f)*x - d)*sinh(f*x + e)^2 - 2*(d*cosh(f*x + e)^4 + 4*d*cosh(f* x + e)*sinh(f*x + e)^3 + d*sinh(f*x + e)^4 + 2*d*cosh(f*x + e)^2 + 2*(3*d* cosh(f*x + e)^2 + d)*sinh(f*x + e)^2 + 4*(d*cosh(f*x + e)^3 + d*cosh(f*x + e))*sinh(f*x + e) + d)*dilog(I*cosh(f*x + e) + I*sinh(f*x + e)) - 2*(d*co sh(f*x + e)^4 + 4*d*cosh(f*x + e)*sinh(f*x + e)^3 + d*sinh(f*x + e)^4 + 2* d*cosh(f*x + e)^2 + 2*(3*d*cosh(f*x + e)^2 + d)*sinh(f*x + e)^2 + 4*(d*cos h(f*x + e)^3 + d*cosh(f*x + e))*sinh(f*x + e) + d)*dilog(-I*cosh(f*x + e) - I*sinh(f*x + e)) + 2*((d*e - c*f)*cosh(f*x + e)^4 + 4*(d*e - c*f)*cosh(f *x + e)*sinh(f*x + e)^3 + (d*e - c*f)*sinh(f*x + e)^4 + 2*(d*e - c*f)*cosh (f*x + e)^2 + 2*(3*(d*e - c*f)*cosh(f*x + e)^2 + d*e - c*f)*sinh(f*x + e)^ 2 + d*e - c*f + 4*((d*e - c*f)*cosh(f*x + e)^3 + (d*e - c*f)*cosh(f*x + e) )*sinh(f*x + e))*log(cosh(f*x + e) + sinh(f*x + e) + I) + 2*((d*e - c*f)*c osh(f*x + e)^4 + 4*(d*e - c*f)*cosh(f*x + e)*sinh(f*x + e)^3 + (d*e - c*f) *sinh(f*x + e)^4 + 2*(d*e - c*f)*cosh(f*x + e)^2 + 2*(3*(d*e - c*f)*cos...
\[ \int (c+d x) \tanh ^3(e+f x) \, dx=\int \left (c + d x\right ) \tanh ^{3}{\left (e + f x \right )}\, dx \]
\[ \int (c+d x) \tanh ^3(e+f x) \, dx=\int { {\left (d x + c\right )} \tanh \left (f x + e\right )^{3} \,d x } \]
c*(x + e/f + log(e^(-2*f*x - 2*e) + 1)/f + 2*e^(-2*f*x - 2*e)/(f*(2*e^(-2* f*x - 2*e) + e^(-4*f*x - 4*e) + 1))) + 1/2*d*((f^2*x^2*e^(4*f*x + 4*e) + f ^2*x^2 + 2*(f^2*x^2*e^(2*e) + 2*f*x*e^(2*e) + e^(2*e))*e^(2*f*x) + 2)/(f^2 *e^(4*f*x + 4*e) + 2*f^2*e^(2*f*x + 2*e) + f^2) - 4*integrate(x/(e^(2*f*x + 2*e) + 1), x))
\[ \int (c+d x) \tanh ^3(e+f x) \, dx=\int { {\left (d x + c\right )} \tanh \left (f x + e\right )^{3} \,d x } \]
Timed out. \[ \int (c+d x) \tanh ^3(e+f x) \, dx=\int {\mathrm {tanh}\left (e+f\,x\right )}^3\,\left (c+d\,x\right ) \,d x \]