\(\int (c+d x) \tanh ^3(e+f x) \, dx\) [13]

Optimal result

Integrand size = 14, antiderivative size = 100 \[ \int (c+d x) \tanh ^3(e+f x) \, dx=\frac {d x}{2 f}-\frac {(c+d x)^2}{2 d}+\frac {(c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {d \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}-\frac {d \tanh (e+f x)}{2 f^2}-\frac {(c+d x) \tanh ^2(e+f x)}{2 f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3801, 3554, 8, 3799, 2221, 2317, 2438} \[ \int (c+d x) \tanh ^3(e+f x) \, dx=\frac {(c+d x) \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {(c+d x) \tanh ^2(e+f x)}{2 f}-\frac {(c+d x)^2}{2 d}+\frac {d \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}-\frac {d \tanh (e+f x)}{2 f^2}+\frac {d x}{2 f} \]

[In]

Int[(c + d*x)*Tanh[e + f*x]^3,x]

[Out]

(d*x)/(2*f) - (c + d*x)^2/(2*d) + ((c + d*x)*Log[1 + E^(2*(e + f*x))])/f + (d*PolyLog[2, -E^(2*(e + f*x))])/(2
*f^2) - (d*Tanh[e + f*x])/(2*f^2) - ((c + d*x)*Tanh[e + f*x]^2)/(2*f)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2221

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]
- Dist[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]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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]

Rule 2438

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]

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3799

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] + Dist[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]

Rule 3801

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(c + d*x)^m*((b*Tan[e
+ f*x])^(n - 1)/(f*(n - 1))), x] + (-Dist[b*d*(m/(f*(n - 1))), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x) \tanh ^2(e+f x)}{2 f}+\frac {d \int \tanh ^2(e+f x) \, dx}{2 f}+\int (c+d x) \tanh (e+f x) \, dx \\ & = -\frac {(c+d x)^2}{2 d}-\frac {d \tanh (e+f x)}{2 f^2}-\frac {(c+d x) \tanh ^2(e+f x)}{2 f}+2 \int \frac {e^{2 (e+f x)} (c+d x)}{1+e^{2 (e+f x)}} \, dx+\frac {d \int 1 \, dx}{2 f} \\ & = \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 \tanh (e+f x)}{2 f^2}-\frac {(c+d x) \tanh ^2(e+f x)}{2 f}-\frac {d \int \log \left (1+e^{2 (e+f x)}\right ) \, dx}{f} \\ & = \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 \tanh (e+f x)}{2 f^2}-\frac {(c+d x) \tanh ^2(e+f x)}{2 f}-\frac {d \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^2} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.81 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.97 \[ \int (c+d x) \tanh ^3(e+f x) \, dx=\frac {-d \operatorname {PolyLog}\left (2,-e^{-2 (e+f x)}\right )+d f x \text {sech}^2(e+f x)-d \text {sech}(e) \text {sech}(e+f x) \sinh (f x)+f \left (d f x^2+2 d x \log \left (1+e^{-2 (e+f x)}\right )+2 c \log (\cosh (e+f x))-c \tanh ^2(e+f x)\right )}{2 f^2} \]

[In]

Integrate[(c + d*x)*Tanh[e + f*x]^3,x]

[Out]

(-(d*PolyLog[2, -E^(-2*(e + f*x))]) + d*f*x*Sech[e + f*x]^2 - d*Sech[e]*Sech[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)

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.66

method	result	size
risch	\(-\frac {d \,x^{2}}{2}+c x +\frac {2 d f x \,{\mathrm e}^{2 f x +2 e}+2 c f \,{\mathrm e}^{2 f x +2 e}+{\mathrm e}^{2 f x +2 e} d +d}{f^{2} \left (1+{\mathrm e}^{2 f x +2 e}\right )^{2}}+\frac {c \ln \left (1+{\mathrm e}^{2 f x +2 e}\right )}{f}-\frac {2 c \ln \left ({\mathrm e}^{f x +e}\right )}{f}-\frac {2 d e x}{f}-\frac {d \,e^{2}}{f^{2}}+\frac {d \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x}{f}+\frac {d \operatorname {polylog}\left (2, -{\mathrm e}^{2 f x +2 e}\right )}{2 f^{2}}+\frac {2 e d \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}\)	\(166\)

[In]

int((d*x+c)*tanh(f*x+e)^3,x,method=_RETURNVERBOSE)

[Out]

-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,
-exp(2*f*x+2*e))/f^2+2/f^2*e*d*ln(exp(f*x+e))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 1462, normalized size of antiderivative = 14.62 \[ \int (c+d x) \tanh ^3(e+f x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)*tanh(f*x+e)^3,x, algorithm="fricas")

[Out]

-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*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*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)*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*f*x + d*e)*cosh(f*x + e)^4 + 4*(d*f*x + d*e)*cosh(f*x + e)*sinh(f*x + e)^3 + (d*f*x + d*e)*sinh(f*x + e)^
4 + d*f*x + 2*(d*f*x + d*e)*cosh(f*x + e)^2 + 2*(d*f*x + 3*(d*f*x + d*e)*cosh(f*x + e)^2 + d*e)*sinh(f*x + e)^
2 + d*e + 4*((d*f*x + d*e)*cosh(f*x + e)^3 + (d*f*x + d*e)*cosh(f*x + e))*sinh(f*x + e))*log(I*cosh(f*x + e) +
I*sinh(f*x + e) + 1) - 2*((d*f*x + d*e)*cosh(f*x + e)^4 + 4*(d*f*x + d*e)*cosh(f*x + e)*sinh(f*x + e)^3 + (d*
f*x + d*e)*sinh(f*x + e)^4 + d*f*x + 2*(d*f*x + d*e)*cosh(f*x + e)^2 + 2*(d*f*x + 3*(d*f*x + d*e)*cosh(f*x + e
)^2 + d*e)*sinh(f*x + e)^2 + d*e + 4*((d*f*x + d*e)*cosh(f*x + e)^3 + (d*f*x + d*e)*cosh(f*x + e))*sinh(f*x +
e))*log(-I*cosh(f*x + e) - I*sinh(f*x + e) + 1) + 4*((d*f^2*x^2 + 2*c*f^2*x - 2*d*e^2 + 4*c*e*f)*cosh(f*x + e)
^3 + (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))*sinh(f*x + e) - 2*d)/(f^2*
cosh(f*x + e)^4 + 4*f^2*cosh(f*x + e)*sinh(f*x + e)^3 + f^2*sinh(f*x + e)^4 + 2*f^2*cosh(f*x + e)^2 + 2*(3*f^2
*cosh(f*x + e)^2 + f^2)*sinh(f*x + e)^2 + f^2 + 4*(f^2*cosh(f*x + e)^3 + f^2*cosh(f*x + e))*sinh(f*x + e))

Sympy [F]

\[ \int (c+d x) \tanh ^3(e+f x) \, dx=\int \left (c + d x\right ) \tanh ^{3}{\left (e + f x \right )}\, dx \]

[In]

integrate((d*x+c)*tanh(f*x+e)**3,x)

[Out]

Integral((c + d*x)*tanh(e + f*x)**3, x)

Maxima [F]

\[ \int (c+d x) \tanh ^3(e+f x) \, dx=\int { {\left (d x + c\right )} \tanh \left (f x + e\right )^{3} \,d x } \]

[In]

integrate((d*x+c)*tanh(f*x+e)^3,x, algorithm="maxima")

[Out]

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))

Giac [F]

\[ \int (c+d x) \tanh ^3(e+f x) \, dx=\int { {\left (d x + c\right )} \tanh \left (f x + e\right )^{3} \,d x } \]

[In]

integrate((d*x+c)*tanh(f*x+e)^3,x, algorithm="giac")

[Out]

integrate((d*x + c)*tanh(f*x + e)^3, x)

Mupad [F(-1)]

Timed out. \[ \int (c+d x) \tanh ^3(e+f x) \, dx=\int {\mathrm {tanh}\left (e+f\,x\right )}^3\,\left (c+d\,x\right ) \,d x \]

[In]

int(tanh(e + f*x)^3*(c + d*x),x)

[Out]

int(tanh(e + f*x)^3*(c + d*x), x)