\(\int (c+d x)^2 \tanh ^3(e+f x) \, dx\) [12]
Optimal result
Integrand size = 16, antiderivative size = 157 \[
\int (c+d x)^2 \tanh ^3(e+f x) \, dx=\frac {c d x}{f}+\frac {d^2 x^2}{2 f}-\frac {(c+d x)^3}{3 d}+\frac {(c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {d^2 \log (\cosh (e+f x))}{f^3}+\frac {d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}-\frac {d^2 \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}-\frac {d (c+d x) \tanh (e+f x)}{f^2}-\frac {(c+d x)^2 \tanh ^2(e+f x)}{2 f}
\]
[Out]
c*d*x/f+1/2*d^2*x^2/f-1/3*(d*x+c)^3/d+(d*x+c)^2*ln(1+exp(2*f*x+2*e))/f+d^2*ln(cosh(f*x+e))/f^3+d*(d*x+c)*polyl
og(2,-exp(2*f*x+2*e))/f^2-1/2*d^2*polylog(3,-exp(2*f*x+2*e))/f^3-d*(d*x+c)*tanh(f*x+e)/f^2-1/2*(d*x+c)^2*tanh(
f*x+e)^2/f
Rubi [A] (verified)
Time = 0.18 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of
steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3801, 3556, 3799, 2221, 2611,
2320, 6724} \[
\int (c+d x)^2 \tanh ^3(e+f x) \, dx=\frac {d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}-\frac {d (c+d x) \tanh (e+f x)}{f^2}+\frac {(c+d x)^2 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {(c+d x)^2 \tanh ^2(e+f x)}{2 f}+\frac {c d x}{f}-\frac {(c+d x)^3}{3 d}-\frac {d^2 \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}+\frac {d^2 \log (\cosh (e+f x))}{f^3}+\frac {d^2 x^2}{2 f}
\]
[In]
Int[(c + d*x)^2*Tanh[e + f*x]^3,x]
[Out]
(c*d*x)/f + (d^2*x^2)/(2*f) - (c + d*x)^3/(3*d) + ((c + d*x)^2*Log[1 + E^(2*(e + f*x))])/f + (d^2*Log[Cosh[e +
f*x]])/f^3 + (d*(c + d*x)*PolyLog[2, -E^(2*(e + f*x))])/f^2 - (d^2*PolyLog[3, -E^(2*(e + f*x))])/(2*f^3) - (d
*(c + d*x)*Tanh[e + f*x])/f^2 - ((c + d*x)^2*Tanh[e + f*x]^2)/(2*f)
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 2320
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]
Rule 2611
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]
Rule 3556
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]
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]
Rule 6724
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]
Rubi steps \begin{align*}
\text {integral}& = -\frac {(c+d x)^2 \tanh ^2(e+f x)}{2 f}+\frac {d \int (c+d x) \tanh ^2(e+f x) \, dx}{f}+\int (c+d x)^2 \tanh (e+f x) \, dx \\ & = -\frac {(c+d x)^3}{3 d}-\frac {d (c+d x) \tanh (e+f x)}{f^2}-\frac {(c+d x)^2 \tanh ^2(e+f x)}{2 f}+2 \int \frac {e^{2 (e+f x)} (c+d x)^2}{1+e^{2 (e+f x)}} \, dx+\frac {d^2 \int \tanh (e+f x) \, dx}{f^2}+\frac {d \int (c+d x) \, dx}{f} \\ & = \frac {c d x}{f}+\frac {d^2 x^2}{2 f}-\frac {(c+d x)^3}{3 d}+\frac {(c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {d^2 \log (\cosh (e+f x))}{f^3}-\frac {d (c+d x) \tanh (e+f x)}{f^2}-\frac {(c+d x)^2 \tanh ^2(e+f x)}{2 f}-\frac {(2 d) \int (c+d x) \log \left (1+e^{2 (e+f x)}\right ) \, dx}{f} \\ & = \frac {c d x}{f}+\frac {d^2 x^2}{2 f}-\frac {(c+d x)^3}{3 d}+\frac {(c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {d^2 \log (\cosh (e+f x))}{f^3}+\frac {d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}-\frac {d (c+d x) \tanh (e+f x)}{f^2}-\frac {(c+d x)^2 \tanh ^2(e+f x)}{2 f}-\frac {d^2 \int \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right ) \, dx}{f^2} \\ & = \frac {c d x}{f}+\frac {d^2 x^2}{2 f}-\frac {(c+d x)^3}{3 d}+\frac {(c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {d^2 \log (\cosh (e+f x))}{f^3}+\frac {d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}-\frac {d (c+d x) \tanh (e+f x)}{f^2}-\frac {(c+d x)^2 \tanh ^2(e+f x)}{2 f}-\frac {d^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^3} \\ & = \frac {c d x}{f}+\frac {d^2 x^2}{2 f}-\frac {(c+d x)^3}{3 d}+\frac {(c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {d^2 \log (\cosh (e+f x))}{f^3}+\frac {d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}-\frac {d^2 \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}-\frac {d (c+d x) \tanh (e+f x)}{f^2}-\frac {(c+d x)^2 \tanh ^2(e+f x)}{2 f} \\
\end{align*}
Mathematica [A] (verified)
Time = 3.00 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.44
\[
\int (c+d x)^2 \tanh ^3(e+f x) \, dx=\frac {1}{6} \left (\frac {4 x \left (-3 c^2 e^{2 e} f^2+3 c d f^2 x+d^2 \left (-3 e^{2 e}+f^2 x^2\right )\right )}{\left (1+e^{2 e}\right ) f^2}+\frac {6 d x (2 c+d x) \log \left (1+e^{-2 (e+f x)}\right )}{f}+\frac {6 \left (d^2+c^2 f^2\right ) \log \left (1+e^{2 (e+f x)}\right )}{f^3}-\frac {6 d (c+d x) \operatorname {PolyLog}\left (2,-e^{-2 (e+f x)}\right )}{f^2}-\frac {3 d^2 \operatorname {PolyLog}\left (3,-e^{-2 (e+f x)}\right )}{f^3}+\frac {3 (c+d x)^2 \text {sech}^2(e+f x)}{f}-\frac {6 d (c+d x) \text {sech}(e) \text {sech}(e+f x) \sinh (f x)}{f^2}+2 x \left (3 c^2+3 c d x+d^2 x^2\right ) \tanh (e)\right )
\]
[In]
Integrate[(c + d*x)^2*Tanh[e + f*x]^3,x]
[Out]
((4*x*(-3*c^2*E^(2*e)*f^2 + 3*c*d*f^2*x + d^2*(-3*E^(2*e) + f^2*x^2)))/((1 + E^(2*e))*f^2) + (6*d*x*(2*c + d*x
)*Log[1 + E^(-2*(e + f*x))])/f + (6*(d^2 + c^2*f^2)*Log[1 + E^(2*(e + f*x))])/f^3 - (6*d*(c + d*x)*PolyLog[2,
-E^(-2*(e + f*x))])/f^2 - (3*d^2*PolyLog[3, -E^(-2*(e + f*x))])/f^3 + (3*(c + d*x)^2*Sech[e + f*x]^2)/f - (6*d
*(c + d*x)*Sech[e]*Sech[e + f*x]*Sinh[f*x])/f^2 + 2*x*(3*c^2 + 3*c*d*x + d^2*x^2)*Tanh[e])/6
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(374\) vs. \(2(149)=298\).
Time = 0.24 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.39
| | |
method | result | size |
| | |
risch |
\(-\frac {d^{2} x^{3}}{3}-d c \,x^{2}+c^{2} x +\frac {c^{3}}{3 d}+\frac {2 d^{2} f \,x^{2} {\mathrm e}^{2 f x +2 e}+4 c d f x \,{\mathrm e}^{2 f x +2 e}+2 c^{2} f \,{\mathrm e}^{2 f x +2 e}+2 d^{2} x \,{\mathrm e}^{2 f x +2 e}+2 c d \,{\mathrm e}^{2 f x +2 e}+2 x \,d^{2}+2 c d}{f^{2} \left (1+{\mathrm e}^{2 f x +2 e}\right )^{2}}+\frac {2 d^{2} e^{2} x}{f^{2}}+\frac {d^{2} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x^{2}}{f}+\frac {d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{2 f x +2 e}\right ) x}{f^{2}}-\frac {2 d c \,e^{2}}{f^{2}}+\frac {d c \operatorname {polylog}\left (2, -{\mathrm e}^{2 f x +2 e}\right )}{f^{2}}-\frac {2 e^{2} d^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {c^{2} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right )}{f}-\frac {2 c^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f}+\frac {4 c e d \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}-\frac {4 d c e x}{f}+\frac {2 d c \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x}{f}+\frac {4 d^{2} e^{3}}{3 f^{3}}-\frac {d^{2} \operatorname {polylog}\left (3, -{\mathrm e}^{2 f x +2 e}\right )}{2 f^{3}}+\frac {d^{2} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right )}{f^{3}}-\frac {2 d^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}\) |
\(375\) |
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[In]
int((d*x+c)^2*tanh(f*x+e)^3,x,method=_RETURNVERBOSE)
[Out]
-1/3*d^2*x^3-d*c*x^2+c^2*x+1/3/d*c^3+2*(d^2*f*x^2*exp(2*f*x+2*e)+2*c*d*f*x*exp(2*f*x+2*e)+c^2*f*exp(2*f*x+2*e)
+d^2*x*exp(2*f*x+2*e)+c*d*exp(2*f*x+2*e)+x*d^2+c*d)/f^2/(1+exp(2*f*x+2*e))^2+2/f^2*d^2*e^2*x+1/f*d^2*ln(1+exp(
2*f*x+2*e))*x^2+1/f^2*d^2*polylog(2,-exp(2*f*x+2*e))*x-2/f^2*d*c*e^2+1/f^2*d*c*polylog(2,-exp(2*f*x+2*e))-2/f^
3*e^2*d^2*ln(exp(f*x+e))+1/f*c^2*ln(1+exp(2*f*x+2*e))-2/f*c^2*ln(exp(f*x+e))+4/f^2*c*e*d*ln(exp(f*x+e))-4/f*d*
c*e*x+2/f*d*c*ln(1+exp(2*f*x+2*e))*x+4/3/f^3*d^2*e^3-1/2*d^2*polylog(3,-exp(2*f*x+2*e))/f^3+1/f^3*d^2*ln(1+exp
(2*f*x+2*e))-2/f^3*d^2*ln(exp(f*x+e))
Fricas [C] (verification not implemented)
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 3071, normalized size of antiderivative = 19.56
\[
\int (c+d x)^2 \tanh ^3(e+f x) \, dx=\text {Too large to display}
\]
[In]
integrate((d*x+c)^2*tanh(f*x+e)^3,x, algorithm="fricas")
[Out]
-1/3*(d^2*f^3*x^3 + 3*c*d*f^3*x^2 + 3*c^2*f^3*x + 2*d^2*e^3 + 6*c^2*e*f^2 + (d^2*f^3*x^3 + 3*c*d*f^3*x^2 + 2*d
^2*e^3 - 6*c*d*e^2*f + 6*c^2*e*f^2 + 6*d^2*e + 3*(c^2*f^3 + 2*d^2*f)*x)*cosh(f*x + e)^4 + 4*(d^2*f^3*x^3 + 3*c
*d*f^3*x^2 + 2*d^2*e^3 - 6*c*d*e^2*f + 6*c^2*e*f^2 + 6*d^2*e + 3*(c^2*f^3 + 2*d^2*f)*x)*cosh(f*x + e)*sinh(f*x
+ e)^3 + (d^2*f^3*x^3 + 3*c*d*f^3*x^2 + 2*d^2*e^3 - 6*c*d*e^2*f + 6*c^2*e*f^2 + 6*d^2*e + 3*(c^2*f^3 + 2*d^2*
f)*x)*sinh(f*x + e)^4 + 6*d^2*e + 2*(d^2*f^3*x^3 + 2*d^2*e^3 + 6*d^2*e + 3*(2*c^2*e - c^2)*f^2 + 3*(c*d*f^3 -
d^2*f^2)*x^2 - 3*(2*c*d*e^2 + c*d)*f + 3*(c^2*f^3 - 2*c*d*f^2 + d^2*f)*x)*cosh(f*x + e)^2 + 2*(d^2*f^3*x^3 + 2
*d^2*e^3 + 6*d^2*e + 3*(2*c^2*e - c^2)*f^2 + 3*(c*d*f^3 - d^2*f^2)*x^2 + 3*(d^2*f^3*x^3 + 3*c*d*f^3*x^2 + 2*d^
2*e^3 - 6*c*d*e^2*f + 6*c^2*e*f^2 + 6*d^2*e + 3*(c^2*f^3 + 2*d^2*f)*x)*cosh(f*x + e)^2 - 3*(2*c*d*e^2 + c*d)*f
+ 3*(c^2*f^3 - 2*c*d*f^2 + d^2*f)*x)*sinh(f*x + e)^2 - 6*(c*d*e^2 + c*d)*f - 6*((d^2*f*x + c*d*f)*cosh(f*x +
e)^4 + 4*(d^2*f*x + c*d*f)*cosh(f*x + e)*sinh(f*x + e)^3 + (d^2*f*x + c*d*f)*sinh(f*x + e)^4 + d^2*f*x + c*d*f
+ 2*(d^2*f*x + c*d*f)*cosh(f*x + e)^2 + 2*(d^2*f*x + c*d*f + 3*(d^2*f*x + c*d*f)*cosh(f*x + e)^2)*sinh(f*x +
e)^2 + 4*((d^2*f*x + c*d*f)*cosh(f*x + e)^3 + (d^2*f*x + c*d*f)*cosh(f*x + e))*sinh(f*x + e))*dilog(I*cosh(f*x
+ e) + I*sinh(f*x + e)) - 6*((d^2*f*x + c*d*f)*cosh(f*x + e)^4 + 4*(d^2*f*x + c*d*f)*cosh(f*x + e)*sinh(f*x +
e)^3 + (d^2*f*x + c*d*f)*sinh(f*x + e)^4 + d^2*f*x + c*d*f + 2*(d^2*f*x + c*d*f)*cosh(f*x + e)^2 + 2*(d^2*f*x
+ c*d*f + 3*(d^2*f*x + c*d*f)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 4*((d^2*f*x + c*d*f)*cosh(f*x + e)^3 + (d^2*
f*x + c*d*f)*cosh(f*x + e))*sinh(f*x + e))*dilog(-I*cosh(f*x + e) - I*sinh(f*x + e)) - 3*((d^2*e^2 - 2*c*d*e*f
+ c^2*f^2 + d^2)*cosh(f*x + e)^4 + 4*(d^2*e^2 - 2*c*d*e*f + c^2*f^2 + d^2)*cosh(f*x + e)*sinh(f*x + e)^3 + (d
^2*e^2 - 2*c*d*e*f + c^2*f^2 + d^2)*sinh(f*x + e)^4 + d^2*e^2 - 2*c*d*e*f + c^2*f^2 + 2*(d^2*e^2 - 2*c*d*e*f +
c^2*f^2 + d^2)*cosh(f*x + e)^2 + 2*(d^2*e^2 - 2*c*d*e*f + c^2*f^2 + 3*(d^2*e^2 - 2*c*d*e*f + c^2*f^2 + d^2)*c
osh(f*x + e)^2 + d^2)*sinh(f*x + e)^2 + d^2 + 4*((d^2*e^2 - 2*c*d*e*f + c^2*f^2 + d^2)*cosh(f*x + e)^3 + (d^2*
e^2 - 2*c*d*e*f + c^2*f^2 + d^2)*cosh(f*x + e))*sinh(f*x + e))*log(cosh(f*x + e) + sinh(f*x + e) + I) - 3*((d^
2*e^2 - 2*c*d*e*f + c^2*f^2 + d^2)*cosh(f*x + e)^4 + 4*(d^2*e^2 - 2*c*d*e*f + c^2*f^2 + d^2)*cosh(f*x + e)*sin
h(f*x + e)^3 + (d^2*e^2 - 2*c*d*e*f + c^2*f^2 + d^2)*sinh(f*x + e)^4 + d^2*e^2 - 2*c*d*e*f + c^2*f^2 + 2*(d^2*
e^2 - 2*c*d*e*f + c^2*f^2 + d^2)*cosh(f*x + e)^2 + 2*(d^2*e^2 - 2*c*d*e*f + c^2*f^2 + 3*(d^2*e^2 - 2*c*d*e*f +
c^2*f^2 + d^2)*cosh(f*x + e)^2 + d^2)*sinh(f*x + e)^2 + d^2 + 4*((d^2*e^2 - 2*c*d*e*f + c^2*f^2 + d^2)*cosh(f
*x + e)^3 + (d^2*e^2 - 2*c*d*e*f + c^2*f^2 + d^2)*cosh(f*x + e))*sinh(f*x + e))*log(cosh(f*x + e) + sinh(f*x +
e) - I) - 3*(d^2*f^2*x^2 + 2*c*d*f^2*x + (d^2*f^2*x^2 + 2*c*d*f^2*x - d^2*e^2 + 2*c*d*e*f)*cosh(f*x + e)^4 +
4*(d^2*f^2*x^2 + 2*c*d*f^2*x - d^2*e^2 + 2*c*d*e*f)*cosh(f*x + e)*sinh(f*x + e)^3 + (d^2*f^2*x^2 + 2*c*d*f^2*x
- d^2*e^2 + 2*c*d*e*f)*sinh(f*x + e)^4 - d^2*e^2 + 2*c*d*e*f + 2*(d^2*f^2*x^2 + 2*c*d*f^2*x - d^2*e^2 + 2*c*d
*e*f)*cosh(f*x + e)^2 + 2*(d^2*f^2*x^2 + 2*c*d*f^2*x - d^2*e^2 + 2*c*d*e*f + 3*(d^2*f^2*x^2 + 2*c*d*f^2*x - d^
2*e^2 + 2*c*d*e*f)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 4*((d^2*f^2*x^2 + 2*c*d*f^2*x - d^2*e^2 + 2*c*d*e*f)*cos
h(f*x + e)^3 + (d^2*f^2*x^2 + 2*c*d*f^2*x - d^2*e^2 + 2*c*d*e*f)*cosh(f*x + e))*sinh(f*x + e))*log(I*cosh(f*x
+ e) + I*sinh(f*x + e) + 1) - 3*(d^2*f^2*x^2 + 2*c*d*f^2*x + (d^2*f^2*x^2 + 2*c*d*f^2*x - d^2*e^2 + 2*c*d*e*f)
*cosh(f*x + e)^4 + 4*(d^2*f^2*x^2 + 2*c*d*f^2*x - d^2*e^2 + 2*c*d*e*f)*cosh(f*x + e)*sinh(f*x + e)^3 + (d^2*f^
2*x^2 + 2*c*d*f^2*x - d^2*e^2 + 2*c*d*e*f)*sinh(f*x + e)^4 - d^2*e^2 + 2*c*d*e*f + 2*(d^2*f^2*x^2 + 2*c*d*f^2*
x - d^2*e^2 + 2*c*d*e*f)*cosh(f*x + e)^2 + 2*(d^2*f^2*x^2 + 2*c*d*f^2*x - d^2*e^2 + 2*c*d*e*f + 3*(d^2*f^2*x^2
+ 2*c*d*f^2*x - d^2*e^2 + 2*c*d*e*f)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 4*((d^2*f^2*x^2 + 2*c*d*f^2*x - d^2*e
^2 + 2*c*d*e*f)*cosh(f*x + e)^3 + (d^2*f^2*x^2 + 2*c*d*f^2*x - d^2*e^2 + 2*c*d*e*f)*cosh(f*x + e))*sinh(f*x +
e))*log(-I*cosh(f*x + e) - I*sinh(f*x + e) + 1) + 6*(d^2*cosh(f*x + e)^4 + 4*d^2*cosh(f*x + e)*sinh(f*x + e)^3
+ d^2*sinh(f*x + e)^4 + 2*d^2*cosh(f*x + e)^2 + 2*(3*d^2*cosh(f*x + e)^2 + d^2)*sinh(f*x + e)^2 + d^2 + 4*(d^
2*cosh(f*x + e)^3 + d^2*cosh(f*x + e))*sinh(f*x + e))*polylog(3, I*cosh(f*x + e) + I*sinh(f*x + e)) + 6*(d^2*c
osh(f*x + e)^4 + 4*d^2*cosh(f*x + e)*sinh(f*x + e)^3 + d^2*sinh(f*x + e)^4 + 2*d^2*cosh(f*x + e)^2 + 2*(3*d^2*
cosh(f*x + e)^2 + d^2)*sinh(f*x + e)^2 + d^2 + 4*(d^2*cosh(f*x + e)^3 + d^2*cosh(f*x + e))*sinh(f*x + e))*poly
log(3, -I*cosh(f*x + e) - I*sinh(f*x + e)) + 4*((d^2*f^3*x^3 + 3*c*d*f^3*x^2 + 2*d^2*e^3 - 6*c*d*e^2*f + 6*c^2
*e*f^2 + 6*d^2*e + 3*(c^2*f^3 + 2*d^2*f)*x)*cosh(f*x + e)^3 + (d^2*f^3*x^3 + 2*d^2*e^3 + 6*d^2*e + 3*(2*c^2*e
- c^2)*f^2 + 3*(c*d*f^3 - d^2*f^2)*x^2 - 3*(2*c*d*e^2 + c*d)*f + 3*(c^2*f^3 - 2*c*d*f^2 + d^2*f)*x)*cosh(f*x +
e))*sinh(f*x + e))/(f^3*cosh(f*x + e)^4 + 4*f^3*cosh(f*x + e)*sinh(f*x + e)^3 + f^3*sinh(f*x + e)^4 + 2*f^3*c
osh(f*x + e)^2 + f^3 + 2*(3*f^3*cosh(f*x + e)^2 + f^3)*sinh(f*x + e)^2 + 4*(f^3*cosh(f*x + e)^3 + f^3*cosh(f*x
+ e))*sinh(f*x + e))
Sympy [F]
\[
\int (c+d x)^2 \tanh ^3(e+f x) \, dx=\int \left (c + d x\right )^{2} \tanh ^{3}{\left (e + f x \right )}\, dx
\]
[In]
integrate((d*x+c)**2*tanh(f*x+e)**3,x)
[Out]
Integral((c + d*x)**2*tanh(e + f*x)**3, x)
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 392 vs. \(2 (148) = 296\).
Time = 0.38 (sec) , antiderivative size = 392, normalized size of antiderivative = 2.50
\[
\int (c+d x)^2 \tanh ^3(e+f x) \, dx=c^{2} {\left (x + \frac {e}{f} + \frac {\log \left (e^{\left (-2 \, f x - 2 \, e\right )} + 1\right )}{f} + \frac {2 \, e^{\left (-2 \, f x - 2 \, e\right )}}{f {\left (2 \, e^{\left (-2 \, f x - 2 \, e\right )} + e^{\left (-4 \, f x - 4 \, e\right )} + 1\right )}}\right )} + \frac {{\left (2 \, f x \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right )\right )} c d}{f^{2}} - \frac {2 \, d^{2} x}{f^{2}} + \frac {d^{2} f^{2} x^{3} + 3 \, c d f^{2} x^{2} + 6 \, d^{2} x + 6 \, c d + {\left (d^{2} f^{2} x^{3} e^{\left (4 \, e\right )} + 3 \, c d f^{2} x^{2} e^{\left (4 \, e\right )}\right )} e^{\left (4 \, f x\right )} + 2 \, {\left (d^{2} f^{2} x^{3} e^{\left (2 \, e\right )} + 3 \, {\left (c d f^{2} e^{\left (2 \, e\right )} + d^{2} f e^{\left (2 \, e\right )}\right )} x^{2} + 3 \, c d e^{\left (2 \, e\right )} + 3 \, {\left (2 \, c d f e^{\left (2 \, e\right )} + d^{2} e^{\left (2 \, e\right )}\right )} x\right )} e^{\left (2 \, f x\right )}}{3 \, {\left (f^{2} e^{\left (4 \, f x + 4 \, e\right )} + 2 \, f^{2} e^{\left (2 \, f x + 2 \, e\right )} + f^{2}\right )}} + \frac {{\left (2 \, f^{2} x^{2} \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right ) - {\rm Li}_{3}(-e^{\left (2 \, f x + 2 \, e\right )})\right )} d^{2}}{2 \, f^{3}} + \frac {d^{2} \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right )}{f^{3}} - \frac {2 \, {\left (d^{2} f^{3} x^{3} + 3 \, c d f^{3} x^{2}\right )}}{3 \, f^{3}}
\]
[In]
integrate((d*x+c)^2*tanh(f*x+e)^3,x, algorithm="maxima")
[Out]
c^2*(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)
)) + (2*f*x*log(e^(2*f*x + 2*e) + 1) + dilog(-e^(2*f*x + 2*e)))*c*d/f^2 - 2*d^2*x/f^2 + 1/3*(d^2*f^2*x^3 + 3*c
*d*f^2*x^2 + 6*d^2*x + 6*c*d + (d^2*f^2*x^3*e^(4*e) + 3*c*d*f^2*x^2*e^(4*e))*e^(4*f*x) + 2*(d^2*f^2*x^3*e^(2*e
) + 3*(c*d*f^2*e^(2*e) + d^2*f*e^(2*e))*x^2 + 3*c*d*e^(2*e) + 3*(2*c*d*f*e^(2*e) + d^2*e^(2*e))*x)*e^(2*f*x))/
(f^2*e^(4*f*x + 4*e) + 2*f^2*e^(2*f*x + 2*e) + f^2) + 1/2*(2*f^2*x^2*log(e^(2*f*x + 2*e) + 1) + 2*f*x*dilog(-e
^(2*f*x + 2*e)) - polylog(3, -e^(2*f*x + 2*e)))*d^2/f^3 + d^2*log(e^(2*f*x + 2*e) + 1)/f^3 - 2/3*(d^2*f^3*x^3
+ 3*c*d*f^3*x^2)/f^3
Giac [F]
\[
\int (c+d x)^2 \tanh ^3(e+f x) \, dx=\int { {\left (d x + c\right )}^{2} \tanh \left (f x + e\right )^{3} \,d x }
\]
[In]
integrate((d*x+c)^2*tanh(f*x+e)^3,x, algorithm="giac")
[Out]
integrate((d*x + c)^2*tanh(f*x + e)^3, x)
Mupad [F(-1)]
Timed out. \[
\int (c+d x)^2 \tanh ^3(e+f x) \, dx=\int {\mathrm {tanh}\left (e+f\,x\right )}^3\,{\left (c+d\,x\right )}^2 \,d x
\]
[In]
int(tanh(e + f*x)^3*(c + d*x)^2,x)
[Out]
int(tanh(e + f*x)^3*(c + d*x)^2, x)