\(\int \tanh ^4(c+d x) (a+b \tanh ^2(c+d x))^2 \, dx\) [144]

Optimal result

Integrand size = 23, antiderivative size = 83 \[ \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=(a+b)^2 x-\frac {(a+b)^2 \tanh (c+d x)}{d}-\frac {(a+b)^2 \tanh ^3(c+d x)}{3 d}-\frac {b (2 a+b) \tanh ^5(c+d x)}{5 d}-\frac {b^2 \tanh ^7(c+d x)}{7 d} \]

[Out]

(a+b)^2*x-(a+b)^2*tanh(d*x+c)/d-1/3*(a+b)^2*tanh(d*x+c)^3/d-1/5*b*(2*a+b)*tanh(d*x+c)^5/d-1/7*b^2*tanh(d*x+c)^
7/d

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3751, 472, 212} \[ \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=-\frac {b (2 a+b) \tanh ^5(c+d x)}{5 d}-\frac {(a+b)^2 \tanh ^3(c+d x)}{3 d}-\frac {(a+b)^2 \tanh (c+d x)}{d}+x (a+b)^2-\frac {b^2 \tanh ^7(c+d x)}{7 d} \]

[In]

Int[Tanh[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^2,x]

[Out]

(a + b)^2*x - ((a + b)^2*Tanh[c + d*x])/d - ((a + b)^2*Tanh[c + d*x]^3)/(3*d) - (b*(2*a + b)*Tanh[c + d*x]^5)/
(5*d) - (b^2*Tanh[c + d*x]^7)/(7*d)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 472

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[(e*x)^m*((a + b*x^n)^p/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

Rule 3751

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[c*(ff/f), Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2
+ ff^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^4 \left (a+b x^2\right )^2}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (-(a+b)^2-(a+b)^2 x^2-b (2 a+b) x^4-b^2 x^6+\frac {a^2+2 a b+b^2}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d} \\ & = -\frac {(a+b)^2 \tanh (c+d x)}{d}-\frac {(a+b)^2 \tanh ^3(c+d x)}{3 d}-\frac {b (2 a+b) \tanh ^5(c+d x)}{5 d}-\frac {b^2 \tanh ^7(c+d x)}{7 d}+\frac {(a+b)^2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = (a+b)^2 x-\frac {(a+b)^2 \tanh (c+d x)}{d}-\frac {(a+b)^2 \tanh ^3(c+d x)}{3 d}-\frac {b (2 a+b) \tanh ^5(c+d x)}{5 d}-\frac {b^2 \tanh ^7(c+d x)}{7 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(190\) vs. \(2(83)=166\).

Time = 0.10 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.29 \[ \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {a^2 \text {arctanh}(\tanh (c+d x))}{d}+\frac {2 a b \text {arctanh}(\tanh (c+d x))}{d}+\frac {b^2 \text {arctanh}(\tanh (c+d x))}{d}-\frac {a^2 \tanh (c+d x)}{d}-\frac {2 a b \tanh (c+d x)}{d}-\frac {b^2 \tanh (c+d x)}{d}-\frac {a^2 \tanh ^3(c+d x)}{3 d}-\frac {2 a b \tanh ^3(c+d x)}{3 d}-\frac {b^2 \tanh ^3(c+d x)}{3 d}-\frac {2 a b \tanh ^5(c+d x)}{5 d}-\frac {b^2 \tanh ^5(c+d x)}{5 d}-\frac {b^2 \tanh ^7(c+d x)}{7 d} \]

[In]

Integrate[Tanh[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^2,x]

[Out]

(a^2*ArcTanh[Tanh[c + d*x]])/d + (2*a*b*ArcTanh[Tanh[c + d*x]])/d + (b^2*ArcTanh[Tanh[c + d*x]])/d - (a^2*Tanh
[c + d*x])/d - (2*a*b*Tanh[c + d*x])/d - (b^2*Tanh[c + d*x])/d - (a^2*Tanh[c + d*x]^3)/(3*d) - (2*a*b*Tanh[c +
d*x]^3)/(3*d) - (b^2*Tanh[c + d*x]^3)/(3*d) - (2*a*b*Tanh[c + d*x]^5)/(5*d) - (b^2*Tanh[c + d*x]^5)/(5*d) - (
b^2*Tanh[c + d*x]^7)/(7*d)

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.63

method	result	size
parallelrisch	\(-\frac {15 \tanh \left (d x +c \right )^{7} b^{2}+42 \tanh \left (d x +c \right )^{5} a b +21 \tanh \left (d x +c \right )^{5} b^{2}+35 \tanh \left (d x +c \right )^{3} a^{2}+70 \tanh \left (d x +c \right )^{3} a b +35 b^{2} \tanh \left (d x +c \right )^{3}-105 a^{2} d x -210 a b d x -105 b^{2} d x +105 a^{2} \tanh \left (d x +c \right )+210 a b \tanh \left (d x +c \right )+105 b^{2} \tanh \left (d x +c \right )}{105 d}\)	\(135\)
derivativedivides	\(\frac {-2 a b \tanh \left (d x +c \right )-\frac {2 \tanh \left (d x +c \right )^{5} a b}{5}-\frac {2 \tanh \left (d x +c \right )^{3} a b}{3}-\frac {\left (a^{2}+2 a b +b^{2}\right ) \ln \left (\tanh \left (d x +c \right )-1\right )}{2}-\frac {\tanh \left (d x +c \right )^{5} b^{2}}{5}-\frac {\tanh \left (d x +c \right )^{3} a^{2}}{3}-\frac {b^{2} \tanh \left (d x +c \right )^{3}}{3}-b^{2} \tanh \left (d x +c \right )-a^{2} \tanh \left (d x +c \right )-\frac {\tanh \left (d x +c \right )^{7} b^{2}}{7}+\frac {\left (a^{2}+2 a b +b^{2}\right ) \ln \left (\tanh \left (d x +c \right )+1\right )}{2}}{d}\)	\(158\)
default	\(\frac {-2 a b \tanh \left (d x +c \right )-\frac {2 \tanh \left (d x +c \right )^{5} a b}{5}-\frac {2 \tanh \left (d x +c \right )^{3} a b}{3}-\frac {\left (a^{2}+2 a b +b^{2}\right ) \ln \left (\tanh \left (d x +c \right )-1\right )}{2}-\frac {\tanh \left (d x +c \right )^{5} b^{2}}{5}-\frac {\tanh \left (d x +c \right )^{3} a^{2}}{3}-\frac {b^{2} \tanh \left (d x +c \right )^{3}}{3}-b^{2} \tanh \left (d x +c \right )-a^{2} \tanh \left (d x +c \right )-\frac {\tanh \left (d x +c \right )^{7} b^{2}}{7}+\frac {\left (a^{2}+2 a b +b^{2}\right ) \ln \left (\tanh \left (d x +c \right )+1\right )}{2}}{d}\)	\(158\)
parts	\(\frac {b^{2} \left (-\frac {\tanh \left (d x +c \right )^{7}}{7}-\frac {\tanh \left (d x +c \right )^{5}}{5}-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {a^{2} \left (-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {2 a b \left (-\frac {\tanh \left (d x +c \right )^{5}}{5}-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}\)	\(176\)
risch	\(a^{2} x +2 a b x +b^{2} x +\frac {4 a^{2} {\mathrm e}^{12 d x +12 c}+12 a b \,{\mathrm e}^{12 d x +12 c}+8 b^{2} {\mathrm e}^{12 d x +12 c}+20 a^{2} {\mathrm e}^{10 d x +10 c}+48 a b \,{\mathrm e}^{10 d x +10 c}+24 b^{2} {\mathrm e}^{10 d x +10 c}+\frac {128 a^{2} {\mathrm e}^{8 d x +8 c}}{3}+\frac {292 a b \,{\mathrm e}^{8 d x +8 c}}{3}+\frac {176 b^{2} {\mathrm e}^{8 d x +8 c}}{3}+\frac {152 a^{2} {\mathrm e}^{6 d x +6 c}}{3}+\frac {352 a b \,{\mathrm e}^{6 d x +6 c}}{3}+\frac {176 b^{2} {\mathrm e}^{6 d x +6 c}}{3}+36 a^{2} {\mathrm e}^{4 d x +4 c}+\frac {404 a b \,{\mathrm e}^{4 d x +4 c}}{5}+\frac {232 \,{\mathrm e}^{4 d x +4 c} b^{2}}{5}+\frac {44 a^{2} {\mathrm e}^{2 d x +2 c}}{3}+\frac {464 a b \,{\mathrm e}^{2 d x +2 c}}{15}+\frac {232 \,{\mathrm e}^{2 d x +2 c} b^{2}}{15}+\frac {8 a^{2}}{3}+\frac {92 a b}{15}+\frac {352 b^{2}}{105}}{d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{7}}\)	\(296\)

[In]

int(tanh(d*x+c)^4*(a+b*tanh(d*x+c)^2)^2,x,method=_RETURNVERBOSE)

[Out]

-1/105*(15*tanh(d*x+c)^7*b^2+42*tanh(d*x+c)^5*a*b+21*tanh(d*x+c)^5*b^2+35*tanh(d*x+c)^3*a^2+70*tanh(d*x+c)^3*a
*b+35*b^2*tanh(d*x+c)^3-105*a^2*d*x-210*a*b*d*x-105*b^2*d*x+105*a^2*tanh(d*x+c)+210*a*b*tanh(d*x+c)+105*b^2*ta
nh(d*x+c))/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 796 vs. \(2 (77) = 154\).

Time = 0.26 (sec) , antiderivative size = 796, normalized size of antiderivative = 9.59 \[ \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {{\left (105 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 140 \, a^{2} + 322 \, a b + 176 \, b^{2}\right )} \cosh \left (d x + c\right )^{7} + 7 \, {\left (105 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 140 \, a^{2} + 322 \, a b + 176 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} - 2 \, {\left (70 \, a^{2} + 161 \, a b + 88 \, b^{2}\right )} \sinh \left (d x + c\right )^{7} + 7 \, {\left (105 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 140 \, a^{2} + 322 \, a b + 176 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} - 14 \, {\left (3 \, {\left (70 \, a^{2} + 161 \, a b + 88 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 40 \, a^{2} + 71 \, a b + 28 \, b^{2}\right )} \sinh \left (d x + c\right )^{5} + 35 \, {\left ({\left (105 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 140 \, a^{2} + 322 \, a b + 176 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (105 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 140 \, a^{2} + 322 \, a b + 176 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 21 \, {\left (105 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 140 \, a^{2} + 322 \, a b + 176 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - 14 \, {\left (5 \, {\left (70 \, a^{2} + 161 \, a b + 88 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 10 \, {\left (40 \, a^{2} + 71 \, a b + 28 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 60 \, a^{2} + 123 \, a b + 84 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} + 7 \, {\left (3 \, {\left (105 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 140 \, a^{2} + 322 \, a b + 176 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} + 10 \, {\left (105 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 140 \, a^{2} + 322 \, a b + 176 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 9 \, {\left (105 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 140 \, a^{2} + 322 \, a b + 176 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 35 \, {\left (105 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 140 \, a^{2} + 322 \, a b + 176 \, b^{2}\right )} \cosh \left (d x + c\right ) - 14 \, {\left ({\left (70 \, a^{2} + 161 \, a b + 88 \, b^{2}\right )} \cosh \left (d x + c\right )^{6} + 5 \, {\left (40 \, a^{2} + 71 \, a b + 28 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 9 \, {\left (20 \, a^{2} + 41 \, a b + 28 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 30 \, a^{2} + 75 \, a b\right )} \sinh \left (d x + c\right )}{105 \, {\left (d \cosh \left (d x + c\right )^{7} + 7 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} + 7 \, d \cosh \left (d x + c\right )^{5} + 35 \, {\left (d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 21 \, d \cosh \left (d x + c\right )^{3} + 7 \, {\left (3 \, d \cosh \left (d x + c\right )^{5} + 10 \, d \cosh \left (d x + c\right )^{3} + 9 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 35 \, d \cosh \left (d x + c\right )\right )}} \]

[In]

integrate(tanh(d*x+c)^4*(a+b*tanh(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

1/105*((105*(a^2 + 2*a*b + b^2)*d*x + 140*a^2 + 322*a*b + 176*b^2)*cosh(d*x + c)^7 + 7*(105*(a^2 + 2*a*b + b^2
)*d*x + 140*a^2 + 322*a*b + 176*b^2)*cosh(d*x + c)*sinh(d*x + c)^6 - 2*(70*a^2 + 161*a*b + 88*b^2)*sinh(d*x +
c)^7 + 7*(105*(a^2 + 2*a*b + b^2)*d*x + 140*a^2 + 322*a*b + 176*b^2)*cosh(d*x + c)^5 - 14*(3*(70*a^2 + 161*a*b
+ 88*b^2)*cosh(d*x + c)^2 + 40*a^2 + 71*a*b + 28*b^2)*sinh(d*x + c)^5 + 35*((105*(a^2 + 2*a*b + b^2)*d*x + 14
0*a^2 + 322*a*b + 176*b^2)*cosh(d*x + c)^3 + (105*(a^2 + 2*a*b + b^2)*d*x + 140*a^2 + 322*a*b + 176*b^2)*cosh(
d*x + c))*sinh(d*x + c)^4 + 21*(105*(a^2 + 2*a*b + b^2)*d*x + 140*a^2 + 322*a*b + 176*b^2)*cosh(d*x + c)^3 - 1
4*(5*(70*a^2 + 161*a*b + 88*b^2)*cosh(d*x + c)^4 + 10*(40*a^2 + 71*a*b + 28*b^2)*cosh(d*x + c)^2 + 60*a^2 + 12
3*a*b + 84*b^2)*sinh(d*x + c)^3 + 7*(3*(105*(a^2 + 2*a*b + b^2)*d*x + 140*a^2 + 322*a*b + 176*b^2)*cosh(d*x +
c)^5 + 10*(105*(a^2 + 2*a*b + b^2)*d*x + 140*a^2 + 322*a*b + 176*b^2)*cosh(d*x + c)^3 + 9*(105*(a^2 + 2*a*b +
b^2)*d*x + 140*a^2 + 322*a*b + 176*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 + 35*(105*(a^2 + 2*a*b + b^2)*d*x + 140
*a^2 + 322*a*b + 176*b^2)*cosh(d*x + c) - 14*((70*a^2 + 161*a*b + 88*b^2)*cosh(d*x + c)^6 + 5*(40*a^2 + 71*a*b
+ 28*b^2)*cosh(d*x + c)^4 + 9*(20*a^2 + 41*a*b + 28*b^2)*cosh(d*x + c)^2 + 30*a^2 + 75*a*b)*sinh(d*x + c))/(d
*cosh(d*x + c)^7 + 7*d*cosh(d*x + c)*sinh(d*x + c)^6 + 7*d*cosh(d*x + c)^5 + 35*(d*cosh(d*x + c)^3 + d*cosh(d*
x + c))*sinh(d*x + c)^4 + 21*d*cosh(d*x + c)^3 + 7*(3*d*cosh(d*x + c)^5 + 10*d*cosh(d*x + c)^3 + 9*d*cosh(d*x
+ c))*sinh(d*x + c)^2 + 35*d*cosh(d*x + c))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (70) = 140\).

Time = 0.21 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.99 \[ \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\begin {cases} a^{2} x - \frac {a^{2} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {a^{2} \tanh {\left (c + d x \right )}}{d} + 2 a b x - \frac {2 a b \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac {2 a b \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {2 a b \tanh {\left (c + d x \right )}}{d} + b^{2} x - \frac {b^{2} \tanh ^{7}{\left (c + d x \right )}}{7 d} - \frac {b^{2} \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac {b^{2} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {b^{2} \tanh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tanh ^{2}{\left (c \right )}\right )^{2} \tanh ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(tanh(d*x+c)**4*(a+b*tanh(d*x+c)**2)**2,x)

[Out]

Piecewise((a**2*x - a**2*tanh(c + d*x)**3/(3*d) - a**2*tanh(c + d*x)/d + 2*a*b*x - 2*a*b*tanh(c + d*x)**5/(5*d
) - 2*a*b*tanh(c + d*x)**3/(3*d) - 2*a*b*tanh(c + d*x)/d + b**2*x - b**2*tanh(c + d*x)**7/(7*d) - b**2*tanh(c
+ d*x)**5/(5*d) - b**2*tanh(c + d*x)**3/(3*d) - b**2*tanh(c + d*x)/d, Ne(d, 0)), (x*(a + b*tanh(c)**2)**2*tanh
(c)**4, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 369 vs. \(2 (77) = 154\).

Time = 0.21 (sec) , antiderivative size = 369, normalized size of antiderivative = 4.45 \[ \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {1}{105} \, b^{2} {\left (105 \, x + \frac {105 \, c}{d} - \frac {8 \, {\left (203 \, e^{\left (-2 \, d x - 2 \, c\right )} + 609 \, e^{\left (-4 \, d x - 4 \, c\right )} + 770 \, e^{\left (-6 \, d x - 6 \, c\right )} + 770 \, e^{\left (-8 \, d x - 8 \, c\right )} + 315 \, e^{\left (-10 \, d x - 10 \, c\right )} + 105 \, e^{\left (-12 \, d x - 12 \, c\right )} + 44\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} + 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} + 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} + 1\right )}}\right )} + \frac {2}{15} \, a b {\left (15 \, x + \frac {15 \, c}{d} - \frac {2 \, {\left (70 \, e^{\left (-2 \, d x - 2 \, c\right )} + 140 \, e^{\left (-4 \, d x - 4 \, c\right )} + 90 \, e^{\left (-6 \, d x - 6 \, c\right )} + 45 \, e^{\left (-8 \, d x - 8 \, c\right )} + 23\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} + \frac {1}{3} \, a^{2} {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 2\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} \]

[In]

integrate(tanh(d*x+c)^4*(a+b*tanh(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

1/105*b^2*(105*x + 105*c/d - 8*(203*e^(-2*d*x - 2*c) + 609*e^(-4*d*x - 4*c) + 770*e^(-6*d*x - 6*c) + 770*e^(-8
*d*x - 8*c) + 315*e^(-10*d*x - 10*c) + 105*e^(-12*d*x - 12*c) + 44)/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*
c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 1
4*c) + 1))) + 2/15*a*b*(15*x + 15*c/d - 2*(70*e^(-2*d*x - 2*c) + 140*e^(-4*d*x - 4*c) + 90*e^(-6*d*x - 6*c) +
45*e^(-8*d*x - 8*c) + 23)/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8
*c) + e^(-10*d*x - 10*c) + 1))) + 1/3*a^2*(3*x + 3*c/d - 4*(3*e^(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + 2)/(d*(3
*e^(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) + 1)))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 300 vs. \(2 (77) = 154\).

Time = 0.38 (sec) , antiderivative size = 300, normalized size of antiderivative = 3.61 \[ \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {105 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} {\left (d x + c\right )} + \frac {4 \, {\left (105 \, a^{2} e^{\left (12 \, d x + 12 \, c\right )} + 315 \, a b e^{\left (12 \, d x + 12 \, c\right )} + 210 \, b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 525 \, a^{2} e^{\left (10 \, d x + 10 \, c\right )} + 1260 \, a b e^{\left (10 \, d x + 10 \, c\right )} + 630 \, b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 1120 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} + 2555 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 1540 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 1330 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} + 3080 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 1540 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 945 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 2121 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 1218 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 385 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 812 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 406 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 70 \, a^{2} + 161 \, a b + 88 \, b^{2}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{7}}}{105 \, d} \]

[In]

integrate(tanh(d*x+c)^4*(a+b*tanh(d*x+c)^2)^2,x, algorithm="giac")

[Out]

1/105*(105*(a^2 + 2*a*b + b^2)*(d*x + c) + 4*(105*a^2*e^(12*d*x + 12*c) + 315*a*b*e^(12*d*x + 12*c) + 210*b^2*
e^(12*d*x + 12*c) + 525*a^2*e^(10*d*x + 10*c) + 1260*a*b*e^(10*d*x + 10*c) + 630*b^2*e^(10*d*x + 10*c) + 1120*
a^2*e^(8*d*x + 8*c) + 2555*a*b*e^(8*d*x + 8*c) + 1540*b^2*e^(8*d*x + 8*c) + 1330*a^2*e^(6*d*x + 6*c) + 3080*a*
b*e^(6*d*x + 6*c) + 1540*b^2*e^(6*d*x + 6*c) + 945*a^2*e^(4*d*x + 4*c) + 2121*a*b*e^(4*d*x + 4*c) + 1218*b^2*e
^(4*d*x + 4*c) + 385*a^2*e^(2*d*x + 2*c) + 812*a*b*e^(2*d*x + 2*c) + 406*b^2*e^(2*d*x + 2*c) + 70*a^2 + 161*a*
b + 88*b^2)/(e^(2*d*x + 2*c) + 1)^7)/d

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.10 \[ \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=x\,\left (a^2+2\,a\,b+b^2\right )-\frac {\mathrm {tanh}\left (c+d\,x\right )\,{\left (a+b\right )}^2}{d}-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^5\,\left (b^2+2\,a\,b\right )}{5\,d}-\frac {b^2\,{\mathrm {tanh}\left (c+d\,x\right )}^7}{7\,d}-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^3\,\left (a^2+2\,a\,b+b^2\right )}{3\,d} \]

[In]

int(tanh(c + d*x)^4*(a + b*tanh(c + d*x)^2)^2,x)

[Out]

x*(2*a*b + a^2 + b^2) - (tanh(c + d*x)*(a + b)^2)/d - (tanh(c + d*x)^5*(2*a*b + b^2))/(5*d) - (b^2*tanh(c + d*
x)^7)/(7*d) - (tanh(c + d*x)^3*(2*a*b + a^2 + b^2))/(3*d)