\(\int \tanh ^4(c+d x) (a+b \tanh ^2(c+d x))^3 \, dx\) [156]

Optimal result

Integrand size = 23, antiderivative size = 114 \[ \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=(a+b)^3 x-\frac {(a+b)^3 \tanh (c+d x)}{d}-\frac {(a+b)^3 \tanh ^3(c+d x)}{3 d}-\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^2 (3 a+b) \tanh ^7(c+d x)}{7 d}-\frac {b^3 \tanh ^9(c+d x)}{9 d} \]

[Out]

(a+b)^3*x-(a+b)^3*tanh(d*x+c)/d-1/3*(a+b)^3*tanh(d*x+c)^3/d-1/5*b*(3*a^2+3*a*b+b^2)*tanh(d*x+c)^5/d-1/7*b^2*(3
*a+b)*tanh(d*x+c)^7/d-1/9*b^3*tanh(d*x+c)^9/d

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 114, 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} \[ \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=-\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^2 (3 a+b) \tanh ^7(c+d x)}{7 d}-\frac {(a+b)^3 \tanh ^3(c+d x)}{3 d}-\frac {(a+b)^3 \tanh (c+d x)}{d}+x (a+b)^3-\frac {b^3 \tanh ^9(c+d x)}{9 d} \]

[In]

Int[Tanh[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

(a + b)^3*x - ((a + b)^3*Tanh[c + d*x])/d - ((a + b)^3*Tanh[c + d*x]^3)/(3*d) - (b*(3*a^2 + 3*a*b + b^2)*Tanh[
c + d*x]^5)/(5*d) - (b^2*(3*a + b)*Tanh[c + d*x]^7)/(7*d) - (b^3*Tanh[c + d*x]^9)/(9*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 )^3}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (-(a+b)^3-(a+b)^3 x^2-b \left (3 a^2+3 a b+b^2\right ) x^4-b^2 (3 a+b) x^6-b^3 x^8+\frac {a^3+3 a^2 b+3 a b^2+b^3}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d} \\ & = -\frac {(a+b)^3 \tanh (c+d x)}{d}-\frac {(a+b)^3 \tanh ^3(c+d x)}{3 d}-\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^2 (3 a+b) \tanh ^7(c+d x)}{7 d}-\frac {b^3 \tanh ^9(c+d x)}{9 d}+\frac {(a+b)^3 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = (a+b)^3 x-\frac {(a+b)^3 \tanh (c+d x)}{d}-\frac {(a+b)^3 \tanh ^3(c+d x)}{3 d}-\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^2 (3 a+b) \tanh ^7(c+d x)}{7 d}-\frac {b^3 \tanh ^9(c+d x)}{9 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.72 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.08 \[ \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {\tanh (c+d x) \left (-315 (a+b)^3-105 (a+b)^3 \tanh ^2(c+d x)-63 b \left (3 a^2+3 a b+b^2\right ) \tanh ^4(c+d x)-45 b^2 (3 a+b) \tanh ^6(c+d x)-35 b^3 \tanh ^8(c+d x)+\frac {315 (a+b)^3 \text {arctanh}\left (\sqrt {\tanh ^2(c+d x)}\right )}{\sqrt {\tanh ^2(c+d x)}}\right )}{315 d} \]

[In]

Integrate[Tanh[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

(Tanh[c + d*x]*(-315*(a + b)^3 - 105*(a + b)^3*Tanh[c + d*x]^2 - 63*b*(3*a^2 + 3*a*b + b^2)*Tanh[c + d*x]^4 -
45*b^2*(3*a + b)*Tanh[c + d*x]^6 - 35*b^3*Tanh[c + d*x]^8 + (315*(a + b)^3*ArcTanh[Sqrt[Tanh[c + d*x]^2]])/Sqr
t[Tanh[c + d*x]^2]))/(315*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(217\) vs. \(2(106)=212\).

Time = 0.17 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.91

method	result	size
parallelrisch	\(-\frac {35 \tanh \left (d x +c \right )^{9} b^{3}+135 a \,b^{2} \tanh \left (d x +c \right )^{7}+45 \tanh \left (d x +c \right )^{7} b^{3}+189 a^{2} b \tanh \left (d x +c \right )^{5}+189 \tanh \left (d x +c \right )^{5} a \,b^{2}+63 b^{3} \tanh \left (d x +c \right )^{5}+105 a^{3} \tanh \left (d x +c \right )^{3}+315 \tanh \left (d x +c \right )^{3} a^{2} b +315 a \,b^{2} \tanh \left (d x +c \right )^{3}+105 b^{3} \tanh \left (d x +c \right )^{3}-315 a^{3} d x -945 a^{2} b d x -945 a \,b^{2} d x -315 b^{3} d x +315 a^{3} \tanh \left (d x +c \right )+945 a^{2} b \tanh \left (d x +c \right )+945 a \,b^{2} \tanh \left (d x +c \right )+315 b^{3} \tanh \left (d x +c \right )}{315 d}\)	\(218\)
derivativedivides	\(\frac {-3 a^{2} b \tanh \left (d x +c \right )-3 a \,b^{2} \tanh \left (d x +c \right )-\frac {3 a \,b^{2} \tanh \left (d x +c \right )^{7}}{7}-\frac {3 a^{2} b \tanh \left (d x +c \right )^{5}}{5}-\frac {3 \tanh \left (d x +c \right )^{5} a \,b^{2}}{5}-\tanh \left (d x +c \right )^{3} a^{2} b -a \,b^{2} \tanh \left (d x +c \right )^{3}-\frac {a^{3} \tanh \left (d x +c \right )^{3}}{3}-\frac {b^{3} \tanh \left (d x +c \right )^{3}}{3}-\frac {\tanh \left (d x +c \right )^{7} b^{3}}{7}-\frac {b^{3} \tanh \left (d x +c \right )^{5}}{5}-a^{3} \tanh \left (d x +c \right )-b^{3} \tanh \left (d x +c \right )-\frac {\tanh \left (d x +c \right )^{9} b^{3}}{9}-\frac {\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \ln \left (\tanh \left (d x +c \right )+1\right )}{2}}{d}\)	\(247\)
default	\(\frac {-3 a^{2} b \tanh \left (d x +c \right )-3 a \,b^{2} \tanh \left (d x +c \right )-\frac {3 a \,b^{2} \tanh \left (d x +c \right )^{7}}{7}-\frac {3 a^{2} b \tanh \left (d x +c \right )^{5}}{5}-\frac {3 \tanh \left (d x +c \right )^{5} a \,b^{2}}{5}-\tanh \left (d x +c \right )^{3} a^{2} b -a \,b^{2} \tanh \left (d x +c \right )^{3}-\frac {a^{3} \tanh \left (d x +c \right )^{3}}{3}-\frac {b^{3} \tanh \left (d x +c \right )^{3}}{3}-\frac {\tanh \left (d x +c \right )^{7} b^{3}}{7}-\frac {b^{3} \tanh \left (d x +c \right )^{5}}{5}-a^{3} \tanh \left (d x +c \right )-b^{3} \tanh \left (d x +c \right )-\frac {\tanh \left (d x +c \right )^{9} b^{3}}{9}-\frac {\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \ln \left (\tanh \left (d x +c \right )+1\right )}{2}}{d}\)	\(247\)
parts	\(\frac {b^{3} \left (-\frac {\tanh \left (d x +c \right )^{9}}{9}-\frac {\tanh \left (d x +c \right )^{7}}{7}-\frac {\tanh \left (d x +c \right )^{5}}{5}-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {a^{3} \left (-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {3 a \,b^{2} \left (-\frac {\tanh \left (d x +c \right )^{7}}{7}-\frac {\tanh \left (d x +c \right )^{5}}{5}-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {3 a^{2} b \left (-\frac {\tanh \left (d x +c \right )^{5}}{5}-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}\)	\(258\)
risch	\(a^{3} x +3 b \,a^{2} x +3 a \,b^{2} x +b^{3} x +\frac {\frac {352 a \,b^{2}}{35}+10 b^{3} {\mathrm e}^{16 d x +16 c}+\frac {8 a^{3}}{3}+108 a^{2} b \,{\mathrm e}^{14 d x +14 c}+24 a \,b^{2} {\mathrm e}^{16 d x +16 c}+18 a^{2} b \,{\mathrm e}^{16 d x +16 c}+308 a^{2} b \,{\mathrm e}^{12 d x +12 c}+600 a \,b^{2} {\mathrm e}^{10 d x +10 c}+540 a^{2} b \,{\mathrm e}^{10 d x +10 c}+344 a \,b^{2} {\mathrm e}^{12 d x +12 c}+\frac {3104 \,{\mathrm e}^{4 d x +4 c} b^{3}}{35}+120 a \,b^{2} {\mathrm e}^{14 d x +14 c}+\frac {46 a^{2} b}{5}+\frac {2328 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}}{35}+\frac {324 a^{2} b \,{\mathrm e}^{2 d x +2 c}}{5}+\frac {8472 a \,b^{2} {\mathrm e}^{4 d x +4 c}}{35}+\frac {1116 a^{2} b \,{\mathrm e}^{4 d x +4 c}}{5}+\frac {2324 a^{2} b \,{\mathrm e}^{6 d x +6 c}}{5}+\frac {2504 a \,b^{2} {\mathrm e}^{6 d x +6 c}}{5}+\frac {1126 b^{3}}{315}+\frac {3096 a^{2} b \,{\mathrm e}^{8 d x +8 c}}{5}+\frac {3336 a \,b^{2} {\mathrm e}^{8 d x +8 c}}{5}+4 a^{3} {\mathrm e}^{16 d x +16 c}+\frac {400 b^{3} {\mathrm e}^{12 d x +12 c}}{3}+\frac {412 a^{3} {\mathrm e}^{6 d x +6 c}}{3}+\frac {2504 \,{\mathrm e}^{6 d x +6 c} b^{3}}{15}+68 a^{3} {\mathrm e}^{4 d x +4 c}+20 a^{3} {\mathrm e}^{2 d x +2 c}+\frac {776 \,{\mathrm e}^{2 d x +2 c} b^{3}}{35}+200 b^{3} {\mathrm e}^{10 d x +10 c}+28 a^{3} {\mathrm e}^{14 d x +14 c}+\frac {1252 b^{3} {\mathrm e}^{8 d x +8 c}}{5}+156 a^{3} {\mathrm e}^{10 d x +10 c}+\frac {260 a^{3} {\mathrm e}^{12 d x +12 c}}{3}+180 a^{3} {\mathrm e}^{8 d x +8 c}+40 b^{3} {\mathrm e}^{14 d x +14 c}}{d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{9}}\)	\(531\)

[In]

int(tanh(d*x+c)^4*(a+b*tanh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

-1/315*(35*tanh(d*x+c)^9*b^3+135*a*b^2*tanh(d*x+c)^7+45*tanh(d*x+c)^7*b^3+189*a^2*b*tanh(d*x+c)^5+189*tanh(d*x
+c)^5*a*b^2+63*b^3*tanh(d*x+c)^5+105*a^3*tanh(d*x+c)^3+315*tanh(d*x+c)^3*a^2*b+315*a*b^2*tanh(d*x+c)^3+105*b^3
*tanh(d*x+c)^3-315*a^3*d*x-945*a^2*b*d*x-945*a*b^2*d*x-315*b^3*d*x+315*a^3*tanh(d*x+c)+945*a^2*b*tanh(d*x+c)+9
45*a*b^2*tanh(d*x+c)+315*b^3*tanh(d*x+c))/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1563 vs. \(2 (106) = 212\).

Time = 0.27 (sec) , antiderivative size = 1563, normalized size of antiderivative = 13.71 \[ \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\text {Too large to display} \]

[In]

integrate(tanh(d*x+c)^4*(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

1/315*((420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^9
+ 9*(420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)*sin
h(d*x + c)^8 - (420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3)*sinh(d*x + c)^9 + 9*(420*a^3 + 1449*a^2*b + 1584*
a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^7 - 9*(280*a^3 + 819*a^2*b + 744*a*b^
2 + 213*b^3 + 4*(420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 21*(4*(420*a^
3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^3 + 3*(420*a^3
+ 1449*a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^6
+ 36*(420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^5 -
9*(14*(420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3)*cosh(d*x + c)^4 + 700*a^3 + 2016*a^2*b + 2136*a*b^2 + 852
*b^3 + 21*(280*a^3 + 819*a^2*b + 744*a*b^2 + 213*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 9*(14*(420*a^3 + 1449
*a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^5 + 35*(420*a^3 + 1449*
a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^3 + 20*(420*a^3 + 1449*a
^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^4 + 84*(42
0*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^3 - 3*(28*(
420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3)*cosh(d*x + c)^6 + 105*(280*a^3 + 819*a^2*b + 744*a*b^2 + 213*b^3)
*cosh(d*x + c)^4 + 2660*a^3 + 8232*a^2*b + 8232*a*b^2 + 1764*b^3 + 120*(175*a^3 + 504*a^2*b + 534*a*b^2 + 213*
b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 9*(4*(420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b
+ 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^7 + 21*(420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b
+ 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^5 + 40*(420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b +
3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^3 + 28*(420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b +
3*a*b^2 + b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^2 + 126*(420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a
^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c) - 9*((420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3)*cosh(d*x +
c)^8 + 7*(280*a^3 + 819*a^2*b + 744*a*b^2 + 213*b^3)*cosh(d*x + c)^6 + 20*(175*a^3 + 504*a^2*b + 534*a*b^2 +
213*b^3)*cosh(d*x + c)^4 + 420*a^3 + 1386*a^2*b + 1176*a*b^2 + 882*b^3 + 28*(95*a^3 + 294*a^2*b + 294*a*b^2 +
63*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/(d*cosh(d*x + c)^9 + 9*d*cosh(d*x + c)*sinh(d*x + c)^8 + 9*d*cosh(d*x
+ c)^7 + 21*(4*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^6 + 36*d*cosh(d*x + c)^5 + 9*(14*d*cosh(d*
x + c)^5 + 35*d*cosh(d*x + c)^3 + 20*d*cosh(d*x + c))*sinh(d*x + c)^4 + 84*d*cosh(d*x + c)^3 + 9*(4*d*cosh(d*x
+ c)^7 + 21*d*cosh(d*x + c)^5 + 40*d*cosh(d*x + c)^3 + 28*d*cosh(d*x + c))*sinh(d*x + c)^2 + 126*d*cosh(d*x +
c))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 260 vs. \(2 (99) = 198\).

Time = 0.30 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.28 \[ \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\begin {cases} a^{3} x - \frac {a^{3} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {a^{3} \tanh {\left (c + d x \right )}}{d} + 3 a^{2} b x - \frac {3 a^{2} b \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac {a^{2} b \tanh ^{3}{\left (c + d x \right )}}{d} - \frac {3 a^{2} b \tanh {\left (c + d x \right )}}{d} + 3 a b^{2} x - \frac {3 a b^{2} \tanh ^{7}{\left (c + d x \right )}}{7 d} - \frac {3 a b^{2} \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac {a b^{2} \tanh ^{3}{\left (c + d x \right )}}{d} - \frac {3 a b^{2} \tanh {\left (c + d x \right )}}{d} + b^{3} x - \frac {b^{3} \tanh ^{9}{\left (c + d x \right )}}{9 d} - \frac {b^{3} \tanh ^{7}{\left (c + d x \right )}}{7 d} - \frac {b^{3} \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac {b^{3} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {b^{3} \tanh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tanh ^{2}{\left (c \right )}\right )^{3} \tanh ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(tanh(d*x+c)**4*(a+b*tanh(d*x+c)**2)**3,x)

[Out]

Piecewise((a**3*x - a**3*tanh(c + d*x)**3/(3*d) - a**3*tanh(c + d*x)/d + 3*a**2*b*x - 3*a**2*b*tanh(c + d*x)**
5/(5*d) - a**2*b*tanh(c + d*x)**3/d - 3*a**2*b*tanh(c + d*x)/d + 3*a*b**2*x - 3*a*b**2*tanh(c + d*x)**7/(7*d)
- 3*a*b**2*tanh(c + d*x)**5/(5*d) - a*b**2*tanh(c + d*x)**3/d - 3*a*b**2*tanh(c + d*x)/d + b**3*x - b**3*tanh(
c + d*x)**9/(9*d) - b**3*tanh(c + d*x)**7/(7*d) - b**3*tanh(c + d*x)**5/(5*d) - b**3*tanh(c + d*x)**3/(3*d) -
b**3*tanh(c + d*x)/d, Ne(d, 0)), (x*(a + b*tanh(c)**2)**3*tanh(c)**4, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 583 vs. \(2 (106) = 212\).

Time = 0.21 (sec) , antiderivative size = 583, normalized size of antiderivative = 5.11 \[ \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {1}{315} \, b^{3} {\left (315 \, x + \frac {315 \, c}{d} - \frac {2 \, {\left (3492 \, e^{\left (-2 \, d x - 2 \, c\right )} + 13968 \, e^{\left (-4 \, d x - 4 \, c\right )} + 26292 \, e^{\left (-6 \, d x - 6 \, c\right )} + 39438 \, e^{\left (-8 \, d x - 8 \, c\right )} + 31500 \, e^{\left (-10 \, d x - 10 \, c\right )} + 21000 \, e^{\left (-12 \, d x - 12 \, c\right )} + 6300 \, e^{\left (-14 \, d x - 14 \, c\right )} + 1575 \, e^{\left (-16 \, d x - 16 \, c\right )} + 563\right )}}{d {\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} + 36 \, e^{\left (-4 \, d x - 4 \, c\right )} + 84 \, e^{\left (-6 \, d x - 6 \, c\right )} + 126 \, e^{\left (-8 \, d x - 8 \, c\right )} + 126 \, e^{\left (-10 \, d x - 10 \, c\right )} + 84 \, e^{\left (-12 \, d x - 12 \, c\right )} + 36 \, e^{\left (-14 \, d x - 14 \, c\right )} + 9 \, e^{\left (-16 \, d x - 16 \, c\right )} + e^{\left (-18 \, d x - 18 \, c\right )} + 1\right )}}\right )} + \frac {1}{35} \, a 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 {1}{5} \, a^{2} 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^{3} {\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)^3,x, algorithm="maxima")

[Out]

1/315*b^3*(315*x + 315*c/d - 2*(3492*e^(-2*d*x - 2*c) + 13968*e^(-4*d*x - 4*c) + 26292*e^(-6*d*x - 6*c) + 3943
8*e^(-8*d*x - 8*c) + 31500*e^(-10*d*x - 10*c) + 21000*e^(-12*d*x - 12*c) + 6300*e^(-14*d*x - 14*c) + 1575*e^(-
16*d*x - 16*c) + 563)/(d*(9*e^(-2*d*x - 2*c) + 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) + 126*e^(-8*d*x - 8*c
) + 126*e^(-10*d*x - 10*c) + 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) + 9*e^(-16*d*x - 16*c) + e^(-18*d*x
- 18*c) + 1))) + 1/35*a*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 - 14*c) + 1))) + 1/5*a^2*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^3*(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. 534 vs. \(2 (106) = 212\).

Time = 0.51 (sec) , antiderivative size = 534, normalized size of antiderivative = 4.68 \[ \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {315 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (630 \, a^{3} e^{\left (16 \, d x + 16 \, c\right )} + 2835 \, a^{2} b e^{\left (16 \, d x + 16 \, c\right )} + 3780 \, a b^{2} e^{\left (16 \, d x + 16 \, c\right )} + 1575 \, b^{3} e^{\left (16 \, d x + 16 \, c\right )} + 4410 \, a^{3} e^{\left (14 \, d x + 14 \, c\right )} + 17010 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} + 18900 \, a b^{2} e^{\left (14 \, d x + 14 \, c\right )} + 6300 \, b^{3} e^{\left (14 \, d x + 14 \, c\right )} + 13650 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 48510 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 54180 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 21000 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 24570 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 85050 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 94500 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 31500 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 28350 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 97524 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 105084 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 39438 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 21630 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 73206 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 78876 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 26292 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 10710 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 35154 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 38124 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 13968 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 3150 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 10206 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 10476 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3492 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 420 \, a^{3} + 1449 \, a^{2} b + 1584 \, a b^{2} + 563 \, b^{3}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{9}}}{315 \, d} \]

[In]

integrate(tanh(d*x+c)^4*(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

1/315*(315*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*(d*x + c) + 2*(630*a^3*e^(16*d*x + 16*c) + 2835*a^2*b*e^(16*d*x + 1
6*c) + 3780*a*b^2*e^(16*d*x + 16*c) + 1575*b^3*e^(16*d*x + 16*c) + 4410*a^3*e^(14*d*x + 14*c) + 17010*a^2*b*e^
(14*d*x + 14*c) + 18900*a*b^2*e^(14*d*x + 14*c) + 6300*b^3*e^(14*d*x + 14*c) + 13650*a^3*e^(12*d*x + 12*c) + 4
8510*a^2*b*e^(12*d*x + 12*c) + 54180*a*b^2*e^(12*d*x + 12*c) + 21000*b^3*e^(12*d*x + 12*c) + 24570*a^3*e^(10*d
*x + 10*c) + 85050*a^2*b*e^(10*d*x + 10*c) + 94500*a*b^2*e^(10*d*x + 10*c) + 31500*b^3*e^(10*d*x + 10*c) + 283
50*a^3*e^(8*d*x + 8*c) + 97524*a^2*b*e^(8*d*x + 8*c) + 105084*a*b^2*e^(8*d*x + 8*c) + 39438*b^3*e^(8*d*x + 8*c
) + 21630*a^3*e^(6*d*x + 6*c) + 73206*a^2*b*e^(6*d*x + 6*c) + 78876*a*b^2*e^(6*d*x + 6*c) + 26292*b^3*e^(6*d*x
+ 6*c) + 10710*a^3*e^(4*d*x + 4*c) + 35154*a^2*b*e^(4*d*x + 4*c) + 38124*a*b^2*e^(4*d*x + 4*c) + 13968*b^3*e^
(4*d*x + 4*c) + 3150*a^3*e^(2*d*x + 2*c) + 10206*a^2*b*e^(2*d*x + 2*c) + 10476*a*b^2*e^(2*d*x + 2*c) + 3492*b^
3*e^(2*d*x + 2*c) + 420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3)/(e^(2*d*x + 2*c) + 1)^9)/d

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.21 \[ \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=x\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )-\frac {\mathrm {tanh}\left (c+d\,x\right )\,{\left (a+b\right )}^3}{d}-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^5\,\left (3\,a^2\,b+3\,a\,b^2+b^3\right )}{5\,d}-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^7\,\left (b^3+3\,a\,b^2\right )}{7\,d}-\frac {b^3\,{\mathrm {tanh}\left (c+d\,x\right )}^9}{9\,d}-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^3\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}{3\,d} \]

[In]

int(tanh(c + d*x)^4*(a + b*tanh(c + d*x)^2)^3,x)

[Out]

x*(3*a*b^2 + 3*a^2*b + a^3 + b^3) - (tanh(c + d*x)*(a + b)^3)/d - (tanh(c + d*x)^5*(3*a*b^2 + 3*a^2*b + b^3))/
(5*d) - (tanh(c + d*x)^7*(3*a*b^2 + b^3))/(7*d) - (b^3*tanh(c + d*x)^9)/(9*d) - (tanh(c + d*x)^3*(3*a*b^2 + 3*
a^2*b + a^3 + b^3))/(3*d)