Integrand size = 23, antiderivative size = 114 \[ \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=(a+b)^3 x-\frac {(a+b)^3 \tanh (c+d x)}{d}-\frac {(a+b)^3 \tanh ^3(c+d x)}{3 d}-\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^2 (3 a+b) \tanh ^7(c+d x)}{7 d}-\frac {b^3 \tanh ^9(c+d x)}{9 d} \] Output:
(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
Time = 1.00 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.08 \[ \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {\tanh (c+d x) \left (-315 (a+b)^3-105 (a+b)^3 \tanh ^2(c+d x)-63 b \left (3 a^2+3 a b+b^2\right ) \tanh ^4(c+d x)-45 b^2 (3 a+b) \tanh ^6(c+d x)-35 b^3 \tanh ^8(c+d x)+\frac {315 (a+b)^3 \text {arctanh}\left (\sqrt {\tanh ^2(c+d x)}\right )}{\sqrt {\tanh ^2(c+d x)}}\right )}{315 d} \] Input:
Integrate[Tanh[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^3,x]
Output:
(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]])/Sqrt[ Tanh[c + d*x]^2]))/(315*d)
Time = 0.32 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 4153, 364, 2009}
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.
| \(\displaystyle \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (i c+i d x)^4 \left (a-b \tan (i c+i d x)^2\right )^3dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle \frac {\int \frac {\tanh ^4(c+d x) \left (b \tanh ^2(c+d x)+a\right )^3}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 364 |
| \(\displaystyle \frac {\int \left (-b^3 \tanh ^8(c+d x)-b^2 (3 a+b) \tanh ^6(c+d x)-b \left (3 a^2+3 b a+b^2\right ) \tanh ^4(c+d x)-(a+b)^3 \tanh ^2(c+d x)-(a+b)^3+\frac {a^3+3 b a^2+3 b^2 a+b^3}{1-\tanh ^2(c+d x)}\right )d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {1}{5} b \left (3 a^2+3 a b+b^2\right ) \tanh ^5(c+d x)+(a+b)^3 \text {arctanh}(\tanh (c+d x))-\frac {1}{7} b^2 (3 a+b) \tanh ^7(c+d x)-\frac {1}{3} (a+b)^3 \tanh ^3(c+d x)-(a+b)^3 \tanh (c+d x)-\frac {1}{9} b^3 \tanh ^9(c+d x)}{d}\) |
Input:
Int[Tanh[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^3,x]
Output:
((a + b)^3*ArcTanh[Tanh[c + d*x]] - (a + b)^3*Tanh[c + d*x] - ((a + b)^3*T anh[c + d*x]^3)/3 - (b*(3*a^2 + 3*a*b + b^2)*Tanh[c + d*x]^5)/5 - (b^2*(3* a + b)*Tanh[c + d*x]^7)/7 - (b^3*Tanh[c + d*x]^9)/9)/d
Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*((a + b*x^2)^p/(c + d*x^2)), x], x ] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && (In tegerQ[m] || IGtQ[2*(m + 1), 0] || !RationalQ[m])
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]}, Simp[c*(ff/f) Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2 + f f^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] && Ratio nalQ[n]))
Leaf count of result is larger than twice the leaf count of optimal. \(217\) vs. \(2(106)=212\).
Time = 0.12 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.91
| method | result | size |
| parallelrisch | \(-\frac {35 b^{3} \tanh \left (d x +c \right )^{9}+135 a \,b^{2} \tanh \left (d x +c \right )^{7}+45 b^{3} \tanh \left (d x +c \right )^{7}+189 a^{2} b \tanh \left (d x +c \right )^{5}+189 a \,b^{2} \tanh \left (d x +c \right )^{5}+63 b^{3} \tanh \left (d x +c \right )^{5}+105 a^{3} \tanh \left (d x +c \right )^{3}+315 a^{2} b \tanh \left (d x +c \right )^{3}+315 a \,b^{2} \tanh \left (d x +c \right )^{3}+105 b^{3} \tanh \left (d x +c \right )^{3}-315 a^{3} d x -945 a^{2} b d x -945 a \,b^{2} d x -315 b^{3} d x +315 a^{3} \tanh \left (d x +c \right )+945 a^{2} b \tanh \left (d x +c \right )+945 a \,b^{2} \tanh \left (d x +c \right )+315 b^{3} \tanh \left (d x +c \right )}{315 d}\) | \(218\) |
| derivativedivides | \(\frac {-3 a^{2} b \tanh \left (d x +c \right )-3 a \,b^{2} \tanh \left (d x +c \right )-\frac {3 a \,b^{2} \tanh \left (d x +c \right )^{7}}{7}-\frac {3 a^{2} b \tanh \left (d x +c \right )^{5}}{5}-\frac {3 a \,b^{2} \tanh \left (d x +c \right )^{5}}{5}-a^{2} b \tanh \left (d x +c \right )^{3}-a \,b^{2} \tanh \left (d x +c \right )^{3}+\frac {\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \ln \left (1+\tanh \left (d x +c \right )\right )}{2}-\frac {\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \ln \left (-1+\tanh \left (d x +c \right )\right )}{2}-\frac {b^{3} \tanh \left (d x +c \right )^{7}}{7}-\frac {b^{3} \tanh \left (d x +c \right )^{5}}{5}-\frac {a^{3} \tanh \left (d x +c \right )^{3}}{3}-\frac {b^{3} \tanh \left (d x +c \right )^{3}}{3}-a^{3} \tanh \left (d x +c \right )-b^{3} \tanh \left (d x +c \right )-\frac {b^{3} \tanh \left (d x +c \right )^{9}}{9}}{d}\) | \(247\) |
| default | \(\frac {-3 a^{2} b \tanh \left (d x +c \right )-3 a \,b^{2} \tanh \left (d x +c \right )-\frac {3 a \,b^{2} \tanh \left (d x +c \right )^{7}}{7}-\frac {3 a^{2} b \tanh \left (d x +c \right )^{5}}{5}-\frac {3 a \,b^{2} \tanh \left (d x +c \right )^{5}}{5}-a^{2} b \tanh \left (d x +c \right )^{3}-a \,b^{2} \tanh \left (d x +c \right )^{3}+\frac {\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \ln \left (1+\tanh \left (d x +c \right )\right )}{2}-\frac {\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \ln \left (-1+\tanh \left (d x +c \right )\right )}{2}-\frac {b^{3} \tanh \left (d x +c \right )^{7}}{7}-\frac {b^{3} \tanh \left (d x +c \right )^{5}}{5}-\frac {a^{3} \tanh \left (d x +c \right )^{3}}{3}-\frac {b^{3} \tanh \left (d x +c \right )^{3}}{3}-a^{3} \tanh \left (d x +c \right )-b^{3} \tanh \left (d x +c \right )-\frac {b^{3} \tanh \left (d x +c \right )^{9}}{9}}{d}\) | \(247\) |
| parts | \(\frac {a^{3} \left (-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (-1+\tanh \left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tanh \left (d x +c \right )\right )}{2}\right )}{d}+\frac {b^{3} \left (-\frac {\tanh \left (d x +c \right )^{9}}{9}-\frac {\tanh \left (d x +c \right )^{7}}{7}-\frac {\tanh \left (d x +c \right )^{5}}{5}-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (-1+\tanh \left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tanh \left (d x +c \right )\right )}{2}\right )}{d}+\frac {3 a \,b^{2} \left (-\frac {\tanh \left (d x +c \right )^{7}}{7}-\frac {\tanh \left (d x +c \right )^{5}}{5}-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (-1+\tanh \left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tanh \left (d x +c \right )\right )}{2}\right )}{d}+\frac {3 a^{2} b \left (-\frac {\tanh \left (d x +c \right )^{5}}{5}-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (-1+\tanh \left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tanh \left (d x +c \right )\right )}{2}\right )}{d}\) | \(258\) |
| risch | \(a^{3} x +3 a^{2} b x +3 a \,b^{2} x +b^{3} x +\frac {\frac {8 a^{3}}{3}+\frac {46 a^{2} b}{5}+\frac {352 a \,b^{2}}{35}+\frac {1126 b^{3}}{315}+\frac {3104 b^{3} {\mathrm e}^{4 d x +4 c}}{35}+\frac {1252 b^{3} {\mathrm e}^{8 d x +8 c}}{5}+\frac {412 a^{3} {\mathrm e}^{6 d x +6 c}}{3}+\frac {2504 b^{3} {\mathrm e}^{6 d x +6 c}}{15}+68 a^{3} {\mathrm e}^{4 d x +4 c}+\frac {400 b^{3} {\mathrm e}^{12 d x +12 c}}{3}+156 a^{3} {\mathrm e}^{10 d x +10 c}+200 b^{3} {\mathrm e}^{10 d x +10 c}+180 a^{3} {\mathrm e}^{8 d x +8 c}+10 b^{3} {\mathrm e}^{16 d x +16 c}+28 a^{3} {\mathrm e}^{14 d x +14 c}+40 b^{3} {\mathrm e}^{14 d x +14 c}+\frac {260 a^{3} {\mathrm e}^{12 d x +12 c}}{3}+\frac {776 b^{3} {\mathrm e}^{2 d x +2 c}}{35}+4 a^{3} {\mathrm e}^{16 d x +16 c}+20 a^{3} {\mathrm e}^{2 d x +2 c}+\frac {2324 a^{2} b \,{\mathrm e}^{6 d x +6 c}}{5}+\frac {2504 a \,b^{2} {\mathrm e}^{6 d x +6 c}}{5}+\frac {1116 a^{2} b \,{\mathrm e}^{4 d x +4 c}}{5}+\frac {8472 a \,b^{2} {\mathrm e}^{4 d x +4 c}}{35}+\frac {324 a^{2} b \,{\mathrm e}^{2 d x +2 c}}{5}+\frac {2328 a \,b^{2} {\mathrm e}^{2 d x +2 c}}{35}+120 a \,b^{2} {\mathrm e}^{14 d x +14 c}+308 a^{2} b \,{\mathrm e}^{12 d x +12 c}+344 a \,b^{2} {\mathrm e}^{12 d x +12 c}+540 a^{2} b \,{\mathrm e}^{10 d x +10 c}+600 a \,b^{2} {\mathrm e}^{10 d x +10 c}+\frac {3096 a^{2} b \,{\mathrm e}^{8 d x +8 c}}{5}+\frac {3336 a \,b^{2} {\mathrm e}^{8 d x +8 c}}{5}+18 a^{2} b \,{\mathrm e}^{16 d x +16 c}+24 a \,b^{2} {\mathrm e}^{16 d x +16 c}+108 a^{2} b \,{\mathrm e}^{14 d x +14 c}}{d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{9}}\) | \(531\) |
Input:
int(tanh(d*x+c)^4*(a+b*tanh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)
Output:
-1/315*(35*b^3*tanh(d*x+c)^9+135*a*b^2*tanh(d*x+c)^7+45*b^3*tanh(d*x+c)^7+ 189*a^2*b*tanh(d*x+c)^5+189*a*b^2*tanh(d*x+c)^5+63*b^3*tanh(d*x+c)^5+105*a ^3*tanh(d*x+c)^3+315*a^2*b*tanh(d*x+c)^3+315*a*b^2*tanh(d*x+c)^3+105*b^3*t anh(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*t anh(d*x+c)+945*a^2*b*tanh(d*x+c)+945*a*b^2*tanh(d*x+c)+315*b^3*tanh(d*x+c) )/d
Leaf count of result is larger than twice the leaf count of optimal. 1563 vs. \(2 (106) = 212\).
Time = 0.11 (sec) , antiderivative size = 1563, normalized size of antiderivative = 13.71 \[ \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:
integrate(tanh(d*x+c)^4*(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")
Output:
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)*sinh( 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 + 31 5*(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)*c osh(d*x + c))*sinh(d*x + c)^6 + 36*(420*a^3 + 1449*a^2*b + 1584*a*b^2 + 56 3*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 + 2 016*a^2*b + 2136*a*b^2 + 852*b^3 + 21*(280*a^3 + 819*a^2*b + 744*a*b^2 + 2 13*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 9*(14*(420*a^3 + 1449*a^2*b + 1 584*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*(420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3 + 315...
Leaf count of result is larger than twice the leaf count of optimal. 260 vs. \(2 (99) = 198\).
Time = 0.31 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.28 \[ \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\begin {cases} a^{3} x - \frac {a^{3} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {a^{3} \tanh {\left (c + d x \right )}}{d} + 3 a^{2} b x - \frac {3 a^{2} b \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac {a^{2} b \tanh ^{3}{\left (c + d x \right )}}{d} - \frac {3 a^{2} b \tanh {\left (c + d x \right )}}{d} + 3 a b^{2} x - \frac {3 a b^{2} \tanh ^{7}{\left (c + d x \right )}}{7 d} - \frac {3 a b^{2} \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac {a b^{2} \tanh ^{3}{\left (c + d x \right )}}{d} - \frac {3 a b^{2} \tanh {\left (c + d x \right )}}{d} + b^{3} x - \frac {b^{3} \tanh ^{9}{\left (c + d x \right )}}{9 d} - \frac {b^{3} \tanh ^{7}{\left (c + d x \right )}}{7 d} - \frac {b^{3} \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac {b^{3} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {b^{3} \tanh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tanh ^{2}{\left (c \right )}\right )^{3} \tanh ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(tanh(d*x+c)**4*(a+b*tanh(d*x+c)**2)**3,x)
Output:
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*tan h(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))
Leaf count of result is larger than twice the leaf count of optimal. 583 vs. \(2 (106) = 212\).
Time = 0.06 (sec) , antiderivative size = 583, normalized size of antiderivative = 5.11 \[ \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx =\text {Too large to display} \] Input:
integrate(tanh(d*x+c)^4*(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")
Output:
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) + 39438*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 - 1 8*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^(-1 0*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)))
Leaf count of result is larger than twice the leaf count of optimal. 534 vs. \(2 (106) = 212\).
Time = 0.29 (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} \] Input:
integrate(tanh(d*x+c)^4*(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")
Output:
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 + 16*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^(1 4*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) + 48510*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) + 3 1500*b^3*e^(10*d*x + 10*c) + 28350*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) + 21 630*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 + 158 4*a*b^2 + 563*b^3)/(e^(2*d*x + 2*c) + 1)^9)/d
Time = 0.24 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.21 \[ \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=x\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )-\frac {\mathrm {tanh}\left (c+d\,x\right )\,{\left (a+b\right )}^3}{d}-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^5\,\left (3\,a^2\,b+3\,a\,b^2+b^3\right )}{5\,d}-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^7\,\left (b^3+3\,a\,b^2\right )}{7\,d}-\frac {b^3\,{\mathrm {tanh}\left (c+d\,x\right )}^9}{9\,d}-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^3\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}{3\,d} \] Input:
int(tanh(c + d*x)^4*(a + b*tanh(c + d*x)^2)^3,x)
Output:
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)
Time = 0.25 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.90 \[ \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {-35 \tanh \left (d x +c \right )^{9} b^{3}-135 \tanh \left (d x +c \right )^{7} a \,b^{2}-45 \tanh \left (d x +c \right )^{7} b^{3}-189 \tanh \left (d x +c \right )^{5} a^{2} b -189 \tanh \left (d x +c \right )^{5} a \,b^{2}-63 \tanh \left (d x +c \right )^{5} b^{3}-105 \tanh \left (d x +c \right )^{3} a^{3}-315 \tanh \left (d x +c \right )^{3} a^{2} b -315 \tanh \left (d x +c \right )^{3} a \,b^{2}-105 \tanh \left (d x +c \right )^{3} b^{3}-315 \tanh \left (d x +c \right ) a^{3}-945 \tanh \left (d x +c \right ) a^{2} b -945 \tanh \left (d x +c \right ) a \,b^{2}-315 \tanh \left (d x +c \right ) b^{3}+315 a^{3} d x +945 a^{2} b d x +945 a \,b^{2} d x +315 b^{3} d x}{315 d} \] Input:
int(tanh(d*x+c)^4*(a+b*tanh(d*x+c)^2)^3,x)
Output:
( - 35*tanh(c + d*x)**9*b**3 - 135*tanh(c + d*x)**7*a*b**2 - 45*tanh(c + d *x)**7*b**3 - 189*tanh(c + d*x)**5*a**2*b - 189*tanh(c + d*x)**5*a*b**2 - 63*tanh(c + d*x)**5*b**3 - 105*tanh(c + d*x)**3*a**3 - 315*tanh(c + d*x)** 3*a**2*b - 315*tanh(c + d*x)**3*a*b**2 - 105*tanh(c + d*x)**3*b**3 - 315*t anh(c + d*x)*a**3 - 945*tanh(c + d*x)*a**2*b - 945*tanh(c + d*x)*a*b**2 - 315*tanh(c + d*x)*b**3 + 315*a**3*d*x + 945*a**2*b*d*x + 945*a*b**2*d*x + 315*b**3*d*x)/(315*d)