Integrand size = 23, antiderivative size = 94 \[ \int \tanh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=(a+b)^3 x-\frac {(a+b)^3 \tanh (c+d x)}{d}-\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^3(c+d x)}{3 d}-\frac {b^2 (3 a+b) \tanh ^5(c+d x)}{5 d}-\frac {b^3 \tanh ^7(c+d x)}{7 d} \]
(a+b)^3*x-(a+b)^3*tanh(d*x+c)/d-1/3*b*(3*a^2+3*a*b+b^2)*tanh(d*x+c)^3/d-1/ 5*b^2*(3*a+b)*tanh(d*x+c)^5/d-1/7*b^3*tanh(d*x+c)^7/d
Time = 2.00 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.15 \[ \int \tanh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {\tanh (c+d x) \left (-105 (a+b)^3-35 b \left (3 a^2+3 a b+b^2\right ) \tanh ^2(c+d x)-21 b^2 (3 a+b) \tanh ^4(c+d x)-15 b^3 \tanh ^6(c+d x)+\frac {105 (a+b)^3 \text {arctanh}\left (\sqrt {\tanh ^2(c+d x)}\right )}{\sqrt {\tanh ^2(c+d x)}}\right )}{105 d} \]
(Tanh[c + d*x]*(-105*(a + b)^3 - 35*b*(3*a^2 + 3*a*b + b^2)*Tanh[c + d*x]^ 2 - 21*b^2*(3*a + b)*Tanh[c + d*x]^4 - 15*b^3*Tanh[c + d*x]^6 + (105*(a + b)^3*ArcTanh[Sqrt[Tanh[c + d*x]^2]])/Sqrt[Tanh[c + d*x]^2]))/(105*d)
Time = 0.30 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 25, 4153, 25, 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.
| \(\displaystyle \int \tanh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\tan (i c+i d x)^2 \left (a-b \tan (i c+i d x)^2\right )^3dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \tan (i c+i d x)^2 \left (a-b \tan (i c+i d x)^2\right )^3dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle -\frac {\int -\frac {\tanh ^2(c+d x) \left (b \tanh ^2(c+d x)+a\right )^3}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\tanh ^2(c+d x) \left (b \tanh ^2(c+d x)+a\right )^3}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 364 |
| \(\displaystyle \frac {\int \left (-b^3 \tanh ^6(c+d x)-b^2 (3 a+b) \tanh ^4(c+d x)-b \left (3 a^2+3 b a+b^2\right ) \tanh ^2(c+d x)-(a+b)^3+\frac {a^3+3 b a^2+3 b^2 a+b^3}{1-\tanh ^2(c+d x)}\right )d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {1}{3} b \left (3 a^2+3 a b+b^2\right ) \tanh ^3(c+d x)-(a+b)^3 \text {arctanh}(\tanh (c+d x))+\frac {1}{5} b^2 (3 a+b) \tanh ^5(c+d x)+(a+b)^3 \tanh (c+d x)+\frac {1}{7} b^3 \tanh ^7(c+d x)}{d}\) |
-((-((a + b)^3*ArcTanh[Tanh[c + d*x]]) + (a + b)^3*Tanh[c + d*x] + (b*(3*a ^2 + 3*a*b + b^2)*Tanh[c + d*x]^3)/3 + (b^2*(3*a + b)*Tanh[c + d*x]^5)/5 + (b^3*Tanh[c + d*x]^7)/7)/d)
3.2.58.3.1 Defintions of rubi rules used
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]))
Time = 0.13 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.74
| method | result | size |
| parallelrisch | \(-\frac {15 \tanh \left (d x +c \right )^{7} b^{3}+63 \tanh \left (d x +c \right )^{5} a \,b^{2}+21 b^{3} \tanh \left (d x +c \right )^{5}+105 \tanh \left (d x +c \right )^{3} a^{2} b +105 a \,b^{2} \tanh \left (d x +c \right )^{3}+35 b^{3} \tanh \left (d x +c \right )^{3}-105 a^{3} d x -315 a^{2} b d x -315 a \,b^{2} d x -105 b^{3} d x +105 a^{3} \tanh \left (d x +c \right )+315 a^{2} b \tanh \left (d x +c \right )+315 a \,b^{2} \tanh \left (d x +c \right )+105 b^{3} \tanh \left (d x +c \right )}{105 d}\) | \(164\) |
| derivativedivides | \(\frac {-3 a^{2} b \tanh \left (d x +c \right )-3 a \,b^{2} \tanh \left (d x +c \right )-\frac {3 \tanh \left (d x +c \right )^{5} a \,b^{2}}{5}-\tanh \left (d x +c \right )^{3} a^{2} b -a \,b^{2} \tanh \left (d x +c \right )^{3}-\frac {b^{3} \tanh \left (d x +c \right )^{5}}{5}-\frac {b^{3} \tanh \left (d x +c \right )^{3}}{3}-a^{3} \tanh \left (d x +c \right )-b^{3} \tanh \left (d x +c \right )-\frac {\tanh \left (d x +c \right )^{7} b^{3}}{7}+\frac {\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \ln \left (\tanh \left (d x +c \right )+1\right )}{2}-\frac {\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \ln \left (\tanh \left (d x +c \right )-1\right )}{2}}{d}\) | \(193\) |
| default | \(\frac {-3 a^{2} b \tanh \left (d x +c \right )-3 a \,b^{2} \tanh \left (d x +c \right )-\frac {3 \tanh \left (d x +c \right )^{5} a \,b^{2}}{5}-\tanh \left (d x +c \right )^{3} a^{2} b -a \,b^{2} \tanh \left (d x +c \right )^{3}-\frac {b^{3} \tanh \left (d x +c \right )^{5}}{5}-\frac {b^{3} \tanh \left (d x +c \right )^{3}}{3}-a^{3} \tanh \left (d x +c \right )-b^{3} \tanh \left (d x +c \right )-\frac {\tanh \left (d x +c \right )^{7} b^{3}}{7}+\frac {\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \ln \left (\tanh \left (d x +c \right )+1\right )}{2}-\frac {\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \ln \left (\tanh \left (d x +c \right )-1\right )}{2}}{d}\) | \(193\) |
| parts | \(\frac {a^{3} \left (-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {b^{3} \left (-\frac {\tanh \left (d x +c \right )^{7}}{7}-\frac {\tanh \left (d x +c \right )^{5}}{5}-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {3 a^{2} b \left (-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {3 a \,b^{2} \left (-\frac {\tanh \left (d x +c \right )^{5}}{5}-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}\) | \(218\) |
| risch | \(a^{3} x +3 b \,a^{2} x +3 a \,b^{2} x +b^{3} x +\frac {\frac {46 a \,b^{2}}{5}+2 a^{3}+12 a^{2} b \,{\mathrm e}^{12 d x +12 c}+72 a \,b^{2} {\mathrm e}^{10 d x +10 c}+60 a^{2} b \,{\mathrm e}^{10 d x +10 c}+18 a \,b^{2} {\mathrm e}^{12 d x +12 c}+\frac {232 \,{\mathrm e}^{4 d x +4 c} b^{3}}{5}+8 a^{2} b +\frac {232 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}}{5}+44 a^{2} b \,{\mathrm e}^{2 d x +2 c}+\frac {606 a \,b^{2} {\mathrm e}^{4 d x +4 c}}{5}+108 a^{2} b \,{\mathrm e}^{4 d x +4 c}+152 a^{2} b \,{\mathrm e}^{6 d x +6 c}+176 a \,b^{2} {\mathrm e}^{6 d x +6 c}+\frac {352 b^{3}}{105}+128 a^{2} b \,{\mathrm e}^{8 d x +8 c}+146 a \,b^{2} {\mathrm e}^{8 d x +8 c}+8 b^{3} {\mathrm e}^{12 d x +12 c}+40 a^{3} {\mathrm e}^{6 d x +6 c}+\frac {176 \,{\mathrm e}^{6 d x +6 c} b^{3}}{3}+30 a^{3} {\mathrm e}^{4 d x +4 c}+12 a^{3} {\mathrm e}^{2 d x +2 c}+\frac {232 \,{\mathrm e}^{2 d x +2 c} b^{3}}{15}+24 b^{3} {\mathrm e}^{10 d x +10 c}+\frac {176 b^{3} {\mathrm e}^{8 d x +8 c}}{3}+12 a^{3} {\mathrm e}^{10 d x +10 c}+2 a^{3} {\mathrm e}^{12 d x +12 c}+30 a^{3} {\mathrm e}^{8 d x +8 c}}{d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{7}}\) | \(415\) |
-1/105*(15*tanh(d*x+c)^7*b^3+63*tanh(d*x+c)^5*a*b^2+21*b^3*tanh(d*x+c)^5+1 05*tanh(d*x+c)^3*a^2*b+105*a*b^2*tanh(d*x+c)^3+35*b^3*tanh(d*x+c)^3-105*a^ 3*d*x-315*a^2*b*d*x-315*a*b^2*d*x-105*b^3*d*x+105*a^3*tanh(d*x+c)+315*a^2* b*tanh(d*x+c)+315*a*b^2*tanh(d*x+c)+105*b^3*tanh(d*x+c))/d
Leaf count of result is larger than twice the leaf count of optimal. 1036 vs. \(2 (88) = 176\).
Time = 0.27 (sec) , antiderivative size = 1036, normalized size of antiderivative = 11.02 \[ \int \tanh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\text {Too large to display} \]
1/105*((105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3 + 105*(a^3 + 3*a^2*b + 3 *a*b^2 + b^3)*d*x)*cosh(d*x + c)^7 + 7*(105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3 + 105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)*sinh(d*x + c)^6 - (105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3)*sinh(d*x + c)^7 + 7*( 105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3 + 105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^5 - 7*(75*a^3 + 240*a^2*b + 213*a*b^2 + 56*b^3 + 3*(105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 35*((105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3 + 105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^3 + (105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3 + 105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c))*sinh(d *x + c)^4 + 21*(105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3 + 105*(a^3 + 3*a ^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^3 - 7*(5*(105*a^3 + 420*a^2*b + 4 83*a*b^2 + 176*b^3)*cosh(d*x + c)^4 + 135*a^3 + 360*a^2*b + 369*a*b^2 + 16 8*b^3 + 10*(75*a^3 + 240*a^2*b + 213*a*b^2 + 56*b^3)*cosh(d*x + c)^2)*sinh (d*x + c)^3 + 7*(3*(105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3 + 105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^5 + 10*(105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3 + 105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^3 + 9*(105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3 + 105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^2 + 35*(105*a^3 + 420*a ^2*b + 483*a*b^2 + 176*b^3 + 105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*c...
Leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (82) = 164\).
Time = 0.22 (sec) , antiderivative size = 192, normalized size of antiderivative = 2.04 \[ \int \tanh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\begin {cases} a^{3} x - \frac {a^{3} \tanh {\left (c + d x \right )}}{d} + 3 a^{2} b x - \frac {a^{2} b \tanh ^{3}{\left (c + d x \right )}}{d} - \frac {3 a^{2} b \tanh {\left (c + d x \right )}}{d} + 3 a b^{2} x - \frac {3 a b^{2} \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac {a b^{2} \tanh ^{3}{\left (c + d x \right )}}{d} - \frac {3 a b^{2} \tanh {\left (c + d x \right )}}{d} + b^{3} x - \frac {b^{3} \tanh ^{7}{\left (c + d x \right )}}{7 d} - \frac {b^{3} \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac {b^{3} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {b^{3} \tanh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tanh ^{2}{\left (c \right )}\right )^{3} \tanh ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((a**3*x - a**3*tanh(c + d*x)/d + 3*a**2*b*x - 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)** 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)**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)**2, True))
Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (88) = 176\).
Time = 0.21 (sec) , antiderivative size = 400, normalized size of antiderivative = 4.26 \[ \int \tanh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {1}{105} \, b^{3} {\left (105 \, x + \frac {105 \, c}{d} - \frac {8 \, {\left (203 \, e^{\left (-2 \, d x - 2 \, c\right )} + 609 \, e^{\left (-4 \, d x - 4 \, c\right )} + 770 \, e^{\left (-6 \, d x - 6 \, c\right )} + 770 \, e^{\left (-8 \, d x - 8 \, c\right )} + 315 \, e^{\left (-10 \, d x - 10 \, c\right )} + 105 \, e^{\left (-12 \, d x - 12 \, c\right )} + 44\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} + 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} + 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} + 1\right )}}\right )} + \frac {1}{5} \, a b^{2} {\left (15 \, x + \frac {15 \, c}{d} - \frac {2 \, {\left (70 \, e^{\left (-2 \, d x - 2 \, c\right )} + 140 \, e^{\left (-4 \, d x - 4 \, c\right )} + 90 \, e^{\left (-6 \, d x - 6 \, c\right )} + 45 \, e^{\left (-8 \, d x - 8 \, c\right )} + 23\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} + a^{2} b {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 2\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} + a^{3} {\left (x + \frac {c}{d} - \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} \]
1/105*b^3*(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*b^2*(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))) + a^2*b*( 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))) + a^3*(x + c/d - 2/(d*(e^(-2*d*x - 2*c) + 1)))
Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (88) = 176\).
Time = 0.43 (sec) , antiderivative size = 418, normalized size of antiderivative = 4.45 \[ \int \tanh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {105 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (105 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 630 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 945 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 420 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 630 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 3150 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 3780 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 1260 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 1575 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 6720 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 7665 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 3080 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 2100 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 7980 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 9240 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 3080 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 1575 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 5670 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 6363 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 2436 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 630 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2310 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 2436 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 812 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 105 \, a^{3} + 420 \, a^{2} b + 483 \, a b^{2} + 176 \, b^{3}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{7}}}{105 \, d} \]
1/105*(105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*(d*x + c) + 2*(105*a^3*e^(12*d* x + 12*c) + 630*a^2*b*e^(12*d*x + 12*c) + 945*a*b^2*e^(12*d*x + 12*c) + 42 0*b^3*e^(12*d*x + 12*c) + 630*a^3*e^(10*d*x + 10*c) + 3150*a^2*b*e^(10*d*x + 10*c) + 3780*a*b^2*e^(10*d*x + 10*c) + 1260*b^3*e^(10*d*x + 10*c) + 157 5*a^3*e^(8*d*x + 8*c) + 6720*a^2*b*e^(8*d*x + 8*c) + 7665*a*b^2*e^(8*d*x + 8*c) + 3080*b^3*e^(8*d*x + 8*c) + 2100*a^3*e^(6*d*x + 6*c) + 7980*a^2*b*e ^(6*d*x + 6*c) + 9240*a*b^2*e^(6*d*x + 6*c) + 3080*b^3*e^(6*d*x + 6*c) + 1 575*a^3*e^(4*d*x + 4*c) + 5670*a^2*b*e^(4*d*x + 4*c) + 6363*a*b^2*e^(4*d*x + 4*c) + 2436*b^3*e^(4*d*x + 4*c) + 630*a^3*e^(2*d*x + 2*c) + 2310*a^2*b* e^(2*d*x + 2*c) + 2436*a*b^2*e^(2*d*x + 2*c) + 812*b^3*e^(2*d*x + 2*c) + 1 05*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3)/(e^(2*d*x + 2*c) + 1)^7)/d
Time = 1.87 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.13 \[ \int \tanh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=x\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )-\frac {\mathrm {tanh}\left (c+d\,x\right )\,{\left (a+b\right )}^3}{d}-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^3\,\left (3\,a^2\,b+3\,a\,b^2+b^3\right )}{3\,d}-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^5\,\left (b^3+3\,a\,b^2\right )}{5\,d}-\frac {b^3\,{\mathrm {tanh}\left (c+d\,x\right )}^7}{7\,d} \]