Integrand size = 14, antiderivative size = 110 \[ \int \left (a+b \tanh ^2(c+d x)\right )^4 \, dx=(a+b)^4 x-\frac {b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \tanh (c+d x)}{d}-\frac {b^2 \left (6 a^2+4 a b+b^2\right ) \tanh ^3(c+d x)}{3 d}-\frac {b^3 (4 a+b) \tanh ^5(c+d x)}{5 d}-\frac {b^4 \tanh ^7(c+d x)}{7 d} \]
(a+b)^4*x-b*(2*a+b)*(2*a^2+2*a*b+b^2)*tanh(d*x+c)/d-1/3*b^2*(6*a^2+4*a*b+b ^2)*tanh(d*x+c)^3/d-1/5*b^3*(4*a+b)*tanh(d*x+c)^5/d-1/7*b^4*tanh(d*x+c)^7/ d
Time = 1.20 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.16 \[ \int \left (a+b \tanh ^2(c+d x)\right )^4 \, dx=\frac {\tanh (c+d x) \left (\frac {105 (a+b)^4 \text {arctanh}\left (\sqrt {\tanh ^2(c+d x)}\right )}{\sqrt {\tanh ^2(c+d x)}}-b \left (105 \left (4 a^3+6 a^2 b+4 a b^2+b^3\right )+35 b \left (6 a^2+4 a b+b^2\right ) \tanh ^2(c+d x)+21 b^2 (4 a+b) \tanh ^4(c+d x)+15 b^3 \tanh ^6(c+d x)\right )\right )}{105 d} \]
(Tanh[c + d*x]*((105*(a + b)^4*ArcTanh[Sqrt[Tanh[c + d*x]^2]])/Sqrt[Tanh[c + d*x]^2] - b*(105*(4*a^3 + 6*a^2*b + 4*a*b^2 + b^3) + 35*b*(6*a^2 + 4*a* b + b^2)*Tanh[c + d*x]^2 + 21*b^2*(4*a + b)*Tanh[c + d*x]^4 + 15*b^3*Tanh[ c + d*x]^6)))/(105*d)
Time = 0.29 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 4144, 300, 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.
| \(\displaystyle \int \left (a+b \tanh ^2(c+d x)\right )^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a-b \tan (i c+i d x)^2\right )^4dx\) |
| \(\Big \downarrow \) 4144 |
| \(\displaystyle \frac {\int \frac {\left (b \tanh ^2(c+d x)+a\right )^4}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 300 |
| \(\displaystyle \frac {\int \left (-b^4 \tanh ^6(c+d x)-b^3 (4 a+b) \tanh ^4(c+d x)-b^2 \left (6 a^2+4 b a+b^2\right ) \tanh ^2(c+d x)-b (2 a+b) \left (2 a^2+2 b a+b^2\right )+\frac {(a+b)^4}{1-\tanh ^2(c+d x)}\right )d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {1}{3} b^2 \left (6 a^2+4 a b+b^2\right ) \tanh ^3(c+d x)-b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \tanh (c+d x)+(a+b)^4 \text {arctanh}(\tanh (c+d x))-\frac {1}{5} b^3 (4 a+b) \tanh ^5(c+d x)-\frac {1}{7} b^4 \tanh ^7(c+d x)}{d}\) |
((a + b)^4*ArcTanh[Tanh[c + d*x]] - b*(2*a + b)*(2*a^2 + 2*a*b + b^2)*Tanh [c + d*x] - (b^2*(6*a^2 + 4*a*b + b^2)*Tanh[c + d*x]^3)/3 - (b^3*(4*a + b) *Tanh[c + d*x]^5)/5 - (b^4*Tanh[c + d*x]^7)/7)/d
3.2.68.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int [PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c , d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
Int[((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[(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, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])
Time = 0.13 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.63
| method | result | size |
| parallelrisch | \(-\frac {15 b^{4} \tanh \left (d x +c \right )^{7}+84 a \,b^{3} \tanh \left (d x +c \right )^{5}+21 b^{4} \tanh \left (d x +c \right )^{5}+210 a^{2} b^{2} \tanh \left (d x +c \right )^{3}+140 a \,b^{3} \tanh \left (d x +c \right )^{3}+35 b^{4} \tanh \left (d x +c \right )^{3}-105 a^{4} d x -420 a^{3} b d x -630 a^{2} b^{2} d x -420 a \,b^{3} d x -105 b^{4} d x +420 a^{3} b \tanh \left (d x +c \right )+630 a^{2} b^{2} \tanh \left (d x +c \right )+420 a \,b^{3} \tanh \left (d x +c \right )+105 b^{4} \tanh \left (d x +c \right )}{105 d}\) | \(179\) |
| derivativedivides | \(\frac {-4 a^{3} b \tanh \left (d x +c \right )-6 a^{2} b^{2} \tanh \left (d x +c \right )-4 a \,b^{3} \tanh \left (d x +c \right )-\frac {4 a \,b^{3} \tanh \left (d x +c \right )^{5}}{5}-2 a^{2} b^{2} \tanh \left (d x +c \right )^{3}-\frac {4 a \,b^{3} \tanh \left (d x +c \right )^{3}}{3}-\frac {b^{4} \tanh \left (d x +c \right )^{5}}{5}-\frac {b^{4} \tanh \left (d x +c \right )^{3}}{3}-b^{4} \tanh \left (d x +c \right )-\frac {b^{4} \tanh \left (d x +c \right )^{7}}{7}+\frac {\left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) \ln \left (\tanh \left (d x +c \right )+1\right )}{2}-\frac {\left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) \ln \left (\tanh \left (d x +c \right )-1\right )}{2}}{d}\) | \(214\) |
| default | \(\frac {-4 a^{3} b \tanh \left (d x +c \right )-6 a^{2} b^{2} \tanh \left (d x +c \right )-4 a \,b^{3} \tanh \left (d x +c \right )-\frac {4 a \,b^{3} \tanh \left (d x +c \right )^{5}}{5}-2 a^{2} b^{2} \tanh \left (d x +c \right )^{3}-\frac {4 a \,b^{3} \tanh \left (d x +c \right )^{3}}{3}-\frac {b^{4} \tanh \left (d x +c \right )^{5}}{5}-\frac {b^{4} \tanh \left (d x +c \right )^{3}}{3}-b^{4} \tanh \left (d x +c \right )-\frac {b^{4} \tanh \left (d x +c \right )^{7}}{7}+\frac {\left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) \ln \left (\tanh \left (d x +c \right )+1\right )}{2}-\frac {\left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) \ln \left (\tanh \left (d x +c \right )-1\right )}{2}}{d}\) | \(214\) |
| parts | \(x \,a^{4}+\frac {b^{4} \left (-\frac {\tanh \left (d x +c \right )^{7}}{7}-\frac {\tanh \left (d x +c \right )^{5}}{5}-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {4 a \,b^{3} \left (-\frac {\tanh \left (d x +c \right )^{5}}{5}-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {6 a^{2} b^{2} \left (-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {4 a^{3} b \left (-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}\) | \(227\) |
| risch | \(x \,a^{4}+4 b \,a^{3} x +6 a^{2} b^{2} x +4 a \,b^{3} x +b^{4} x +\frac {8 b \left (161 a \,b^{2}+105 a^{3}+315 a^{2} b \,{\mathrm e}^{12 d x +12 c}+1575 a^{2} b \,{\mathrm e}^{10 d x +10 c}+315 a \,b^{2} {\mathrm e}^{12 d x +12 c}+609 \,{\mathrm e}^{4 d x +4 c} b^{3}+210 a^{2} b +812 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}+1155 a^{2} b \,{\mathrm e}^{2 d x +2 c}+2121 a \,b^{2} {\mathrm e}^{4 d x +4 c}+1260 a \,b^{2} {\mathrm e}^{10 d x +10 c}+2835 a^{2} b \,{\mathrm e}^{4 d x +4 c}+3990 a^{2} b \,{\mathrm e}^{6 d x +6 c}+3080 a \,b^{2} {\mathrm e}^{6 d x +6 c}+770 b^{3} {\mathrm e}^{8 d x +8 c}+44 b^{3}+3360 a^{2} b \,{\mathrm e}^{8 d x +8 c}+2555 a \,b^{2} {\mathrm e}^{8 d x +8 c}+2100 a^{3} {\mathrm e}^{6 d x +6 c}+770 \,{\mathrm e}^{6 d x +6 c} b^{3}+1575 a^{3} {\mathrm e}^{4 d x +4 c}+630 a^{3} {\mathrm e}^{2 d x +2 c}+203 \,{\mathrm e}^{2 d x +2 c} b^{3}+315 b^{3} {\mathrm e}^{10 d x +10 c}+105 b^{3} {\mathrm e}^{12 d x +12 c}+630 a^{3} {\mathrm e}^{10 d x +10 c}+105 a^{3} {\mathrm e}^{12 d x +12 c}+1575 a^{3} {\mathrm e}^{8 d x +8 c}\right )}{105 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{7}}\) | \(425\) |
-1/105*(15*b^4*tanh(d*x+c)^7+84*a*b^3*tanh(d*x+c)^5+21*b^4*tanh(d*x+c)^5+2 10*a^2*b^2*tanh(d*x+c)^3+140*a*b^3*tanh(d*x+c)^3+35*b^4*tanh(d*x+c)^3-105* a^4*d*x-420*a^3*b*d*x-630*a^2*b^2*d*x-420*a*b^3*d*x-105*b^4*d*x+420*a^3*b* tanh(d*x+c)+630*a^2*b^2*tanh(d*x+c)+420*a*b^3*tanh(d*x+c)+105*b^4*tanh(d*x +c))/d
Leaf count of result is larger than twice the leaf count of optimal. 1176 vs. \(2 (104) = 208\).
Time = 0.28 (sec) , antiderivative size = 1176, normalized size of antiderivative = 10.69 \[ \int \left (a+b \tanh ^2(c+d x)\right )^4 \, dx=\text {Too large to display} \]
1/105*((420*a^3*b + 840*a^2*b^2 + 644*a*b^3 + 176*b^4 + 105*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*x)*cosh(d*x + c)^7 + 7*(420*a^3*b + 840*a^ 2*b^2 + 644*a*b^3 + 176*b^4 + 105*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b ^4)*d*x)*cosh(d*x + c)*sinh(d*x + c)^6 - 4*(105*a^3*b + 210*a^2*b^2 + 161* a*b^3 + 44*b^4)*sinh(d*x + c)^7 + 7*(420*a^3*b + 840*a^2*b^2 + 644*a*b^3 + 176*b^4 + 105*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*x)*cosh(d*x + c)^5 - 28*(75*a^3*b + 120*a^2*b^2 + 71*a*b^3 + 14*b^4 + 3*(105*a^3*b + 21 0*a^2*b^2 + 161*a*b^3 + 44*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 35*((42 0*a^3*b + 840*a^2*b^2 + 644*a*b^3 + 176*b^4 + 105*(a^4 + 4*a^3*b + 6*a^2*b ^2 + 4*a*b^3 + b^4)*d*x)*cosh(d*x + c)^3 + (420*a^3*b + 840*a^2*b^2 + 644* a*b^3 + 176*b^4 + 105*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*x)*cos h(d*x + c))*sinh(d*x + c)^4 + 21*(420*a^3*b + 840*a^2*b^2 + 644*a*b^3 + 17 6*b^4 + 105*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*x)*cosh(d*x + c) ^3 - 28*(5*(105*a^3*b + 210*a^2*b^2 + 161*a*b^3 + 44*b^4)*cosh(d*x + c)^4 + 135*a^3*b + 180*a^2*b^2 + 123*a*b^3 + 42*b^4 + 10*(75*a^3*b + 120*a^2*b^ 2 + 71*a*b^3 + 14*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 7*(3*(420*a^3*b + 840*a^2*b^2 + 644*a*b^3 + 176*b^4 + 105*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a *b^3 + b^4)*d*x)*cosh(d*x + c)^5 + 10*(420*a^3*b + 840*a^2*b^2 + 644*a*b^3 + 176*b^4 + 105*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*x)*cosh(d*x + c)^3 + 9*(420*a^3*b + 840*a^2*b^2 + 644*a*b^3 + 176*b^4 + 105*(a^4 +...
Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (99) = 198\).
Time = 0.23 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.90 \[ \int \left (a+b \tanh ^2(c+d x)\right )^4 \, dx=\begin {cases} a^{4} x + 4 a^{3} b x - \frac {4 a^{3} b \tanh {\left (c + d x \right )}}{d} + 6 a^{2} b^{2} x - \frac {2 a^{2} b^{2} \tanh ^{3}{\left (c + d x \right )}}{d} - \frac {6 a^{2} b^{2} \tanh {\left (c + d x \right )}}{d} + 4 a b^{3} x - \frac {4 a b^{3} \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac {4 a b^{3} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {4 a b^{3} \tanh {\left (c + d x \right )}}{d} + b^{4} x - \frac {b^{4} \tanh ^{7}{\left (c + d x \right )}}{7 d} - \frac {b^{4} \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac {b^{4} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {b^{4} \tanh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tanh ^{2}{\left (c \right )}\right )^{4} & \text {otherwise} \end {cases} \]
Piecewise((a**4*x + 4*a**3*b*x - 4*a**3*b*tanh(c + d*x)/d + 6*a**2*b**2*x - 2*a**2*b**2*tanh(c + d*x)**3/d - 6*a**2*b**2*tanh(c + d*x)/d + 4*a*b**3* x - 4*a*b**3*tanh(c + d*x)**5/(5*d) - 4*a*b**3*tanh(c + d*x)**3/(3*d) - 4* a*b**3*tanh(c + d*x)/d + b**4*x - b**4*tanh(c + d*x)**7/(7*d) - b**4*tanh( c + d*x)**5/(5*d) - b**4*tanh(c + d*x)**3/(3*d) - b**4*tanh(c + d*x)/d, Ne (d, 0)), (x*(a + b*tanh(c)**2)**4, True))
Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (104) = 208\).
Time = 0.21 (sec) , antiderivative size = 410, normalized size of antiderivative = 3.73 \[ \int \left (a+b \tanh ^2(c+d x)\right )^4 \, dx=\frac {1}{105} \, b^{4} {\left (105 \, x + \frac {105 \, c}{d} - \frac {8 \, {\left (203 \, e^{\left (-2 \, d x - 2 \, c\right )} + 609 \, e^{\left (-4 \, d x - 4 \, c\right )} + 770 \, e^{\left (-6 \, d x - 6 \, c\right )} + 770 \, e^{\left (-8 \, d x - 8 \, c\right )} + 315 \, e^{\left (-10 \, d x - 10 \, c\right )} + 105 \, e^{\left (-12 \, d x - 12 \, c\right )} + 44\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} + 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} + 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} + 1\right )}}\right )} + \frac {4}{15} \, a b^{3} {\left (15 \, x + \frac {15 \, c}{d} - \frac {2 \, {\left (70 \, e^{\left (-2 \, d x - 2 \, c\right )} + 140 \, e^{\left (-4 \, d x - 4 \, c\right )} + 90 \, e^{\left (-6 \, d x - 6 \, c\right )} + 45 \, e^{\left (-8 \, d x - 8 \, c\right )} + 23\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} + 2 \, a^{2} b^{2} {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 2\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} + 4 \, a^{3} b {\left (x + \frac {c}{d} - \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} + a^{4} x \]
1/105*b^4*(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))) + 4/15*a*b^3*(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))) + 2*a^2* b^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))) + 4*a^3*b*( x + c/d - 2/(d*(e^(-2*d*x - 2*c) + 1))) + a^4*x
Leaf count of result is larger than twice the leaf count of optimal. 447 vs. \(2 (104) = 208\).
Time = 0.31 (sec) , antiderivative size = 447, normalized size of antiderivative = 4.06 \[ \int \left (a+b \tanh ^2(c+d x)\right )^4 \, dx=\frac {105 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} {\left (d x + c\right )} + \frac {8 \, {\left (105 \, a^{3} b e^{\left (12 \, d x + 12 \, c\right )} + 315 \, a^{2} b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 315 \, a b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 105 \, b^{4} e^{\left (12 \, d x + 12 \, c\right )} + 630 \, a^{3} b e^{\left (10 \, d x + 10 \, c\right )} + 1575 \, a^{2} b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 1260 \, a b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 315 \, b^{4} e^{\left (10 \, d x + 10 \, c\right )} + 1575 \, a^{3} b e^{\left (8 \, d x + 8 \, c\right )} + 3360 \, a^{2} b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 2555 \, a b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 770 \, b^{4} e^{\left (8 \, d x + 8 \, c\right )} + 2100 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} + 3990 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 3080 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 770 \, b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 1575 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} + 2835 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 2121 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 609 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 630 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} + 1155 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 812 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 203 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 105 \, a^{3} b + 210 \, a^{2} b^{2} + 161 \, a b^{3} + 44 \, b^{4}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{7}}}{105 \, d} \]
1/105*(105*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*(d*x + c) + 8*(105* a^3*b*e^(12*d*x + 12*c) + 315*a^2*b^2*e^(12*d*x + 12*c) + 315*a*b^3*e^(12* d*x + 12*c) + 105*b^4*e^(12*d*x + 12*c) + 630*a^3*b*e^(10*d*x + 10*c) + 15 75*a^2*b^2*e^(10*d*x + 10*c) + 1260*a*b^3*e^(10*d*x + 10*c) + 315*b^4*e^(1 0*d*x + 10*c) + 1575*a^3*b*e^(8*d*x + 8*c) + 3360*a^2*b^2*e^(8*d*x + 8*c) + 2555*a*b^3*e^(8*d*x + 8*c) + 770*b^4*e^(8*d*x + 8*c) + 2100*a^3*b*e^(6*d *x + 6*c) + 3990*a^2*b^2*e^(6*d*x + 6*c) + 3080*a*b^3*e^(6*d*x + 6*c) + 77 0*b^4*e^(6*d*x + 6*c) + 1575*a^3*b*e^(4*d*x + 4*c) + 2835*a^2*b^2*e^(4*d*x + 4*c) + 2121*a*b^3*e^(4*d*x + 4*c) + 609*b^4*e^(4*d*x + 4*c) + 630*a^3*b *e^(2*d*x + 2*c) + 1155*a^2*b^2*e^(2*d*x + 2*c) + 812*a*b^3*e^(2*d*x + 2*c ) + 203*b^4*e^(2*d*x + 2*c) + 105*a^3*b + 210*a^2*b^2 + 161*a*b^3 + 44*b^4 )/(e^(2*d*x + 2*c) + 1)^7)/d
Time = 0.18 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.21 \[ \int \left (a+b \tanh ^2(c+d x)\right )^4 \, dx=x\,\left (a^4+4\,a^3\,b+6\,a^2\,b^2+4\,a\,b^3+b^4\right )-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^3\,\left (6\,a^2\,b^2+4\,a\,b^3+b^4\right )}{3\,d}-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^5\,\left (b^4+4\,a\,b^3\right )}{5\,d}-\frac {b^4\,{\mathrm {tanh}\left (c+d\,x\right )}^7}{7\,d}-\frac {b\,\mathrm {tanh}\left (c+d\,x\right )\,\left (4\,a^3+6\,a^2\,b+4\,a\,b^2+b^3\right )}{d} \]