Integrand size = 14, antiderivative size = 160 \[ \int \left (a+b \tanh ^2(c+d x)\right )^5 \, dx=(a+b)^5 x-\frac {b \left (5 a^4+10 a^3 b+10 a^2 b^2+5 a b^3+b^4\right ) \tanh (c+d x)}{d}-\frac {b^2 \left (10 a^3+10 a^2 b+5 a b^2+b^3\right ) \tanh ^3(c+d x)}{3 d}-\frac {b^3 \left (10 a^2+5 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^4 (5 a+b) \tanh ^7(c+d x)}{7 d}-\frac {b^5 \tanh ^9(c+d x)}{9 d} \] Output:
(a+b)^5*x-b*(5*a^4+10*a^3*b+10*a^2*b^2+5*a*b^3+b^4)*tanh(d*x+c)/d-1/3*b^2* (10*a^3+10*a^2*b+5*a*b^2+b^3)*tanh(d*x+c)^3/d-1/5*b^3*(10*a^2+5*a*b+b^2)*t anh(d*x+c)^5/d-1/7*b^4*(5*a+b)*tanh(d*x+c)^7/d-1/9*b^5*tanh(d*x+c)^9/d
Time = 1.58 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.06 \[ \int \left (a+b \tanh ^2(c+d x)\right )^5 \, dx=\frac {\tanh (c+d x) \left (\frac {315 (a+b)^5 \text {arctanh}\left (\sqrt {\tanh ^2(c+d x)}\right )}{\sqrt {\tanh ^2(c+d x)}}-b \left (315 \left (5 a^4+10 a^3 b+10 a^2 b^2+5 a b^3+b^4\right )+105 b \left (10 a^3+10 a^2 b+5 a b^2+b^3\right ) \tanh ^2(c+d x)+63 b^2 \left (10 a^2+5 a b+b^2\right ) \tanh ^4(c+d x)+45 b^3 (5 a+b) \tanh ^6(c+d x)+35 b^4 \tanh ^8(c+d x)\right )\right )}{315 d} \] Input:
Integrate[(a + b*Tanh[c + d*x]^2)^5,x]
Output:
(Tanh[c + d*x]*((315*(a + b)^5*ArcTanh[Sqrt[Tanh[c + d*x]^2]])/Sqrt[Tanh[c + d*x]^2] - b*(315*(5*a^4 + 10*a^3*b + 10*a^2*b^2 + 5*a*b^3 + b^4) + 105* b*(10*a^3 + 10*a^2*b + 5*a*b^2 + b^3)*Tanh[c + d*x]^2 + 63*b^2*(10*a^2 + 5 *a*b + b^2)*Tanh[c + d*x]^4 + 45*b^3*(5*a + b)*Tanh[c + d*x]^6 + 35*b^4*Ta nh[c + d*x]^8)))/(315*d)
Time = 0.35 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.97, 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.
| \(\displaystyle \int \left (a+b \tanh ^2(c+d x)\right )^5 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a-b \tan (i c+i d x)^2\right )^5dx\) |
| \(\Big \downarrow \) 4144 |
| \(\displaystyle \frac {\int \frac {\left (b \tanh ^2(c+d x)+a\right )^5}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 300 |
| \(\displaystyle \frac {\int \left (-b^5 \tanh ^8(c+d x)-b^4 (5 a+b) \tanh ^6(c+d x)-b^3 \left (10 a^2+5 b a+b^2\right ) \tanh ^4(c+d x)-b^2 \left (10 a^3+10 b a^2+5 b^2 a+b^3\right ) \tanh ^2(c+d x)-b \left (5 a^4+10 b a^3+10 b^2 a^2+5 b^3 a+b^4\right )+\frac {(a+b)^5}{1-\tanh ^2(c+d x)}\right )d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {1}{5} b^3 \left (10 a^2+5 a b+b^2\right ) \tanh ^5(c+d x)-\frac {1}{3} b^2 \left (10 a^3+10 a^2 b+5 a b^2+b^3\right ) \tanh ^3(c+d x)-b \left (5 a^4+10 a^3 b+10 a^2 b^2+5 a b^3+b^4\right ) \tanh (c+d x)+(a+b)^5 \text {arctanh}(\tanh (c+d x))-\frac {1}{7} b^4 (5 a+b) \tanh ^7(c+d x)-\frac {1}{9} b^5 \tanh ^9(c+d x)}{d}\) |
Input:
Int[(a + b*Tanh[c + d*x]^2)^5,x]
Output:
((a + b)^5*ArcTanh[Tanh[c + d*x]] - b*(5*a^4 + 10*a^3*b + 10*a^2*b^2 + 5*a *b^3 + b^4)*Tanh[c + d*x] - (b^2*(10*a^3 + 10*a^2*b + 5*a*b^2 + b^3)*Tanh[ c + d*x]^3)/3 - (b^3*(10*a^2 + 5*a*b + b^2)*Tanh[c + d*x]^5)/5 - (b^4*(5*a + b)*Tanh[c + d*x]^7)/7 - (b^5*Tanh[c + d*x]^9)/9)/d
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 = 262, normalized size of antiderivative = 1.64
| method | result | size |
| parallelrisch | \(-\frac {1575 a^{4} b \tanh \left (d x +c \right )+3150 a^{3} b^{2} \tanh \left (d x +c \right )+3150 a^{2} b^{3} \tanh \left (d x +c \right )+1575 a \,b^{4} \tanh \left (d x +c \right )+225 a \,b^{4} \tanh \left (d x +c \right )^{7}+630 a^{2} b^{3} \tanh \left (d x +c \right )^{5}+315 a \,b^{4} \tanh \left (d x +c \right )^{5}+1050 a^{3} b^{2} \tanh \left (d x +c \right )^{3}+1050 a^{2} b^{3} \tanh \left (d x +c \right )^{3}+525 a \,b^{4} \tanh \left (d x +c \right )^{3}+45 b^{5} \tanh \left (d x +c \right )^{7}+63 b^{5} \tanh \left (d x +c \right )^{5}+105 b^{5} \tanh \left (d x +c \right )^{3}+315 b^{5} \tanh \left (d x +c \right )+35 b^{5} \tanh \left (d x +c \right )^{9}-1575 a^{4} b d x -3150 a^{3} b^{2} d x -3150 a^{2} b^{3} d x -1575 a \,b^{4} d x -315 a^{5} d x -315 b^{5} d x}{315 d}\) | \(262\) |
| derivativedivides | \(\frac {-5 a^{4} b \tanh \left (d x +c \right )-10 a^{3} b^{2} \tanh \left (d x +c \right )-10 a^{2} b^{3} \tanh \left (d x +c \right )-5 a \,b^{4} \tanh \left (d x +c \right )-\frac {5 a \,b^{4} \tanh \left (d x +c \right )^{7}}{7}-2 a^{2} b^{3} \tanh \left (d x +c \right )^{5}-a \,b^{4} \tanh \left (d x +c \right )^{5}-\frac {10 a^{3} b^{2} \tanh \left (d x +c \right )^{3}}{3}-\frac {10 a^{2} b^{3} \tanh \left (d x +c \right )^{3}}{3}-\frac {5 a \,b^{4} \tanh \left (d x +c \right )^{3}}{3}+\frac {\left (a^{5}+5 a^{4} b +10 a^{3} b^{2}+10 a^{2} b^{3}+5 a \,b^{4}+b^{5}\right ) \ln \left (1+\tanh \left (d x +c \right )\right )}{2}-\frac {\left (a^{5}+5 a^{4} b +10 a^{3} b^{2}+10 a^{2} b^{3}+5 a \,b^{4}+b^{5}\right ) \ln \left (-1+\tanh \left (d x +c \right )\right )}{2}-\frac {b^{5} \tanh \left (d x +c \right )^{7}}{7}-\frac {b^{5} \tanh \left (d x +c \right )^{5}}{5}-\frac {b^{5} \tanh \left (d x +c \right )^{3}}{3}-b^{5} \tanh \left (d x +c \right )-\frac {b^{5} \tanh \left (d x +c \right )^{9}}{9}}{d}\) | \(303\) |
| default | \(\frac {-5 a^{4} b \tanh \left (d x +c \right )-10 a^{3} b^{2} \tanh \left (d x +c \right )-10 a^{2} b^{3} \tanh \left (d x +c \right )-5 a \,b^{4} \tanh \left (d x +c \right )-\frac {5 a \,b^{4} \tanh \left (d x +c \right )^{7}}{7}-2 a^{2} b^{3} \tanh \left (d x +c \right )^{5}-a \,b^{4} \tanh \left (d x +c \right )^{5}-\frac {10 a^{3} b^{2} \tanh \left (d x +c \right )^{3}}{3}-\frac {10 a^{2} b^{3} \tanh \left (d x +c \right )^{3}}{3}-\frac {5 a \,b^{4} \tanh \left (d x +c \right )^{3}}{3}+\frac {\left (a^{5}+5 a^{4} b +10 a^{3} b^{2}+10 a^{2} b^{3}+5 a \,b^{4}+b^{5}\right ) \ln \left (1+\tanh \left (d x +c \right )\right )}{2}-\frac {\left (a^{5}+5 a^{4} b +10 a^{3} b^{2}+10 a^{2} b^{3}+5 a \,b^{4}+b^{5}\right ) \ln \left (-1+\tanh \left (d x +c \right )\right )}{2}-\frac {b^{5} \tanh \left (d x +c \right )^{7}}{7}-\frac {b^{5} \tanh \left (d x +c \right )^{5}}{5}-\frac {b^{5} \tanh \left (d x +c \right )^{3}}{3}-b^{5} \tanh \left (d x +c \right )-\frac {b^{5} \tanh \left (d x +c \right )^{9}}{9}}{d}\) | \(303\) |
| parts | \(a^{5} x +\frac {b^{5} \left (-\frac {\tanh \left (d x +c \right )^{9}}{9}-\frac {\tanh \left (d x +c \right )^{7}}{7}-\frac {\tanh \left (d x +c \right )^{5}}{5}-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (-1+\tanh \left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tanh \left (d x +c \right )\right )}{2}\right )}{d}+\frac {5 a \,b^{4} \left (-\frac {\tanh \left (d x +c \right )^{7}}{7}-\frac {\tanh \left (d x +c \right )^{5}}{5}-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (-1+\tanh \left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tanh \left (d x +c \right )\right )}{2}\right )}{d}+\frac {10 a^{2} b^{3} \left (-\frac {\tanh \left (d x +c \right )^{5}}{5}-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (-1+\tanh \left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tanh \left (d x +c \right )\right )}{2}\right )}{d}+\frac {10 a^{3} b^{2} \left (-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (-1+\tanh \left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tanh \left (d x +c \right )\right )}{2}\right )}{d}+\frac {5 a^{4} b \left (-\tanh \left (d x +c \right )-\frac {\ln \left (-1+\tanh \left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tanh \left (d x +c \right )\right )}{2}\right )}{d}\) | \(309\) |
| risch | \(a^{5} x +5 a^{4} b x +10 a^{3} b^{2} x +10 a^{2} b^{3} x +5 a \,b^{4} x +b^{5} x +\frac {2 b \left (4200 a^{3} b +2640 a \,b^{3}+4830 a^{2} b^{2}+26292 b^{4} {\mathrm e}^{6 d x +6 c}+44100 a^{4} {\mathrm e}^{4 d x +4 c}+110250 a^{4} {\mathrm e}^{8 d x +8 c}+39438 b^{4} {\mathrm e}^{8 d x +8 c}+88200 a^{4} {\mathrm e}^{6 d x +6 c}+21000 b^{4} {\mathrm e}^{12 d x +12 c}+88200 a^{4} {\mathrm e}^{10 d x +10 c}+31500 b^{4} {\mathrm e}^{10 d x +10 c}+1575 a^{4} {\mathrm e}^{16 d x +16 c}+1575 b^{4} {\mathrm e}^{16 d x +16 c}+12600 a^{4} {\mathrm e}^{14 d x +14 c}+6300 b^{4} {\mathrm e}^{14 d x +14 c}+44100 a^{4} {\mathrm e}^{12 d x +12 c}+13968 b^{4} {\mathrm e}^{4 d x +4 c}+12600 a^{4} {\mathrm e}^{2 d x +2 c}+3492 b^{4} {\mathrm e}^{2 d x +2 c}+563 b^{4}+1575 a^{4}+9450 a^{2} b^{2} {\mathrm e}^{16 d x +16 c}+6300 a^{3} b \,{\mathrm e}^{16 d x +16 c}+283500 a^{2} b^{2} {\mathrm e}^{10 d x +10 c}+90300 a \,b^{3} {\mathrm e}^{12 d x +12 c}+245700 a^{3} b \,{\mathrm e}^{10 d x +10 c}+161700 a^{2} b^{2} {\mathrm e}^{12 d x +12 c}+136500 a^{3} b \,{\mathrm e}^{12 d x +12 c}+216300 a^{3} b \,{\mathrm e}^{6 d x +6 c}+157500 a \,b^{3} {\mathrm e}^{10 d x +10 c}+6300 a \,b^{3} {\mathrm e}^{16 d x +16 c}+44100 a^{3} b \,{\mathrm e}^{14 d x +14 c}+56700 a^{2} b^{2} {\mathrm e}^{14 d x +14 c}+31500 a \,b^{3} {\mathrm e}^{14 d x +14 c}+31500 a^{3} b \,{\mathrm e}^{2 d x +2 c}+34020 a^{2} b^{2} {\mathrm e}^{2 d x +2 c}+283500 a^{3} b \,{\mathrm e}^{8 d x +8 c}+325080 a^{2} b^{2} {\mathrm e}^{8 d x +8 c}+175140 a \,b^{3} {\mathrm e}^{8 d x +8 c}+63540 a \,b^{3} {\mathrm e}^{4 d x +4 c}+244020 a^{2} b^{2} {\mathrm e}^{6 d x +6 c}+131460 a \,b^{3} {\mathrm e}^{6 d x +6 c}+107100 a^{3} b \,{\mathrm e}^{4 d x +4 c}+117180 a^{2} b^{2} {\mathrm e}^{4 d x +4 c}+17460 a \,b^{3} {\mathrm e}^{2 d x +2 c}\right )}{315 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{9}}\) | \(694\) |
Input:
int((a+b*tanh(d*x+c)^2)^5,x,method=_RETURNVERBOSE)
Output:
-1/315*(1575*a^4*b*tanh(d*x+c)+3150*a^3*b^2*tanh(d*x+c)+3150*a^2*b^3*tanh( d*x+c)+1575*a*b^4*tanh(d*x+c)+225*a*b^4*tanh(d*x+c)^7+630*a^2*b^3*tanh(d*x +c)^5+315*a*b^4*tanh(d*x+c)^5+1050*a^3*b^2*tanh(d*x+c)^3+1050*a^2*b^3*tanh (d*x+c)^3+525*a*b^4*tanh(d*x+c)^3+45*b^5*tanh(d*x+c)^7+63*b^5*tanh(d*x+c)^ 5+105*b^5*tanh(d*x+c)^3+315*b^5*tanh(d*x+c)+35*b^5*tanh(d*x+c)^9-1575*a^4* b*d*x-3150*a^3*b^2*d*x-3150*a^2*b^3*d*x-1575*a*b^4*d*x-315*a^5*d*x-315*b^5 *d*x)/d
Leaf count of result is larger than twice the leaf count of optimal. 2133 vs. \(2 (152) = 304\).
Time = 0.12 (sec) , antiderivative size = 2133, normalized size of antiderivative = 13.33 \[ \int \left (a+b \tanh ^2(c+d x)\right )^5 \, dx=\text {Too large to display} \] Input:
integrate((a+b*tanh(d*x+c)^2)^5,x, algorithm="fricas")
Output:
1/315*((1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cosh(d* x + c)^9 + 9*(1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563* b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*c osh(d*x + c)*sinh(d*x + c)^8 - (1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5)*sinh(d*x + c)^9 + 9*(1575*a^4*b + 4200*a^3*b^2 + 48 30*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a ^2*b^3 + 5*a*b^4 + b^5)*d*x)*cosh(d*x + c)^7 - 9*(1225*a^4*b + 2800*a^3*b^ 2 + 2730*a^2*b^3 + 1240*a*b^4 + 213*b^5 + 4*(1575*a^4*b + 4200*a^3*b^2 + 4 830*a^2*b^3 + 2640*a*b^4 + 563*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 21* (4*(1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315* (a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cosh(d*x + c)^3 + 3*(1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cosh( d*x + c))*sinh(d*x + c)^6 + 36*(1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a *b^4 + b^5)*d*x)*cosh(d*x + c)^5 - 9*(3500*a^4*b + 7000*a^3*b^2 + 6720*a^2 *b^3 + 3560*a*b^4 + 852*b^5 + 14*(1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5)*cosh(d*x + c)^4 + 21*(1225*a^4*b + 2800*a^3*b^2 + 2730*a^2*b^3 + 1240*a*b^4 + 213*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^5 ...
Leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (148) = 296\).
Time = 0.33 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.92 \[ \int \left (a+b \tanh ^2(c+d x)\right )^5 \, dx=\begin {cases} a^{5} x + 5 a^{4} b x - \frac {5 a^{4} b \tanh {\left (c + d x \right )}}{d} + 10 a^{3} b^{2} x - \frac {10 a^{3} b^{2} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {10 a^{3} b^{2} \tanh {\left (c + d x \right )}}{d} + 10 a^{2} b^{3} x - \frac {2 a^{2} b^{3} \tanh ^{5}{\left (c + d x \right )}}{d} - \frac {10 a^{2} b^{3} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {10 a^{2} b^{3} \tanh {\left (c + d x \right )}}{d} + 5 a b^{4} x - \frac {5 a b^{4} \tanh ^{7}{\left (c + d x \right )}}{7 d} - \frac {a b^{4} \tanh ^{5}{\left (c + d x \right )}}{d} - \frac {5 a b^{4} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {5 a b^{4} \tanh {\left (c + d x \right )}}{d} + b^{5} x - \frac {b^{5} \tanh ^{9}{\left (c + d x \right )}}{9 d} - \frac {b^{5} \tanh ^{7}{\left (c + d x \right )}}{7 d} - \frac {b^{5} \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac {b^{5} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {b^{5} \tanh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tanh ^{2}{\left (c \right )}\right )^{5} & \text {otherwise} \end {cases} \] Input:
integrate((a+b*tanh(d*x+c)**2)**5,x)
Output:
Piecewise((a**5*x + 5*a**4*b*x - 5*a**4*b*tanh(c + d*x)/d + 10*a**3*b**2*x - 10*a**3*b**2*tanh(c + d*x)**3/(3*d) - 10*a**3*b**2*tanh(c + d*x)/d + 10 *a**2*b**3*x - 2*a**2*b**3*tanh(c + d*x)**5/d - 10*a**2*b**3*tanh(c + d*x) **3/(3*d) - 10*a**2*b**3*tanh(c + d*x)/d + 5*a*b**4*x - 5*a*b**4*tanh(c + d*x)**7/(7*d) - a*b**4*tanh(c + d*x)**5/d - 5*a*b**4*tanh(c + d*x)**3/(3*d ) - 5*a*b**4*tanh(c + d*x)/d + b**5*x - b**5*tanh(c + d*x)**9/(9*d) - b**5 *tanh(c + d*x)**7/(7*d) - b**5*tanh(c + d*x)**5/(5*d) - b**5*tanh(c + d*x) **3/(3*d) - b**5*tanh(c + d*x)/d, Ne(d, 0)), (x*(a + b*tanh(c)**2)**5, Tru e))
Leaf count of result is larger than twice the leaf count of optimal. 624 vs. \(2 (152) = 304\).
Time = 0.06 (sec) , antiderivative size = 624, normalized size of antiderivative = 3.90 \[ \int \left (a+b \tanh ^2(c+d x)\right )^5 \, dx =\text {Too large to display} \] Input:
integrate((a+b*tanh(d*x+c)^2)^5,x, algorithm="maxima")
Output:
1/315*b^5*(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))) + 1/21*a*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^(-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))) + 2/3*a^2*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))) + 10/3*a^3*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))) + 5*a^4*b*(x + c/d - 2/(d*(e^(-2*d*x - 2*c) + 1))) + a^5*x
Leaf count of result is larger than twice the leaf count of optimal. 721 vs. \(2 (152) = 304\).
Time = 0.17 (sec) , antiderivative size = 721, normalized size of antiderivative = 4.51 \[ \int \left (a+b \tanh ^2(c+d x)\right )^5 \, dx =\text {Too large to display} \] Input:
integrate((a+b*tanh(d*x+c)^2)^5,x, algorithm="giac")
Output:
1/315*(315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*(d*x + c) + 2*(1575*a^4*b*e^(16*d*x + 16*c) + 6300*a^3*b^2*e^(16*d*x + 16*c) + 9450*a^2*b^3*e^(16*d*x + 16*c) + 6300*a*b^4*e^(16*d*x + 16*c) + 1575*b^5*e ^(16*d*x + 16*c) + 12600*a^4*b*e^(14*d*x + 14*c) + 44100*a^3*b^2*e^(14*d*x + 14*c) + 56700*a^2*b^3*e^(14*d*x + 14*c) + 31500*a*b^4*e^(14*d*x + 14*c) + 6300*b^5*e^(14*d*x + 14*c) + 44100*a^4*b*e^(12*d*x + 12*c) + 136500*a^3 *b^2*e^(12*d*x + 12*c) + 161700*a^2*b^3*e^(12*d*x + 12*c) + 90300*a*b^4*e^ (12*d*x + 12*c) + 21000*b^5*e^(12*d*x + 12*c) + 88200*a^4*b*e^(10*d*x + 10 *c) + 245700*a^3*b^2*e^(10*d*x + 10*c) + 283500*a^2*b^3*e^(10*d*x + 10*c) + 157500*a*b^4*e^(10*d*x + 10*c) + 31500*b^5*e^(10*d*x + 10*c) + 110250*a^ 4*b*e^(8*d*x + 8*c) + 283500*a^3*b^2*e^(8*d*x + 8*c) + 325080*a^2*b^3*e^(8 *d*x + 8*c) + 175140*a*b^4*e^(8*d*x + 8*c) + 39438*b^5*e^(8*d*x + 8*c) + 8 8200*a^4*b*e^(6*d*x + 6*c) + 216300*a^3*b^2*e^(6*d*x + 6*c) + 244020*a^2*b ^3*e^(6*d*x + 6*c) + 131460*a*b^4*e^(6*d*x + 6*c) + 26292*b^5*e^(6*d*x + 6 *c) + 44100*a^4*b*e^(4*d*x + 4*c) + 107100*a^3*b^2*e^(4*d*x + 4*c) + 11718 0*a^2*b^3*e^(4*d*x + 4*c) + 63540*a*b^4*e^(4*d*x + 4*c) + 13968*b^5*e^(4*d *x + 4*c) + 12600*a^4*b*e^(2*d*x + 2*c) + 31500*a^3*b^2*e^(2*d*x + 2*c) + 34020*a^2*b^3*e^(2*d*x + 2*c) + 17460*a*b^4*e^(2*d*x + 2*c) + 3492*b^5*e^( 2*d*x + 2*c) + 1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563 *b^5)/(e^(2*d*x + 2*c) + 1)^9)/d
Time = 2.50 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.18 \[ \int \left (a+b \tanh ^2(c+d x)\right )^5 \, dx=x\,\left (a^5+5\,a^4\,b+10\,a^3\,b^2+10\,a^2\,b^3+5\,a\,b^4+b^5\right )-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^3\,\left (10\,a^3\,b^2+10\,a^2\,b^3+5\,a\,b^4+b^5\right )}{3\,d}-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^5\,\left (10\,a^2\,b^3+5\,a\,b^4+b^5\right )}{5\,d}-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^7\,\left (b^5+5\,a\,b^4\right )}{7\,d}-\frac {b^5\,{\mathrm {tanh}\left (c+d\,x\right )}^9}{9\,d}-\frac {b\,\mathrm {tanh}\left (c+d\,x\right )\,\left (5\,a^4+10\,a^3\,b+10\,a^2\,b^2+5\,a\,b^3+b^4\right )}{d} \] Input:
int((a + b*tanh(c + d*x)^2)^5,x)
Output:
x*(5*a*b^4 + 5*a^4*b + a^5 + b^5 + 10*a^2*b^3 + 10*a^3*b^2) - (tanh(c + d* x)^3*(5*a*b^4 + b^5 + 10*a^2*b^3 + 10*a^3*b^2))/(3*d) - (tanh(c + d*x)^5*( 5*a*b^4 + b^5 + 10*a^2*b^3))/(5*d) - (tanh(c + d*x)^7*(5*a*b^4 + b^5))/(7* d) - (b^5*tanh(c + d*x)^9)/(9*d) - (b*tanh(c + d*x)*(5*a*b^3 + 10*a^3*b + 5*a^4 + b^4 + 10*a^2*b^2))/d
Time = 0.23 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.63 \[ \int \left (a+b \tanh ^2(c+d x)\right )^5 \, dx=\frac {-35 \tanh \left (d x +c \right )^{9} b^{5}-225 \tanh \left (d x +c \right )^{7} a \,b^{4}-45 \tanh \left (d x +c \right )^{7} b^{5}-630 \tanh \left (d x +c \right )^{5} a^{2} b^{3}-315 \tanh \left (d x +c \right )^{5} a \,b^{4}-63 \tanh \left (d x +c \right )^{5} b^{5}-1050 \tanh \left (d x +c \right )^{3} a^{3} b^{2}-1050 \tanh \left (d x +c \right )^{3} a^{2} b^{3}-525 \tanh \left (d x +c \right )^{3} a \,b^{4}-105 \tanh \left (d x +c \right )^{3} b^{5}-1575 \tanh \left (d x +c \right ) a^{4} b -3150 \tanh \left (d x +c \right ) a^{3} b^{2}-3150 \tanh \left (d x +c \right ) a^{2} b^{3}-1575 \tanh \left (d x +c \right ) a \,b^{4}-315 \tanh \left (d x +c \right ) b^{5}+315 a^{5} d x +1575 a^{4} b d x +3150 a^{3} b^{2} d x +3150 a^{2} b^{3} d x +1575 a \,b^{4} d x +315 b^{5} d x}{315 d} \] Input:
int((a+b*tanh(d*x+c)^2)^5,x)
Output:
( - 35*tanh(c + d*x)**9*b**5 - 225*tanh(c + d*x)**7*a*b**4 - 45*tanh(c + d *x)**7*b**5 - 630*tanh(c + d*x)**5*a**2*b**3 - 315*tanh(c + d*x)**5*a*b**4 - 63*tanh(c + d*x)**5*b**5 - 1050*tanh(c + d*x)**3*a**3*b**2 - 1050*tanh( c + d*x)**3*a**2*b**3 - 525*tanh(c + d*x)**3*a*b**4 - 105*tanh(c + d*x)**3 *b**5 - 1575*tanh(c + d*x)*a**4*b - 3150*tanh(c + d*x)*a**3*b**2 - 3150*ta nh(c + d*x)*a**2*b**3 - 1575*tanh(c + d*x)*a*b**4 - 315*tanh(c + d*x)*b**5 + 315*a**5*d*x + 1575*a**4*b*d*x + 3150*a**3*b**2*d*x + 3150*a**2*b**3*d* x + 1575*a*b**4*d*x + 315*b**5*d*x)/(315*d)