Integrand size = 21, antiderivative size = 99 \[ \int \cosh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=-\frac {3 b \left (4 (a+b)^2+(2 a+b)^2\right ) \arctan (\sinh (c+d x))}{8 d}+\frac {(a+b)^3 \sinh (c+d x)}{d}+\frac {3 b^2 (4 a+3 b) \text {sech}(c+d x) \tanh (c+d x)}{8 d}-\frac {b^3 \text {sech}^3(c+d x) \tanh (c+d x)}{4 d} \] Output:
-3/8*b*(4*(a+b)^2+(2*a+b)^2)*arctan(sinh(d*x+c))/d+(a+b)^3*sinh(d*x+c)/d+3 /8*b^2*(4*a+3*b)*sech(d*x+c)*tanh(d*x+c)/d-1/4*b^3*sech(d*x+c)^3*tanh(d*x+ c)/d
Time = 0.22 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.90 \[ \int \cosh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {-3 b \left (8 a^2+12 a b+5 b^2\right ) \arctan (\sinh (c+d x))+8 (a+b)^3 \sinh (c+d x)+3 b^2 (4 a+3 b) \text {sech}(c+d x) \tanh (c+d x)-2 b^3 \text {sech}^3(c+d x) \tanh (c+d x)}{8 d} \] Input:
Integrate[Cosh[c + d*x]*(a + b*Tanh[c + d*x]^2)^3,x]
Output:
(-3*b*(8*a^2 + 12*a*b + 5*b^2)*ArcTan[Sinh[c + d*x]] + 8*(a + b)^3*Sinh[c + d*x] + 3*b^2*(4*a + 3*b)*Sech[c + d*x]*Tanh[c + d*x] - 2*b^3*Sech[c + d* x]^3*Tanh[c + d*x])/(8*d)
Time = 0.34 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 4159, 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.
| \(\displaystyle \int \cosh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-b \tan (i c+i d x)^2\right )^3}{\sec (i c+i d x)}dx\) |
| \(\Big \downarrow \) 4159 |
| \(\displaystyle \frac {\int \frac {\left ((a+b) \sinh ^2(c+d x)+a\right )^3}{\left (\sinh ^2(c+d x)+1\right )^3}d\sinh (c+d x)}{d}\) |
| \(\Big \downarrow \) 300 |
| \(\displaystyle \frac {\int \left ((a+b)^3-\frac {3 b (a+b)^2 \sinh ^4(c+d x)+3 b (a+b) (2 a+b) \sinh ^2(c+d x)+b \left (3 a^2+3 b a+b^2\right )}{\left (\sinh ^2(c+d x)+1\right )^3}\right )d\sinh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {3}{8} b \left (4 (a+b)^2+(2 a+b)^2\right ) \arctan (\sinh (c+d x))+\frac {3 b^2 (4 a+3 b) \sinh (c+d x)}{8 \left (\sinh ^2(c+d x)+1\right )}+(a+b)^3 \sinh (c+d x)-\frac {b^3 \sinh (c+d x)}{4 \left (\sinh ^2(c+d x)+1\right )^2}}{d}\) |
Input:
Int[Cosh[c + d*x]*(a + b*Tanh[c + d*x]^2)^3,x]
Output:
((-3*b*(4*(a + b)^2 + (2*a + b)^2)*ArcTan[Sinh[c + d*x]])/8 + (a + b)^3*Si nh[c + d*x] - (b^3*Sinh[c + d*x])/(4*(1 + Sinh[c + d*x]^2)^2) + (3*b^2*(4* a + 3*b)*Sinh[c + d*x])/(8*(1 + Sinh[c + d*x]^2)))/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[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_ ))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[ExpandToSum[b*(ff*x)^n + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 - ff^2 *x^2)^((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f} , x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(192\) vs. \(2(93)=186\).
Time = 6.30 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.95
| method | result | size |
| derivativedivides | \(\frac {\sinh \left (d x +c \right ) a^{3}+3 a^{2} b \left (\sinh \left (d x +c \right )-2 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+3 b^{2} a \left (\frac {\sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{2}}+\frac {3 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}-\frac {3 \,\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}-3 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+b^{3} \left (\frac {\sinh \left (d x +c \right )^{5}}{\cosh \left (d x +c \right )^{4}}+\frac {5 \sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{4}}+\frac {5 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{4}}-5 \left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )-\frac {15 \arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )}{d}\) | \(193\) |
| default | \(\frac {\sinh \left (d x +c \right ) a^{3}+3 a^{2} b \left (\sinh \left (d x +c \right )-2 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+3 b^{2} a \left (\frac {\sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{2}}+\frac {3 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}-\frac {3 \,\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}-3 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+b^{3} \left (\frac {\sinh \left (d x +c \right )^{5}}{\cosh \left (d x +c \right )^{4}}+\frac {5 \sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{4}}+\frac {5 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{4}}-5 \left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )-\frac {15 \arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )}{d}\) | \(193\) |
| risch | \(\frac {{\mathrm e}^{d x +c} a^{3}}{2 d}+\frac {3 \,{\mathrm e}^{d x +c} a^{2} b}{2 d}+\frac {3 \,{\mathrm e}^{d x +c} b^{2} a}{2 d}+\frac {{\mathrm e}^{d x +c} b^{3}}{2 d}-\frac {{\mathrm e}^{-d x -c} a^{3}}{2 d}-\frac {3 \,{\mathrm e}^{-d x -c} a^{2} b}{2 d}-\frac {3 \,{\mathrm e}^{-d x -c} b^{2} a}{2 d}-\frac {{\mathrm e}^{-d x -c} b^{3}}{2 d}+\frac {b^{2} {\mathrm e}^{d x +c} \left (12 \,{\mathrm e}^{6 d x +6 c} a +9 \,{\mathrm e}^{6 d x +6 c} b +12 \,{\mathrm e}^{4 d x +4 c} a +b \,{\mathrm e}^{4 d x +4 c}-12 \,{\mathrm e}^{2 d x +2 c} a -{\mathrm e}^{2 d x +2 c} b -12 a -9 b \right )}{4 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{4}}+\frac {3 i b \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{d}+\frac {9 i b^{2} \ln \left ({\mathrm e}^{d x +c}-i\right ) a}{2 d}+\frac {15 i b^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{8 d}-\frac {3 i b \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{d}-\frac {9 i b^{2} \ln \left ({\mathrm e}^{d x +c}+i\right ) a}{2 d}-\frac {15 i b^{3} \ln \left ({\mathrm e}^{d x +c}+i\right )}{8 d}\) | \(353\) |
Input:
int(cosh(d*x+c)*(a+tanh(d*x+c)^2*b)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(sinh(d*x+c)*a^3+3*a^2*b*(sinh(d*x+c)-2*arctan(exp(d*x+c)))+3*b^2*a*(s inh(d*x+c)^3/cosh(d*x+c)^2+3*sinh(d*x+c)/cosh(d*x+c)^2-3/2*sech(d*x+c)*tan h(d*x+c)-3*arctan(exp(d*x+c)))+b^3*(sinh(d*x+c)^5/cosh(d*x+c)^4+5/cosh(d*x +c)^4*sinh(d*x+c)^3+5/cosh(d*x+c)^4*sinh(d*x+c)-5*(1/4*sech(d*x+c)^3+3/8*s ech(d*x+c))*tanh(d*x+c)-15/4*arctan(exp(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 2411 vs. \(2 (93) = 186\).
Time = 0.10 (sec) , antiderivative size = 2411, normalized size of antiderivative = 24.35 \[ \int \cosh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:
integrate(cosh(d*x+c)*(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")
Output:
1/4*(2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^10 + 20*(a^3 + 3*a^2* b + 3*a*b^2 + b^3)*cosh(d*x + c)*sinh(d*x + c)^9 + 2*(a^3 + 3*a^2*b + 3*a* b^2 + b^3)*sinh(d*x + c)^10 + 3*(2*a^3 + 6*a^2*b + 10*a*b^2 + 5*b^3)*cosh( d*x + c)^8 + 3*(2*a^3 + 6*a^2*b + 10*a*b^2 + 5*b^3 + 30*(a^3 + 3*a^2*b + 3 *a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 24*(10*(a^3 + 3*a^2*b + 3 *a*b^2 + b^3)*cosh(d*x + c)^3 + (2*a^3 + 6*a^2*b + 10*a*b^2 + 5*b^3)*cosh( d*x + c))*sinh(d*x + c)^7 + (4*a^3 + 12*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^6 + (420*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 + 4*a^3 + 1 2*a^2*b + 24*a*b^2 + 5*b^3 + 84*(2*a^3 + 6*a^2*b + 10*a*b^2 + 5*b^3)*cosh( d*x + c)^2)*sinh(d*x + c)^6 + 6*(84*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d *x + c)^5 + 28*(2*a^3 + 6*a^2*b + 10*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + (4*a ^3 + 12*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 - (4*a^3 + 12*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^4 + (420*(a^3 + 3*a^2*b + 3*a *b^2 + b^3)*cosh(d*x + c)^6 + 210*(2*a^3 + 6*a^2*b + 10*a*b^2 + 5*b^3)*cos h(d*x + c)^4 - 4*a^3 - 12*a^2*b - 24*a*b^2 - 5*b^3 + 15*(4*a^3 + 12*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(60*(a^3 + 3*a^2* b + 3*a*b^2 + b^3)*cosh(d*x + c)^7 + 42*(2*a^3 + 6*a^2*b + 10*a*b^2 + 5*b^ 3)*cosh(d*x + c)^5 + 5*(4*a^3 + 12*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c) ^3 - (4*a^3 + 12*a^2*b + 24*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - 2*a^3 - 6*a^2*b - 6*a*b^2 - 2*b^3 - 3*(2*a^3 + 6*a^2*b + 10*a*b^2 + 5...
\[ \int \cosh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3} \cosh {\left (c + d x \right )}\, dx \] Input:
integrate(cosh(d*x+c)*(a+b*tanh(d*x+c)**2)**3,x)
Output:
Integral((a + b*tanh(c + d*x)**2)**3*cosh(c + d*x), x)
Leaf count of result is larger than twice the leaf count of optimal. 295 vs. \(2 (93) = 186\).
Time = 0.17 (sec) , antiderivative size = 295, normalized size of antiderivative = 2.98 \[ \int \cosh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {1}{4} \, b^{3} {\left (\frac {15 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {2 \, e^{\left (-d x - c\right )}}{d} + \frac {17 \, e^{\left (-2 \, d x - 2 \, c\right )} + 13 \, e^{\left (-4 \, d x - 4 \, c\right )} + 7 \, e^{\left (-6 \, d x - 6 \, c\right )} - 7 \, e^{\left (-8 \, d x - 8 \, c\right )} + 2}{d {\left (e^{\left (-d x - c\right )} + 4 \, e^{\left (-3 \, d x - 3 \, c\right )} + 6 \, e^{\left (-5 \, d x - 5 \, c\right )} + 4 \, e^{\left (-7 \, d x - 7 \, c\right )} + e^{\left (-9 \, d x - 9 \, c\right )}\right )}}\right )} + \frac {3}{2} \, a b^{2} {\left (\frac {6 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )}}{d} + \frac {4 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + 1}{d {\left (e^{\left (-d x - c\right )} + 2 \, e^{\left (-3 \, d x - 3 \, c\right )} + e^{\left (-5 \, d x - 5 \, c\right )}\right )}}\right )} + \frac {3}{2} \, a^{2} b {\left (\frac {4 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac {e^{\left (d x + c\right )}}{d} - \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {a^{3} \sinh \left (d x + c\right )}{d} \] Input:
integrate(cosh(d*x+c)*(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")
Output:
1/4*b^3*(15*arctan(e^(-d*x - c))/d - 2*e^(-d*x - c)/d + (17*e^(-2*d*x - 2* c) + 13*e^(-4*d*x - 4*c) + 7*e^(-6*d*x - 6*c) - 7*e^(-8*d*x - 8*c) + 2)/(d *(e^(-d*x - c) + 4*e^(-3*d*x - 3*c) + 6*e^(-5*d*x - 5*c) + 4*e^(-7*d*x - 7 *c) + e^(-9*d*x - 9*c)))) + 3/2*a*b^2*(6*arctan(e^(-d*x - c))/d - e^(-d*x - c)/d + (4*e^(-2*d*x - 2*c) - e^(-4*d*x - 4*c) + 1)/(d*(e^(-d*x - c) + 2* e^(-3*d*x - 3*c) + e^(-5*d*x - 5*c)))) + 3/2*a^2*b*(4*arctan(e^(-d*x - c)) /d + e^(d*x + c)/d - e^(-d*x - c)/d) + a^3*sinh(d*x + c)/d
Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (93) = 186\).
Time = 0.21 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.75 \[ \int \cosh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {8 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 24 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 24 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 8 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 3 \, {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (8 \, a^{2} b + 12 \, a b^{2} + 5 \, b^{3}\right )} + \frac {4 \, {\left (12 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 9 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 48 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 28 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}^{2}}}{16 \, d} \] Input:
integrate(cosh(d*x+c)*(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")
Output:
1/16*(8*a^3*(e^(d*x + c) - e^(-d*x - c)) + 24*a^2*b*(e^(d*x + c) - e^(-d*x - c)) + 24*a*b^2*(e^(d*x + c) - e^(-d*x - c)) + 8*b^3*(e^(d*x + c) - e^(- d*x - c)) - 3*(pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(8*a ^2*b + 12*a*b^2 + 5*b^3) + 4*(12*a*b^2*(e^(d*x + c) - e^(-d*x - c))^3 + 9* b^3*(e^(d*x + c) - e^(-d*x - c))^3 + 48*a*b^2*(e^(d*x + c) - e^(-d*x - c)) + 28*b^3*(e^(d*x + c) - e^(-d*x - c)))/((e^(d*x + c) - e^(-d*x - c))^2 + 4)^2)/d
Time = 0.29 (sec) , antiderivative size = 355, normalized size of antiderivative = 3.59 \[ \int \cosh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {{\mathrm {e}}^{c+d\,x}\,{\left (a+b\right )}^3}{2\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,{\left (a+b\right )}^3}{2\,d}-\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (5\,b^3\,\sqrt {d^2}+12\,a\,b^2\,\sqrt {d^2}+8\,a^2\,b\,\sqrt {d^2}\right )}{d\,\sqrt {64\,a^4\,b^2+192\,a^3\,b^3+224\,a^2\,b^4+120\,a\,b^5+25\,b^6}}\right )\,\sqrt {64\,a^4\,b^2+192\,a^3\,b^3+224\,a^2\,b^4+120\,a\,b^5+25\,b^6}}{4\,\sqrt {d^2}}+\frac {3\,{\mathrm {e}}^{c+d\,x}\,\left (3\,b^3+4\,a\,b^2\right )}{4\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {6\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (13\,b^3+12\,a\,b^2\right )}{2\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {4\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )} \] Input:
int(cosh(c + d*x)*(a + b*tanh(c + d*x)^2)^3,x)
Output:
(exp(c + d*x)*(a + b)^3)/(2*d) - (exp(- c - d*x)*(a + b)^3)/(2*d) - (3*ata n((exp(d*x)*exp(c)*(5*b^3*(d^2)^(1/2) + 12*a*b^2*(d^2)^(1/2) + 8*a^2*b*(d^ 2)^(1/2)))/(d*(120*a*b^5 + 25*b^6 + 224*a^2*b^4 + 192*a^3*b^3 + 64*a^4*b^2 )^(1/2)))*(120*a*b^5 + 25*b^6 + 224*a^2*b^4 + 192*a^3*b^3 + 64*a^4*b^2)^(1 /2))/(4*(d^2)^(1/2)) + (3*exp(c + d*x)*(4*a*b^2 + 3*b^3))/(4*d*(exp(2*c + 2*d*x) + 1)) + (6*b^3*exp(c + d*x))/(d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4 *d*x) + exp(6*c + 6*d*x) + 1)) - (exp(c + d*x)*(12*a*b^2 + 13*b^3))/(2*d*( 2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) - (4*b^3*exp(c + d*x))/(d*(4*e xp(2*c + 2*d*x) + 6*exp(4*c + 4*d*x) + 4*exp(6*c + 6*d*x) + exp(8*c + 8*d* x) + 1))
Time = 0.24 (sec) , antiderivative size = 743, normalized size of antiderivative = 7.51 \[ \int \cosh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {-15 e^{9 d x +9 c} \mathit {atan} \left (e^{d x +c}\right ) b^{3}-60 e^{7 d x +7 c} \mathit {atan} \left (e^{d x +c}\right ) b^{3}-90 e^{5 d x +5 c} \mathit {atan} \left (e^{d x +c}\right ) b^{3}-60 e^{3 d x +3 c} \mathit {atan} \left (e^{d x +c}\right ) b^{3}-24 e^{4 d x +4 c} a \,b^{2}-2 b^{3}-36 e^{9 d x +9 c} \mathit {atan} \left (e^{d x +c}\right ) a \,b^{2}-24 e^{d x +c} \mathit {atan} \left (e^{d x +c}\right ) a^{2} b +6 e^{10 d x +10 c} a^{2} b +6 e^{10 d x +10 c} a \,b^{2}+18 e^{8 d x +8 c} a^{2} b +30 e^{8 d x +8 c} a \,b^{2}+12 e^{6 d x +6 c} a^{2} b +24 e^{6 d x +6 c} a \,b^{2}-12 e^{4 d x +4 c} a^{2} b -24 e^{9 d x +9 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2} b -96 e^{7 d x +7 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2} b -144 e^{5 d x +5 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2} b -96 e^{3 d x +3 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2} b -6 a \,b^{2}-5 e^{4 d x +4 c} b^{3}-36 e^{d x +c} \mathit {atan} \left (e^{d x +c}\right ) a \,b^{2}-30 e^{2 d x +2 c} a \,b^{2}-2 a^{3}-15 e^{2 d x +2 c} b^{3}-144 e^{7 d x +7 c} \mathit {atan} \left (e^{d x +c}\right ) a \,b^{2}-216 e^{5 d x +5 c} \mathit {atan} \left (e^{d x +c}\right ) a \,b^{2}-144 e^{3 d x +3 c} \mathit {atan} \left (e^{d x +c}\right ) a \,b^{2}-18 e^{2 d x +2 c} a^{2} b -15 e^{d x +c} \mathit {atan} \left (e^{d x +c}\right ) b^{3}-6 a^{2} b +2 e^{10 d x +10 c} a^{3}+2 e^{10 d x +10 c} b^{3}+6 e^{8 d x +8 c} a^{3}+15 e^{8 d x +8 c} b^{3}+4 e^{6 d x +6 c} a^{3}+5 e^{6 d x +6 c} b^{3}-4 e^{4 d x +4 c} a^{3}-6 e^{2 d x +2 c} a^{3}}{4 e^{d x +c} d \left (e^{8 d x +8 c}+4 e^{6 d x +6 c}+6 e^{4 d x +4 c}+4 e^{2 d x +2 c}+1\right )} \] Input:
int(cosh(d*x+c)*(a+b*tanh(d*x+c)^2)^3,x)
Output:
( - 24*e**(9*c + 9*d*x)*atan(e**(c + d*x))*a**2*b - 36*e**(9*c + 9*d*x)*at an(e**(c + d*x))*a*b**2 - 15*e**(9*c + 9*d*x)*atan(e**(c + d*x))*b**3 - 96 *e**(7*c + 7*d*x)*atan(e**(c + d*x))*a**2*b - 144*e**(7*c + 7*d*x)*atan(e* *(c + d*x))*a*b**2 - 60*e**(7*c + 7*d*x)*atan(e**(c + d*x))*b**3 - 144*e** (5*c + 5*d*x)*atan(e**(c + d*x))*a**2*b - 216*e**(5*c + 5*d*x)*atan(e**(c + d*x))*a*b**2 - 90*e**(5*c + 5*d*x)*atan(e**(c + d*x))*b**3 - 96*e**(3*c + 3*d*x)*atan(e**(c + d*x))*a**2*b - 144*e**(3*c + 3*d*x)*atan(e**(c + d*x ))*a*b**2 - 60*e**(3*c + 3*d*x)*atan(e**(c + d*x))*b**3 - 24*e**(c + d*x)* atan(e**(c + d*x))*a**2*b - 36*e**(c + d*x)*atan(e**(c + d*x))*a*b**2 - 15 *e**(c + d*x)*atan(e**(c + d*x))*b**3 + 2*e**(10*c + 10*d*x)*a**3 + 6*e**( 10*c + 10*d*x)*a**2*b + 6*e**(10*c + 10*d*x)*a*b**2 + 2*e**(10*c + 10*d*x) *b**3 + 6*e**(8*c + 8*d*x)*a**3 + 18*e**(8*c + 8*d*x)*a**2*b + 30*e**(8*c + 8*d*x)*a*b**2 + 15*e**(8*c + 8*d*x)*b**3 + 4*e**(6*c + 6*d*x)*a**3 + 12* e**(6*c + 6*d*x)*a**2*b + 24*e**(6*c + 6*d*x)*a*b**2 + 5*e**(6*c + 6*d*x)* b**3 - 4*e**(4*c + 4*d*x)*a**3 - 12*e**(4*c + 4*d*x)*a**2*b - 24*e**(4*c + 4*d*x)*a*b**2 - 5*e**(4*c + 4*d*x)*b**3 - 6*e**(2*c + 2*d*x)*a**3 - 18*e* *(2*c + 2*d*x)*a**2*b - 30*e**(2*c + 2*d*x)*a*b**2 - 15*e**(2*c + 2*d*x)*b **3 - 2*a**3 - 6*a**2*b - 6*a*b**2 - 2*b**3)/(4*e**(c + d*x)*d*(e**(8*c + 8*d*x) + 4*e**(6*c + 6*d*x) + 6*e**(4*c + 4*d*x) + 4*e**(2*c + 2*d*x) + 1) )