Integrand size = 23, antiderivative size = 101 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {a^2 (a-6 b) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {3 a^2 b \text {sech}(c+d x)}{d}-\frac {b^2 (3 a+b) \text {sech}^3(c+d x)}{3 d}+\frac {b^3 \text {sech}^5(c+d x)}{5 d} \] Output:
1/2*a^2*(a-6*b)*arctanh(cosh(d*x+c))/d-1/2*a^3*coth(d*x+c)*csch(d*x+c)/d+3 *a^2*b*sech(d*x+c)/d-1/3*b^2*(3*a+b)*sech(d*x+c)^3/d+1/5*b^3*sech(d*x+c)^5 /d
Time = 16.31 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.57 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=-\frac {a^3 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {\left (a^3-6 a^2 b\right ) \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {\left (-a^3+6 a^2 b\right ) \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {a^3 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {3 a^2 b \text {sech}(c+d x)}{d}-\frac {b^2 (3 a+b) \text {sech}^3(c+d x)}{3 d}+\frac {b^3 \text {sech}^5(c+d x)}{5 d} \] Input:
Integrate[Csch[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^3,x]
Output:
-1/8*(a^3*Csch[(c + d*x)/2]^2)/d + ((a^3 - 6*a^2*b)*Log[Cosh[(c + d*x)/2]] )/(2*d) + ((-a^3 + 6*a^2*b)*Log[Sinh[(c + d*x)/2]])/(2*d) - (a^3*Sech[(c + d*x)/2]^2)/(8*d) + (3*a^2*b*Sech[c + d*x])/d - (b^2*(3*a + b)*Sech[c + d* x]^3)/(3*d) + (b^3*Sech[c + d*x]^5)/(5*d)
Time = 0.41 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.59, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3042, 26, 4147, 369, 403, 25, 403, 25, 299, 219}
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.
| \(\displaystyle \int \text {csch}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \left (a-b \tan (i c+i d x)^2\right )^3}{\sin (i c+i d x)^3}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\left (a-b \tan (i c+i d x)^2\right )^3}{\sin (i c+i d x)^3}dx\) |
| \(\Big \downarrow \) 4147 |
| \(\displaystyle -\frac {\int \frac {\text {sech}^2(c+d x) \left (-b \text {sech}^2(c+d x)+a+b\right )^3}{\left (1-\text {sech}^2(c+d x)\right )^2}d\text {sech}(c+d x)}{d}\) |
| \(\Big \downarrow \) 369 |
| \(\displaystyle -\frac {\frac {\text {sech}(c+d x) \left (a-b \text {sech}^2(c+d x)+b\right )^3}{2 \left (1-\text {sech}^2(c+d x)\right )}-\frac {1}{2} \int \frac {\left (-7 b \text {sech}^2(c+d x)+a+b\right ) \left (-b \text {sech}^2(c+d x)+a+b\right )^2}{1-\text {sech}^2(c+d x)}d\text {sech}(c+d x)}{d}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{5} \int -\frac {\left (-b \text {sech}^2(c+d x)+a+b\right ) \left ((5 a-2 b) (a+b)-(33 a-2 b) b \text {sech}^2(c+d x)\right )}{1-\text {sech}^2(c+d x)}d\text {sech}(c+d x)-\frac {7}{5} b \text {sech}(c+d x) \left (a-b \text {sech}^2(c+d x)+b\right )^2\right )+\frac {\text {sech}(c+d x) \left (a-b \text {sech}^2(c+d x)+b\right )^3}{2 \left (1-\text {sech}^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{2} \left (-\frac {1}{5} \int \frac {\left (-b \text {sech}^2(c+d x)+a+b\right ) \left ((5 a-2 b) (a+b)-(33 a-2 b) b \text {sech}^2(c+d x)\right )}{1-\text {sech}^2(c+d x)}d\text {sech}(c+d x)-\frac {7}{5} b \text {sech}(c+d x) \left (a-b \text {sech}^2(c+d x)+b\right )^2\right )+\frac {\text {sech}(c+d x) \left (a-b \text {sech}^2(c+d x)+b\right )^3}{2 \left (1-\text {sech}^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{3} \int -\frac {(a+b) \left (15 a^2-24 b a-4 b^2\right )-b \left (81 a^2-28 b a-4 b^2\right ) \text {sech}^2(c+d x)}{1-\text {sech}^2(c+d x)}d\text {sech}(c+d x)-\frac {1}{3} b (33 a-2 b) \text {sech}(c+d x) \left (a-b \text {sech}^2(c+d x)+b\right )\right )-\frac {7}{5} b \text {sech}(c+d x) \left (a-b \text {sech}^2(c+d x)+b\right )^2\right )+\frac {\text {sech}(c+d x) \left (a-b \text {sech}^2(c+d x)+b\right )^3}{2 \left (1-\text {sech}^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{5} \left (-\frac {1}{3} \int \frac {(a+b) \left (15 a^2-24 b a-4 b^2\right )-b \left (81 a^2-28 b a-4 b^2\right ) \text {sech}^2(c+d x)}{1-\text {sech}^2(c+d x)}d\text {sech}(c+d x)-\frac {1}{3} b (33 a-2 b) \text {sech}(c+d x) \left (a-b \text {sech}^2(c+d x)+b\right )\right )-\frac {7}{5} b \text {sech}(c+d x) \left (a-b \text {sech}^2(c+d x)+b\right )^2\right )+\frac {\text {sech}(c+d x) \left (a-b \text {sech}^2(c+d x)+b\right )^3}{2 \left (1-\text {sech}^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{3} \left (-15 a^2 (a-6 b) \int \frac {1}{1-\text {sech}^2(c+d x)}d\text {sech}(c+d x)-b \left (81 a^2-28 a b-4 b^2\right ) \text {sech}(c+d x)\right )-\frac {1}{3} b (33 a-2 b) \text {sech}(c+d x) \left (a-b \text {sech}^2(c+d x)+b\right )\right )-\frac {7}{5} b \text {sech}(c+d x) \left (a-b \text {sech}^2(c+d x)+b\right )^2\right )+\frac {\text {sech}(c+d x) \left (a-b \text {sech}^2(c+d x)+b\right )^3}{2 \left (1-\text {sech}^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{3} \left (-15 a^2 (a-6 b) \text {arctanh}(\text {sech}(c+d x))-b \left (81 a^2-28 a b-4 b^2\right ) \text {sech}(c+d x)\right )-\frac {1}{3} b (33 a-2 b) \text {sech}(c+d x) \left (a-b \text {sech}^2(c+d x)+b\right )\right )-\frac {7}{5} b \text {sech}(c+d x) \left (a-b \text {sech}^2(c+d x)+b\right )^2\right )+\frac {\text {sech}(c+d x) \left (a-b \text {sech}^2(c+d x)+b\right )^3}{2 \left (1-\text {sech}^2(c+d x)\right )}}{d}\) |
Input:
Int[Csch[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^3,x]
Output:
-(((Sech[c + d*x]*(a + b - b*Sech[c + d*x]^2)^3)/(2*(1 - Sech[c + d*x]^2)) + ((-7*b*Sech[c + d*x]*(a + b - b*Sech[c + d*x]^2)^2)/5 + ((-15*a^2*(a - 6*b)*ArcTanh[Sech[c + d*x]] - b*(81*a^2 - 28*a*b - 4*b^2)*Sech[c + d*x])/3 - ((33*a - 2*b)*b*Sech[c + d*x]*(a + b - b*Sech[c + d*x]^2))/3)/5)/2)/d)
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2* b*(p + 1))), x] - Simp[e^2/(2*b*(p + 1)) Int[(e*x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(m - 1) + d*(m + 2*q - 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 0 ] && GtQ[m, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Sec[e + f*x], x]}, Simp[1/(f*ff^ m) Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*((a - b + b*ff^2*x^2)^p/x^(m + 1 )), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[( m - 1)/2]
Time = 21.89 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.02
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+3 a^{2} b \left (\frac {1}{\cosh \left (d x +c \right )}-2 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )-\frac {b^{2} a}{\cosh \left (d x +c \right )^{3}}+b^{3} \left (-\frac {\sinh \left (d x +c \right )^{2}}{3 \cosh \left (d x +c \right )^{5}}-\frac {2}{15 \cosh \left (d x +c \right )^{5}}\right )}{d}\) | \(103\) |
| default | \(\frac {a^{3} \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+3 a^{2} b \left (\frac {1}{\cosh \left (d x +c \right )}-2 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )-\frac {b^{2} a}{\cosh \left (d x +c \right )^{3}}+b^{3} \left (-\frac {\sinh \left (d x +c \right )^{2}}{3 \cosh \left (d x +c \right )^{5}}-\frac {2}{15 \cosh \left (d x +c \right )^{5}}\right )}{d}\) | \(103\) |
| risch | \(-\frac {{\mathrm e}^{d x +c} \left (15 a^{3} {\mathrm e}^{12 d x +12 c}-90 a^{2} b \,{\mathrm e}^{12 d x +12 c}+90 \,{\mathrm e}^{10 d x +10 c} a^{3}-180 a^{2} b \,{\mathrm e}^{10 d x +10 c}+120 a \,b^{2} {\mathrm e}^{10 d x +10 c}+40 b^{3} {\mathrm e}^{10 d x +10 c}+225 a^{3} {\mathrm e}^{8 d x +8 c}+90 a^{2} b \,{\mathrm e}^{8 d x +8 c}-96 b^{3} {\mathrm e}^{8 d x +8 c}+300 a^{3} {\mathrm e}^{6 d x +6 c}+360 a^{2} b \,{\mathrm e}^{6 d x +6 c}-240 a \,b^{2} {\mathrm e}^{6 d x +6 c}+112 b^{3} {\mathrm e}^{6 d x +6 c}+225 \,{\mathrm e}^{4 d x +4 c} a^{3}+90 a^{2} b \,{\mathrm e}^{4 d x +4 c}-96 b^{3} {\mathrm e}^{4 d x +4 c}+90 \,{\mathrm e}^{2 d x +2 c} a^{3}-180 a^{2} b \,{\mathrm e}^{2 d x +2 c}+120 a \,b^{2} {\mathrm e}^{2 d x +2 c}+40 b^{3} {\mathrm e}^{2 d x +2 c}+15 a^{3}-90 a^{2} b \right )}{15 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{5} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}-\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d}+\frac {3 a^{2} b \ln \left ({\mathrm e}^{d x +c}-1\right )}{d}+\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d}-\frac {3 a^{2} b \ln \left ({\mathrm e}^{d x +c}+1\right )}{d}\) | \(410\) |
Input:
int(csch(d*x+c)^3*(a+tanh(d*x+c)^2*b)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(a^3*(-1/2*csch(d*x+c)*coth(d*x+c)+arctanh(exp(d*x+c)))+3*a^2*b*(1/cos h(d*x+c)-2*arctanh(exp(d*x+c)))-b^2*a/cosh(d*x+c)^3+b^3*(-1/3*sinh(d*x+c)^ 2/cosh(d*x+c)^5-2/15/cosh(d*x+c)^5))
Leaf count of result is larger than twice the leaf count of optimal. 5037 vs. \(2 (93) = 186\).
Time = 0.14 (sec) , antiderivative size = 5037, normalized size of antiderivative = 49.87 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)^3*(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")
Output:
Too large to include
\[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3} \operatorname {csch}^{3}{\left (c + d x \right )}\, dx \] Input:
integrate(csch(d*x+c)**3*(a+b*tanh(d*x+c)**2)**3,x)
Output:
Integral((a + b*tanh(c + d*x)**2)**3*csch(c + d*x)**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (93) = 186\).
Time = 0.04 (sec) , antiderivative size = 403, normalized size of antiderivative = 3.99 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {1}{2} \, a^{3} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} - 3 \, a^{2} b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {2 \, e^{\left (-d x - c\right )}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} - \frac {8}{15} \, b^{3} {\left (\frac {5 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} - \frac {2 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} - \frac {8 \, a b^{2}}{d {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3}} \] Input:
integrate(csch(d*x+c)^3*(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")
Output:
1/2*a^3*(log(e^(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d + 2*(e^(-d*x - c) + e^(-3*d*x - 3*c))/(d*(2*e^(-2*d*x - 2*c) - e^(-4*d*x - 4*c) - 1))) - 3*a^2*b*(log(e^(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d - 2*e^(-d*x - c )/(d*(e^(-2*d*x - 2*c) + 1))) - 8/15*b^3*(5*e^(-3*d*x - 3*c)/(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*e^(-5*d*x - 5*c)/(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)) + 5*e^(-7*d*x - 7*c)/(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))) - 8*a*b^2/(d*(e^(d*x + c) + e^(-d*x - c))^3)
Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (93) = 186\).
Time = 0.23 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.04 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=-\frac {\frac {60 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4} - 15 \, {\left (a^{3} - 6 \, a^{2} b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) + 15 \, {\left (a^{3} - 6 \, a^{2} b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {8 \, {\left (45 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4} - 60 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 20 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 48 \, b^{3}\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5}}}{60 \, d} \] Input:
integrate(csch(d*x+c)^3*(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")
Output:
-1/60*(60*a^3*(e^(d*x + c) + e^(-d*x - c))/((e^(d*x + c) + e^(-d*x - c))^2 - 4) - 15*(a^3 - 6*a^2*b)*log(e^(d*x + c) + e^(-d*x - c) + 2) + 15*(a^3 - 6*a^2*b)*log(e^(d*x + c) + e^(-d*x - c) - 2) - 8*(45*a^2*b*(e^(d*x + c) + e^(-d*x - c))^4 - 60*a*b^2*(e^(d*x + c) + e^(-d*x - c))^2 - 20*b^3*(e^(d* x + c) + e^(-d*x - c))^2 + 48*b^3)/(e^(d*x + c) + e^(-d*x - c))^5)/d
Time = 0.17 (sec) , antiderivative size = 412, normalized size of antiderivative = 4.08 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a^3\,\sqrt {-d^2}-6\,a^2\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {a^6-12\,a^5\,b+36\,a^4\,b^2}}\right )\,\sqrt {a^6-12\,a^5\,b+36\,a^4\,b^2}}{\sqrt {-d^2}}-\frac {64\,b^3\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}+\frac {8\,{\mathrm {e}}^{c+d\,x}\,\left (17\,b^3+15\,a\,b^2\right )}{15\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}+\frac {32\,b^3\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}-\frac {a^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {8\,{\mathrm {e}}^{c+d\,x}\,\left (b^3+3\,a\,b^2\right )}{3\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {2\,a^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {6\,a^2\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \] Input:
int((a + b*tanh(c + d*x)^2)^3/sinh(c + d*x)^3,x)
Output:
(atan((exp(d*x)*exp(c)*(a^3*(-d^2)^(1/2) - 6*a^2*b*(-d^2)^(1/2)))/(d*(a^6 - 12*a^5*b + 36*a^4*b^2)^(1/2)))*(a^6 - 12*a^5*b + 36*a^4*b^2)^(1/2))/(-d^ 2)^(1/2) - (64*b^3*exp(c + d*x))/(5*d*(4*exp(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)) + (8*exp(c + d*x)*(15*a *b^2 + 17*b^3))/(15*d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) + (32*b^3*exp(c + d*x))/(5*d*(5*exp(2*c + 2*d*x) + 10*exp(4* c + 4*d*x) + 10*exp(6*c + 6*d*x) + 5*exp(8*c + 8*d*x) + exp(10*c + 10*d*x) + 1)) - (a^3*exp(c + d*x))/(d*(exp(2*c + 2*d*x) - 1)) - (8*exp(c + d*x)*( 3*a*b^2 + b^3))/(3*d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) - (2*a^3 *exp(c + d*x))/(d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1)) + (6*a^2*b* exp(c + d*x))/(d*(exp(2*c + 2*d*x) + 1))
Time = 0.29 (sec) , antiderivative size = 1198, normalized size of antiderivative = 11.86 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx =\text {Too large to display} \] Input:
int(csch(d*x+c)^3*(a+b*tanh(d*x+c)^2)^3,x)
Output:
( - 15*e**(14*c + 14*d*x)*log(e**(c + d*x) - 1)*a**3 + 90*e**(14*c + 14*d* x)*log(e**(c + d*x) - 1)*a**2*b + 15*e**(14*c + 14*d*x)*log(e**(c + d*x) + 1)*a**3 - 90*e**(14*c + 14*d*x)*log(e**(c + d*x) + 1)*a**2*b - 30*e**(13* c + 13*d*x)*a**3 + 180*e**(13*c + 13*d*x)*a**2*b - 45*e**(12*c + 12*d*x)*l og(e**(c + d*x) - 1)*a**3 + 270*e**(12*c + 12*d*x)*log(e**(c + d*x) - 1)*a **2*b + 45*e**(12*c + 12*d*x)*log(e**(c + d*x) + 1)*a**3 - 270*e**(12*c + 12*d*x)*log(e**(c + d*x) + 1)*a**2*b - 180*e**(11*c + 11*d*x)*a**3 + 360*e **(11*c + 11*d*x)*a**2*b - 240*e**(11*c + 11*d*x)*a*b**2 - 80*e**(11*c + 1 1*d*x)*b**3 - 15*e**(10*c + 10*d*x)*log(e**(c + d*x) - 1)*a**3 + 90*e**(10 *c + 10*d*x)*log(e**(c + d*x) - 1)*a**2*b + 15*e**(10*c + 10*d*x)*log(e**( c + d*x) + 1)*a**3 - 90*e**(10*c + 10*d*x)*log(e**(c + d*x) + 1)*a**2*b - 450*e**(9*c + 9*d*x)*a**3 - 180*e**(9*c + 9*d*x)*a**2*b + 192*e**(9*c + 9* d*x)*b**3 + 75*e**(8*c + 8*d*x)*log(e**(c + d*x) - 1)*a**3 - 450*e**(8*c + 8*d*x)*log(e**(c + d*x) - 1)*a**2*b - 75*e**(8*c + 8*d*x)*log(e**(c + d*x ) + 1)*a**3 + 450*e**(8*c + 8*d*x)*log(e**(c + d*x) + 1)*a**2*b - 600*e**( 7*c + 7*d*x)*a**3 - 720*e**(7*c + 7*d*x)*a**2*b + 480*e**(7*c + 7*d*x)*a*b **2 - 224*e**(7*c + 7*d*x)*b**3 + 75*e**(6*c + 6*d*x)*log(e**(c + d*x) - 1 )*a**3 - 450*e**(6*c + 6*d*x)*log(e**(c + d*x) - 1)*a**2*b - 75*e**(6*c + 6*d*x)*log(e**(c + d*x) + 1)*a**3 + 450*e**(6*c + 6*d*x)*log(e**(c + d*x) + 1)*a**2*b - 450*e**(5*c + 5*d*x)*a**3 - 180*e**(5*c + 5*d*x)*a**2*b +...