Integrand size = 23, antiderivative size = 166 \[ \int \frac {\sinh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {5 (3 a-4 b) \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{8 (a+b)^{9/2} d}-\frac {(a-2 b) \cosh (c+d x)}{(a+b)^4 d}+\frac {\cosh ^3(c+d x)}{3 (a+b)^3 d}+\frac {a b \text {sech}(c+d x)}{4 (a+b)^3 d \left (a+b-b \text {sech}^2(c+d x)\right )^2}+\frac {(7 a-4 b) b \text {sech}(c+d x)}{8 (a+b)^4 d \left (a+b-b \text {sech}^2(c+d x)\right )} \] Output:
5/8*(3*a-4*b)*b^(1/2)*arctanh(b^(1/2)*sech(d*x+c)/(a+b)^(1/2))/(a+b)^(9/2) /d-(a-2*b)*cosh(d*x+c)/(a+b)^4/d+1/3*cosh(d*x+c)^3/(a+b)^3/d+1/4*a*b*sech( d*x+c)/(a+b)^3/d/(a+b-b*sech(d*x+c)^2)^2+1/8*(7*a-4*b)*b*sech(d*x+c)/(a+b) ^4/d/(a+b-b*sech(d*x+c)^2)
Result contains complex when optimal does not.
Time = 1.89 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.37 \[ \int \frac {\sinh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {\frac {15 i (3 a-4 b) \sqrt {b} \left (\arctan \left (\frac {-i \sqrt {a+b}-\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )+\arctan \left (\frac {-i \sqrt {a+b}+\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )\right )}{(a+b)^{9/2}}-\frac {6 \cosh (c+d x) \left (3 a^3-24 a^2 b+30 a b^2-13 b^3+\left (6 a^3-27 a^2 b-11 a b^2+22 b^3\right ) \cosh (2 (c+d x))+3 (a-3 b) (a+b)^2 \cosh ^2(2 (c+d x))\right )}{(a+b)^4 (a-b+(a+b) \cosh (2 (c+d x)))^2}+\frac {2 \cosh (3 (c+d x))}{(a+b)^3}}{24 d} \] Input:
Integrate[Sinh[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^3,x]
Output:
(((15*I)*(3*a - 4*b)*Sqrt[b]*(ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]] + ArcTan[((-I)*Sqrt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2]) /Sqrt[b]]))/(a + b)^(9/2) - (6*Cosh[c + d*x]*(3*a^3 - 24*a^2*b + 30*a*b^2 - 13*b^3 + (6*a^3 - 27*a^2*b - 11*a*b^2 + 22*b^3)*Cosh[2*(c + d*x)] + 3*(a - 3*b)*(a + b)^2*Cosh[2*(c + d*x)]^2))/((a + b)^4*(a - b + (a + b)*Cosh[2 *(c + d*x)])^2) + (2*Cosh[3*(c + d*x)])/(a + b)^3)/(24*d)
Time = 0.54 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 26, 4147, 25, 361, 1582, 1584, 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.
| \(\displaystyle \int \frac {\sinh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \sin (i c+i d x)^3}{\left (a-b \tan (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\sin (i c+i d x)^3}{\left (a-b \tan (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 4147 |
| \(\displaystyle \frac {\int -\frac {\cosh ^4(c+d x) \left (1-\text {sech}^2(c+d x)\right )}{\left (-b \text {sech}^2(c+d x)+a+b\right )^3}d\text {sech}(c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {\cosh ^4(c+d x) \left (1-\text {sech}^2(c+d x)\right )}{\left (-b \text {sech}^2(c+d x)+a+b\right )^3}d\text {sech}(c+d x)}{d}\) |
| \(\Big \downarrow \) 361 |
| \(\displaystyle \frac {\frac {a b \text {sech}(c+d x)}{4 (a+b)^3 \left (a-b \text {sech}^2(c+d x)+b\right )^2}-\frac {1}{4} b \int \frac {\cosh ^4(c+d x) \left (-\frac {3 a \text {sech}^4(c+d x)}{(a+b)^3}-\frac {4 a \text {sech}^2(c+d x)}{b (a+b)^2}+\frac {4}{b (a+b)}\right )}{\left (-b \text {sech}^2(c+d x)+a+b\right )^2}d\text {sech}(c+d x)}{d}\) |
| \(\Big \downarrow \) 1582 |
| \(\displaystyle \frac {\frac {a b \text {sech}(c+d x)}{4 (a+b)^3 \left (a-b \text {sech}^2(c+d x)+b\right )^2}-\frac {1}{4} b \left (\frac {\int \frac {\cosh ^4(c+d x) \left (-\frac {(7 a-4 b) b^2 \text {sech}^4(c+d x)}{a+b}-8 (a-b) b \text {sech}^2(c+d x)+8 b (a+b)\right )}{-b \text {sech}^2(c+d x)+a+b}d\text {sech}(c+d x)}{2 b^2 (a+b)^3}-\frac {(7 a-4 b) \text {sech}(c+d x)}{2 (a+b)^4 \left (a-b \text {sech}^2(c+d x)+b\right )}\right )}{d}\) |
| \(\Big \downarrow \) 1584 |
| \(\displaystyle \frac {\frac {a b \text {sech}(c+d x)}{4 (a+b)^3 \left (a-b \text {sech}^2(c+d x)+b\right )^2}-\frac {1}{4} b \left (\frac {\int \left (8 b \cosh ^4(c+d x)+\frac {8 b (2 b-a) \cosh ^2(c+d x)}{a+b}-\frac {5 b^2 (4 b-3 a)}{(a+b) \left (b \text {sech}^2(c+d x)-a-b\right )}\right )d\text {sech}(c+d x)}{2 b^2 (a+b)^3}-\frac {(7 a-4 b) \text {sech}(c+d x)}{2 (a+b)^4 \left (a-b \text {sech}^2(c+d x)+b\right )}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a b \text {sech}(c+d x)}{4 (a+b)^3 \left (a-b \text {sech}^2(c+d x)+b\right )^2}-\frac {1}{4} b \left (\frac {-\frac {5 b^{3/2} (3 a-4 b) \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{3/2}}+\frac {8 b (a-2 b) \cosh (c+d x)}{a+b}-\frac {8}{3} b \cosh ^3(c+d x)}{2 b^2 (a+b)^3}-\frac {(7 a-4 b) \text {sech}(c+d x)}{2 (a+b)^4 \left (a-b \text {sech}^2(c+d x)+b\right )}\right )}{d}\) |
Input:
Int[Sinh[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^3,x]
Output:
((a*b*Sech[c + d*x])/(4*(a + b)^3*(a + b - b*Sech[c + d*x]^2)^2) - (b*(((- 5*(3*a - 4*b)*b^(3/2)*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/(a + b )^(3/2) + (8*(a - 2*b)*b*Cosh[c + d*x])/(a + b) - (8*b*Cosh[c + d*x]^3)/3) /(2*b^2*(a + b)^3) - ((7*a - 4*b)*Sech[c + d*x])/(2*(a + b)^4*(a + b - b*S ech[c + d*x]^2))))/4)/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[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : > Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1)) Int[x^m*(a + b*x^2)^(p + 1)*E xpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)] - ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ILtQ[m/ 2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^ 4)^(p_.), x_Symbol] :> Simp[(-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*((d + e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1))), x] + Simp[(-d)^(m/2 - 1)/(2*e^ (2*p)*(q + 1)) Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e *x^2))*(2*(-d)^(-m/2 + 1)*e^(2*p)*(q + 1)*(a + b*x^2 + c*x^4)^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2))], x], x], x] /; Fre eQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* (a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(340\) vs. \(2(150)=300\).
Time = 83.76 (sec) , antiderivative size = 341, normalized size of antiderivative = 2.05
| method | result | size |
| derivativedivides | \(\frac {\frac {1}{3 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {a -5 b}{2 \left (a +b \right )^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {2 b \left (\frac {-\frac {\left (9 a +20 b \right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{8}-\frac {\left (27 a^{3}+66 a^{2} b +56 b^{2} a -16 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 a}+\left (-\frac {27}{8} a^{2}-\frac {11}{2} a b +2 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\frac {9 a^{2}}{8}+\frac {a b}{4}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a \right )^{2}}-\frac {5 \left (3 a -4 b \right ) \operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{16 \sqrt {a b +b^{2}}}\right )}{\left (a +b \right )^{4}}-\frac {1}{3 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a +5 b}{2 \left (a +b \right )^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(341\) |
| default | \(\frac {\frac {1}{3 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {a -5 b}{2 \left (a +b \right )^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {2 b \left (\frac {-\frac {\left (9 a +20 b \right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{8}-\frac {\left (27 a^{3}+66 a^{2} b +56 b^{2} a -16 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 a}+\left (-\frac {27}{8} a^{2}-\frac {11}{2} a b +2 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\frac {9 a^{2}}{8}+\frac {a b}{4}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a \right )^{2}}-\frac {5 \left (3 a -4 b \right ) \operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{16 \sqrt {a b +b^{2}}}\right )}{\left (a +b \right )^{4}}-\frac {1}{3 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a +5 b}{2 \left (a +b \right )^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(341\) |
| risch | \(\frac {{\mathrm e}^{3 d x +3 c}}{24 \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right ) d}-\frac {3 \,{\mathrm e}^{d x +c} a}{8 \left (a +b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right ) d}+\frac {9 \,{\mathrm e}^{d x +c} b}{8 \left (a +b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right ) d}-\frac {3 \,{\mathrm e}^{-d x -c} a}{8 \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) d}+\frac {9 \,{\mathrm e}^{-d x -c} b}{8 \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) d}+\frac {{\mathrm e}^{-3 d x -3 c}}{24 \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right ) d}+\frac {{\mathrm e}^{d x +c} b \left (9 \,{\mathrm e}^{6 d x +6 c} a^{2}+5 \,{\mathrm e}^{6 d x +6 c} a b -4 \,{\mathrm e}^{6 d x +6 c} b^{2}+27 \,{\mathrm e}^{4 d x +4 c} a^{2}-13 \,{\mathrm e}^{4 d x +4 c} a b +4 \,{\mathrm e}^{4 d x +4 c} b^{2}+27 \,{\mathrm e}^{2 d x +2 c} a^{2}-13 \,{\mathrm e}^{2 d x +2 c} b a +4 b^{2} {\mathrm e}^{2 d x +2 c}+9 a^{2}+5 a b -4 b^{2}\right )}{4 \left ({\mathrm e}^{4 d x +4 c} a +b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 \,{\mathrm e}^{2 d x +2 c} b +a +b \right )^{2} d \left (a +b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}+\frac {15 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) a}{16 \left (a +b \right )^{5} d}-\frac {5 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) b}{4 \left (a +b \right )^{5} d}-\frac {15 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) a}{16 \left (a +b \right )^{5} d}+\frac {5 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) b}{4 \left (a +b \right )^{5} d}\) | \(663\) |
Input:
int(sinh(d*x+c)^3/(a+tanh(d*x+c)^2*b)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(1/3/(a+b)^3/(tanh(1/2*d*x+1/2*c)+1)^3-1/2/(a+b)^3/(tanh(1/2*d*x+1/2*c )+1)^2-1/2*(a-5*b)/(a+b)^4/(tanh(1/2*d*x+1/2*c)+1)-2*b/(a+b)^4*((-1/8*(9*a +20*b)*a*tanh(1/2*d*x+1/2*c)^6-1/8*(27*a^3+66*a^2*b+56*a*b^2-16*b^3)/a*tan h(1/2*d*x+1/2*c)^4+(-27/8*a^2-11/2*a*b+2*b^2)*tanh(1/2*d*x+1/2*c)^2-9/8*a^ 2+1/4*a*b)/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*b*tanh(1/2 *d*x+1/2*c)^2+a)^2-5/16*(3*a-4*b)/(a*b+b^2)^(1/2)*arctanh(1/4*(2*tanh(1/2* d*x+1/2*c)^2*a+2*a+4*b)/(a*b+b^2)^(1/2)))-1/3/(a+b)^3/(tanh(1/2*d*x+1/2*c) -1)^3-1/2/(a+b)^3/(tanh(1/2*d*x+1/2*c)-1)^2-1/2/(a+b)^4*(-a+5*b)/(tanh(1/2 *d*x+1/2*c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 7129 vs. \(2 (156) = 312\).
Time = 0.35 (sec) , antiderivative size = 13095, normalized size of antiderivative = 78.89 \[ \int \frac {\sinh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(sinh(d*x+c)^3/(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\sinh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Timed out} \] Input:
integrate(sinh(d*x+c)**3/(a+b*tanh(d*x+c)**2)**3,x)
Output:
Timed out
Exception generated. \[ \int \frac {\sinh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(sinh(d*x+c)^3/(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Exception generated. \[ \int \frac {\sinh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(sinh(d*x+c)^3/(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Timed out. \[ \int \frac {\sinh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^3}{{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \] Input:
int(sinh(c + d*x)^3/(a + b*tanh(c + d*x)^2)^3,x)
Output:
int(sinh(c + d*x)^3/(a + b*tanh(c + d*x)^2)^3, x)
Time = 0.38 (sec) , antiderivative size = 3335, normalized size of antiderivative = 20.09 \[ \int \frac {\sinh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:
int(sinh(d*x+c)^3/(a+b*tanh(d*x+c)^2)^3,x)
Output:
( - 45*e**(11*c + 11*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*a**3 - 30*e**(11*c + 11*d*x)* sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e** (c + d*x)*sqrt(b))*a**2*b + 75*e**(11*c + 11*d*x)*sqrt(b)*sqrt(a + b)*log( e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*a*b** 2 + 60*e**(11*c + 11*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*b**3 + 45*e**(11*c + 11*d*x)* sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) + 2*e** (c + d*x)*sqrt(b))*a**3 + 30*e**(11*c + 11*d*x)*sqrt(b)*sqrt(a + b)*log(e* *(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) + 2*e**(c + d*x)*sqrt(b))*a**2*b - 75*e**(11*c + 11*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) + 2*e**(c + d*x)*sqrt(b))*a*b**2 - 60*e**(11*c + 11*d*x)* sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) + 2*e** (c + d*x)*sqrt(b))*b**3 - 180*e**(9*c + 9*d*x)*sqrt(b)*sqrt(a + b)*log(e** (2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*a**3 + 2 40*e**(9*c + 9*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*a**2*b + 180*e**(9*c + 9*d*x)*sqrt( b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*a*b**2 - 240*e**(9*c + 9*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2* c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*b**3 + 1...