Integrand size = 23, antiderivative size = 240 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {3 \left (a^2-10 a b+5 b^2\right ) x}{8 (a+b)^5}+\frac {3 \sqrt {b} \left (5 a^2-10 a b+b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 \sqrt {a} (a+b)^5 d}-\frac {(5 a-3 b) \cosh (c+d x) \sinh (c+d x)}{8 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {(7 a-5 b) b \tanh (c+d x)}{8 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {3 (a-b) b \tanh (c+d x)}{2 (a+b)^4 d \left (a+b \tanh ^2(c+d x)\right )} \]
3/8*(a^2-10*a*b+5*b^2)*x/(a+b)^5+3/8*(5*a^2-10*a*b+b^2)*arctan(b^(1/2)*tan h(d*x+c)/a^(1/2))*b^(1/2)/(a+b)^5/d/a^(1/2)-1/8*(5*a-3*b)*cosh(d*x+c)*sinh (d*x+c)/(a+b)^2/d/(a+b*tanh(d*x+c)^2)^2+1/4*cosh(d*x+c)^3*sinh(d*x+c)/(a+b )/d/(a+b*tanh(d*x+c)^2)^2+1/8*(7*a-5*b)*b*tanh(d*x+c)/(a+b)^3/d/(a+b*tanh( d*x+c)^2)^2+3/2*(a-b)*b*tanh(d*x+c)/(a+b)^4/d/(a+b*tanh(d*x+c)^2)
Time = 1.64 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.77 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {12 \left (a^2-10 a b+5 b^2\right ) (c+d x)+\frac {12 \sqrt {b} \left (5 a^2-10 a b+b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a}}-8 (a-2 b) (a+b) \sinh (2 (c+d x))+\frac {16 a b^2 (a+b) \sinh (2 (c+d x))}{(a-b+(a+b) \cosh (2 (c+d x)))^2}+\frac {4 (9 a-5 b) b (a+b) \sinh (2 (c+d x))}{a-b+(a+b) \cosh (2 (c+d x))}+(a+b)^2 \sinh (4 (c+d x))}{32 (a+b)^5 d} \]
(12*(a^2 - 10*a*b + 5*b^2)*(c + d*x) + (12*Sqrt[b]*(5*a^2 - 10*a*b + b^2)* ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/Sqrt[a] - 8*(a - 2*b)*(a + b)*Sin h[2*(c + d*x)] + (16*a*b^2*(a + b)*Sinh[2*(c + d*x)])/(a - b + (a + b)*Cos h[2*(c + d*x)])^2 + (4*(9*a - 5*b)*b*(a + b)*Sinh[2*(c + d*x)])/(a - b + ( a + b)*Cosh[2*(c + d*x)]) + (a + b)^2*Sinh[4*(c + d*x)])/(32*(a + b)^5*d)
Time = 0.51 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.14, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {3042, 4146, 372, 402, 25, 402, 27, 402, 27, 397, 218, 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.
| \(\displaystyle \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (i c+i d x)^4}{\left (a-b \tan (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 4146 |
| \(\displaystyle \frac {\int \frac {\tanh ^4(c+d x)}{\left (1-\tanh ^2(c+d x)\right )^3 \left (b \tanh ^2(c+d x)+a\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 (a+b) \left (1-\tanh ^2(c+d x)\right )^2 \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {\int \frac {(4 a-3 b) \tanh ^2(c+d x)+a}{\left (1-\tanh ^2(c+d x)\right )^2 \left (b \tanh ^2(c+d x)+a\right )^3}d\tanh (c+d x)}{4 (a+b)}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 (a+b) \left (1-\tanh ^2(c+d x)\right )^2 \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {\frac {\int -\frac {a (3 a-5 b)-5 (5 a-3 b) b \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )^3}d\tanh (c+d x)}{2 (a+b)}+\frac {(5 a-3 b) \tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )^2}}{4 (a+b)}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 (a+b) \left (1-\tanh ^2(c+d x)\right )^2 \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {\frac {(5 a-3 b) \tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {\int \frac {a (3 a-5 b)-5 (5 a-3 b) b \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )^3}d\tanh (c+d x)}{2 (a+b)}}{4 (a+b)}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 (a+b) \left (1-\tanh ^2(c+d x)\right )^2 \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {\frac {(5 a-3 b) \tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {\frac {b (7 a-5 b) \tanh (c+d x)}{(a+b) \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {\int -\frac {12 a \left (a (a-3 b)-(7 a-5 b) b \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{4 a (a+b)}}{2 (a+b)}}{4 (a+b)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 (a+b) \left (1-\tanh ^2(c+d x)\right )^2 \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {\frac {(5 a-3 b) \tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {\frac {3 \int \frac {a (a-3 b)-(7 a-5 b) b \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{a+b}+\frac {b (7 a-5 b) \tanh (c+d x)}{(a+b) \left (a+b \tanh ^2(c+d x)\right )^2}}{2 (a+b)}}{4 (a+b)}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 (a+b) \left (1-\tanh ^2(c+d x)\right )^2 \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {\frac {(5 a-3 b) \tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {\frac {3 \left (\frac {4 b (a-b) \tanh (c+d x)}{(a+b) \left (a+b \tanh ^2(c+d x)\right )}-\frac {\int -\frac {2 a \left (a^2-6 b a+b^2-4 (a-b) b \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{2 a (a+b)}\right )}{a+b}+\frac {b (7 a-5 b) \tanh (c+d x)}{(a+b) \left (a+b \tanh ^2(c+d x)\right )^2}}{2 (a+b)}}{4 (a+b)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 (a+b) \left (1-\tanh ^2(c+d x)\right )^2 \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {\frac {(5 a-3 b) \tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {\frac {3 \left (\frac {\int \frac {a^2-6 b a+b^2-4 (a-b) b \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{a+b}+\frac {4 b (a-b) \tanh (c+d x)}{(a+b) \left (a+b \tanh ^2(c+d x)\right )}\right )}{a+b}+\frac {b (7 a-5 b) \tanh (c+d x)}{(a+b) \left (a+b \tanh ^2(c+d x)\right )^2}}{2 (a+b)}}{4 (a+b)}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 (a+b) \left (1-\tanh ^2(c+d x)\right )^2 \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {\frac {(5 a-3 b) \tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {\frac {3 \left (\frac {\frac {\left (a^2-10 a b+5 b^2\right ) \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a+b}+\frac {b \left (5 a^2-10 a b+b^2\right ) \int \frac {1}{b \tanh ^2(c+d x)+a}d\tanh (c+d x)}{a+b}}{a+b}+\frac {4 b (a-b) \tanh (c+d x)}{(a+b) \left (a+b \tanh ^2(c+d x)\right )}\right )}{a+b}+\frac {b (7 a-5 b) \tanh (c+d x)}{(a+b) \left (a+b \tanh ^2(c+d x)\right )^2}}{2 (a+b)}}{4 (a+b)}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 (a+b) \left (1-\tanh ^2(c+d x)\right )^2 \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {\frac {(5 a-3 b) \tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {\frac {3 \left (\frac {\frac {\left (a^2-10 a b+5 b^2\right ) \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a+b}+\frac {\sqrt {b} \left (5 a^2-10 a b+b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)}}{a+b}+\frac {4 b (a-b) \tanh (c+d x)}{(a+b) \left (a+b \tanh ^2(c+d x)\right )}\right )}{a+b}+\frac {b (7 a-5 b) \tanh (c+d x)}{(a+b) \left (a+b \tanh ^2(c+d x)\right )^2}}{2 (a+b)}}{4 (a+b)}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 (a+b) \left (1-\tanh ^2(c+d x)\right )^2 \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {\frac {(5 a-3 b) \tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {\frac {3 \left (\frac {\frac {\sqrt {b} \left (5 a^2-10 a b+b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)}+\frac {\left (a^2-10 a b+5 b^2\right ) \text {arctanh}(\tanh (c+d x))}{a+b}}{a+b}+\frac {4 b (a-b) \tanh (c+d x)}{(a+b) \left (a+b \tanh ^2(c+d x)\right )}\right )}{a+b}+\frac {b (7 a-5 b) \tanh (c+d x)}{(a+b) \left (a+b \tanh ^2(c+d x)\right )^2}}{2 (a+b)}}{4 (a+b)}}{d}\) |
(Tanh[c + d*x]/(4*(a + b)*(1 - Tanh[c + d*x]^2)^2*(a + b*Tanh[c + d*x]^2)^ 2) - (((5*a - 3*b)*Tanh[c + d*x])/(2*(a + b)*(1 - Tanh[c + d*x]^2)*(a + b* Tanh[c + d*x]^2)^2) - (((7*a - 5*b)*b*Tanh[c + d*x])/((a + b)*(a + b*Tanh[ c + d*x]^2)^2) + (3*(((Sqrt[b]*(5*a^2 - 10*a*b + b^2)*ArcTan[(Sqrt[b]*Tanh [c + d*x])/Sqrt[a]])/(Sqrt[a]*(a + b)) + ((a^2 - 10*a*b + 5*b^2)*ArcTanh[T anh[c + d*x]])/(a + b))/(a + b) + (4*(a - b)*b*Tanh[c + d*x])/((a + b)*(a + b*Tanh[c + d*x]^2))))/(a + b))/(2*(a + b)))/(4*(a + b)))/d
3.1.41.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 )^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 )) Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a , b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_ )])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim p[c*(ff^(m + 1)/f) Subst[Int[x^m*((a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2)^(m/ 2 + 1)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x ] && IntegerQ[m/2]
Leaf count of result is larger than twice the leaf count of optimal. \(609\) vs. \(2(220)=440\).
Time = 193.92 (sec) , antiderivative size = 610, normalized size of antiderivative = 2.54
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{4 \left (a +b \right )^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {1}{2 \left (a +b \right )^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {3 a -9 b}{8 \left (a +b \right )^{4} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {11 b -a}{8 \left (a +b \right )^{4} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (3 a^{2}-30 a b +15 b^{2}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )^{5}}+\frac {1}{4 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {1}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {3 a -9 b}{8 \left (a +b \right )^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {-11 b +a}{8 \left (a +b \right )^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (-3 a^{2}+30 a b -15 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 \left (a +b \right )^{5}}-\frac {2 b \left (\frac {-\frac {3 a \left (3 a^{2}+2 a b -b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8}+\left (-\frac {27}{8} a^{3}-\frac {23}{4} a^{2} b +\frac {1}{8} a \,b^{2}+\frac {5}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {27}{8} a^{3}-\frac {23}{4} a^{2} b +\frac {1}{8} a \,b^{2}+\frac {5}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-\frac {9}{8} a^{3}-\frac {3}{4} a^{2} b +\frac {3}{8} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\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 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a \right )^{2}}+\frac {\left (15 a^{2}-30 a b +3 b^{2}\right ) a \left (\frac {\left (a +\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (-a +\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{8}\right )}{\left (a +b \right )^{5}}}{d}\) | \(610\) |
| default | \(\frac {-\frac {1}{4 \left (a +b \right )^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {1}{2 \left (a +b \right )^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {3 a -9 b}{8 \left (a +b \right )^{4} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {11 b -a}{8 \left (a +b \right )^{4} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (3 a^{2}-30 a b +15 b^{2}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )^{5}}+\frac {1}{4 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {1}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {3 a -9 b}{8 \left (a +b \right )^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {-11 b +a}{8 \left (a +b \right )^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (-3 a^{2}+30 a b -15 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 \left (a +b \right )^{5}}-\frac {2 b \left (\frac {-\frac {3 a \left (3 a^{2}+2 a b -b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8}+\left (-\frac {27}{8} a^{3}-\frac {23}{4} a^{2} b +\frac {1}{8} a \,b^{2}+\frac {5}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {27}{8} a^{3}-\frac {23}{4} a^{2} b +\frac {1}{8} a \,b^{2}+\frac {5}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-\frac {9}{8} a^{3}-\frac {3}{4} a^{2} b +\frac {3}{8} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\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 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a \right )^{2}}+\frac {\left (15 a^{2}-30 a b +3 b^{2}\right ) a \left (\frac {\left (a +\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (-a +\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{8}\right )}{\left (a +b \right )^{5}}}{d}\) | \(610\) |
| risch | \(\frac {3 x \,a^{2}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \left (a +b \right )^{2}}-\frac {15 x a b}{4 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \left (a +b \right )^{2}}+\frac {15 x \,b^{2}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \left (a +b \right )^{2}}+\frac {{\mathrm e}^{4 d x +4 c}}{64 \left (a +b \right )^{3} d}-\frac {{\mathrm e}^{2 d x +2 c} a}{8 \left (a^{2}+2 a b +b^{2}\right ) \left (a +b \right )^{2} d}+\frac {{\mathrm e}^{2 d x +2 c} b}{4 \left (a^{2}+2 a b +b^{2}\right ) \left (a +b \right )^{2} d}+\frac {{\mathrm e}^{-2 d x -2 c} a}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \left (a +b \right ) d}-\frac {{\mathrm e}^{-2 d x -2 c} b}{4 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \left (a +b \right ) d}-\frac {{\mathrm e}^{-4 d x -4 c}}{64 \left (a^{2}+2 a b +b^{2}\right ) \left (a +b \right ) d}-\frac {b \left (9 a^{3} {\mathrm e}^{6 d x +6 c}-9 a^{2} b \,{\mathrm e}^{6 d x +6 c}-13 a \,b^{2} {\mathrm e}^{6 d x +6 c}+5 \,{\mathrm e}^{6 d x +6 c} b^{3}+27 a^{3} {\mathrm e}^{4 d x +4 c}-33 a^{2} b \,{\mathrm e}^{4 d x +4 c}+37 a \,b^{2} {\mathrm e}^{4 d x +4 c}-15 \,{\mathrm e}^{4 d x +4 c} b^{3}+27 a^{3} {\mathrm e}^{2 d x +2 c}-11 a^{2} b \,{\mathrm e}^{2 d x +2 c}-23 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}+15 \,{\mathrm e}^{2 d x +2 c} b^{3}+9 a^{3}+13 a^{2} b -a \,b^{2}-5 b^{3}\right )}{4 \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )^{2} d \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) \left (a +b \right )}+\frac {15 a \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right )}{16 \left (a +b \right )^{5} d}-\frac {15 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right ) b}{8 \left (a +b \right )^{5} d}+\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right ) b^{2}}{16 a \left (a +b \right )^{5} d}-\frac {15 a \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{16 \left (a +b \right )^{5} d}+\frac {15 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right ) b}{8 \left (a +b \right )^{5} d}-\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right ) b^{2}}{16 a \left (a +b \right )^{5} d}\) | \(879\) |
1/d*(-1/4/(a+b)^3/(1+tanh(1/2*d*x+1/2*c))^4+1/2/(a+b)^3/(1+tanh(1/2*d*x+1/ 2*c))^3-1/8*(3*a-9*b)/(a+b)^4/(1+tanh(1/2*d*x+1/2*c))-1/8*(11*b-a)/(a+b)^4 /(1+tanh(1/2*d*x+1/2*c))^2+1/8/(a+b)^5*(3*a^2-30*a*b+15*b^2)*ln(1+tanh(1/2 *d*x+1/2*c))+1/4/(a+b)^3/(tanh(1/2*d*x+1/2*c)-1)^4+1/2/(a+b)^3/(tanh(1/2*d *x+1/2*c)-1)^3-1/8*(3*a-9*b)/(a+b)^4/(tanh(1/2*d*x+1/2*c)-1)-1/8*(-11*b+a) /(a+b)^4/(tanh(1/2*d*x+1/2*c)-1)^2+1/8/(a+b)^5*(-3*a^2+30*a*b-15*b^2)*ln(t anh(1/2*d*x+1/2*c)-1)-2*b/(a+b)^5*((-3/8*a*(3*a^2+2*a*b-b^2)*tanh(1/2*d*x+ 1/2*c)^7+(-27/8*a^3-23/4*a^2*b+1/8*a*b^2+5/2*b^3)*tanh(1/2*d*x+1/2*c)^5+(- 27/8*a^3-23/4*a^2*b+1/8*a*b^2+5/2*b^3)*tanh(1/2*d*x+1/2*c)^3+(-9/8*a^3-3/4 *a^2*b+3/8*a*b^2)*tanh(1/2*d*x+1/2*c))/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2 *d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2+1/8*(15*a^2-30*a*b+3*b^2)*a *(1/2*(a+((a+b)*b)^(1/2)+b)/a/((a+b)*b)^(1/2)/((2*((a+b)*b)^(1/2)+a+2*b)*a )^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*((a+b)*b)^(1/2)+a+2*b)*a)^(1/2))- 1/2*(-a+((a+b)*b)^(1/2)-b)/a/((a+b)*b)^(1/2)/((2*((a+b)*b)^(1/2)-a-2*b)*a) ^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*((a+b)*b)^(1/2)-a-2*b)*a)^(1/2))) ))
Leaf count of result is larger than twice the leaf count of optimal. 9248 vs. \(2 (220) = 440\).
Time = 0.53 (sec) , antiderivative size = 18818, normalized size of antiderivative = 78.41 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 3392 vs. \(2 (220) = 440\).
Time = 0.73 (sec) , antiderivative size = 3392, normalized size of antiderivative = 14.13 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
-3/8*(a*b - 3*b^2)*log((a + b)*e^(4*d*x + 4*c) + 2*(a - b)*e^(2*d*x + 2*c) + a + b)/((a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d) - 3/4*b*log((a + b)*e^(4*d*x + 4*c) + 2*(a - b)*e^(2*d*x + 2*c) + a + b)/((a ^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d) + 3/8*(a*b - 3*b^2)*log(2*(a - b)*e^(-2*d*x - 2*c) + (a + b)*e^(-4*d*x - 4*c) + a + b)/((a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d) + 3/4*b*log(2*(a - b)*e^(-2* d*x - 2*c) + (a + b)*e^(-4*d*x - 4*c) + a + b)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d) + 3/128*(5*a^4*b - 80*a^3*b^2 + 50*a^2*b^3 + 8*a*b^4 + b^5)*arctan(1/2*((a + b)*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))/((a^7 + 5*a ^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*sqrt(a*b)*d) + 3/32* (5*a^3*b - 15*a^2*b^2 - 5*a*b^3 - b^4)*arctan(1/2*((a + b)*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))/((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*sq rt(a*b)*d) - 3/128*(5*a^4*b - 80*a^3*b^2 + 50*a^2*b^3 + 8*a*b^4 + b^5)*arc tan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sqrt(a*b))/((a^7 + 5*a^6*b + 10 *a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*sqrt(a*b)*d) - 3/32*(5*a^3*b - 15*a^2*b^2 - 5*a*b^3 - b^4)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b )/sqrt(a*b))/((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*sqrt(a*b)* d) - 3/64*(15*a^2*b + 10*a*b^2 + 3*b^3)*arctan(1/2*((a + b)*e^(-2*d*x - 2* c) + a - b)/sqrt(a*b))/((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*sqrt(a*b)*d) - 1/64*(9*a^5*b - 65*a^4*b^2 - 134*a^3*b^3 - 34*a^2*b^4 + 29*a*b^5 + 3...
\[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int { \frac {\sinh \left (d x + c\right )^{4}}{{\left (b \tanh \left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^4}{{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \]