Integrand size = 23, antiderivative size = 192 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {3 \left (a^2-6 a b+b^2\right ) x}{8 (a+b)^4}+\frac {3 \sqrt {a} (a-b) \sqrt {b} \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 (a+b)^4 d}-\frac {(5 a-b) \cosh (c+d x) \sinh (c+d x)}{8 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 (a+b) d \left (a+b \tanh ^2(c+d x)\right )}+\frac {3 (3 a-b) b \tanh (c+d x)}{8 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )} \] Output:
3/8*(a^2-6*a*b+b^2)*x/(a+b)^4+3/2*a^(1/2)*(a-b)*b^(1/2)*arctan(b^(1/2)*tan h(d*x+c)/a^(1/2))/(a+b)^4/d-1/8*(5*a-b)*cosh(d*x+c)*sinh(d*x+c)/(a+b)^2/d/ (a+b*tanh(d*x+c)^2)+1/4*cosh(d*x+c)^3*sinh(d*x+c)/(a+b)/d/(a+b*tanh(d*x+c) ^2)+3/8*(3*a-b)*b*tanh(d*x+c)/(a+b)^3/d/(a+b*tanh(d*x+c)^2)
Time = 1.03 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.69 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {12 \left (a^2-6 a b+b^2\right ) (c+d x)+48 \sqrt {a} (a-b) \sqrt {b} \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )-8 (a-b) (a+b) \sinh (2 (c+d x))+\frac {16 a b (a+b) \sinh (2 (c+d x))}{a-b+(a+b) \cosh (2 (c+d x))}+(a+b)^2 \sinh (4 (c+d x))}{32 (a+b)^4 d} \] Input:
Integrate[Sinh[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^2,x]
Output:
(12*(a^2 - 6*a*b + b^2)*(c + d*x) + 48*Sqrt[a]*(a - b)*Sqrt[b]*ArcTan[(Sqr t[b]*Tanh[c + d*x])/Sqrt[a]] - 8*(a - b)*(a + b)*Sinh[2*(c + d*x)] + (16*a *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)^4*d)
Time = 0.45 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.16, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3042, 4146, 372, 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.
| \(\displaystyle \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (i c+i d x)^4}{\left (a-b \tan (i c+i d x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 4146 |
| \(\displaystyle \frac {\int \frac {\tanh ^4(c+d x)}{\left (1-\tanh ^2(c+d x)\right )^3 \left (b \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 (a+b) \left (1-\tanh ^2(c+d x)\right )^2 \left (a+b \tanh ^2(c+d x)\right )}-\frac {\int \frac {(4 a-b) \tanh ^2(c+d x)+a}{\left (1-\tanh ^2(c+d x)\right )^2 \left (b \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{4 (a+b)}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 (a+b) \left (1-\tanh ^2(c+d x)\right )^2 \left (a+b \tanh ^2(c+d x)\right )}-\frac {\frac {\int -\frac {3 \left (a (a-b)-(5 a-b) b \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{2 (a+b)}+\frac {(5 a-b) \tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )}}{4 (a+b)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 (a+b) \left (1-\tanh ^2(c+d x)\right )^2 \left (a+b \tanh ^2(c+d x)\right )}-\frac {\frac {(5 a-b) \tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )}-\frac {3 \int \frac {a (a-b)-(5 a-b) b \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{2 (a+b)}}{4 (a+b)}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 (a+b) \left (1-\tanh ^2(c+d x)\right )^2 \left (a+b \tanh ^2(c+d x)\right )}-\frac {\frac {(5 a-b) \tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )}-\frac {3 \left (\frac {b (3 a-b) \tanh (c+d x)}{(a+b) \left (a+b \tanh ^2(c+d x)\right )}-\frac {\int -\frac {2 a \left (a (a-3 b)-(3 a-b) b \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{2 a (a+b)}\right )}{2 (a+b)}}{4 (a+b)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 (a+b) \left (1-\tanh ^2(c+d x)\right )^2 \left (a+b \tanh ^2(c+d x)\right )}-\frac {\frac {(5 a-b) \tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )}-\frac {3 \left (\frac {\int \frac {a (a-3 b)-(3 a-b) b \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{a+b}+\frac {b (3 a-b) \tanh (c+d x)}{(a+b) \left (a+b \tanh ^2(c+d x)\right )}\right )}{2 (a+b)}}{4 (a+b)}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 (a+b) \left (1-\tanh ^2(c+d x)\right )^2 \left (a+b \tanh ^2(c+d x)\right )}-\frac {\frac {(5 a-b) \tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )}-\frac {3 \left (\frac {\frac {\left (a^2-6 a b+b^2\right ) \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a+b}+\frac {4 a b (a-b) \int \frac {1}{b \tanh ^2(c+d x)+a}d\tanh (c+d x)}{a+b}}{a+b}+\frac {b (3 a-b) \tanh (c+d x)}{(a+b) \left (a+b \tanh ^2(c+d x)\right )}\right )}{2 (a+b)}}{4 (a+b)}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 (a+b) \left (1-\tanh ^2(c+d x)\right )^2 \left (a+b \tanh ^2(c+d x)\right )}-\frac {\frac {(5 a-b) \tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )}-\frac {3 \left (\frac {\frac {\left (a^2-6 a b+b^2\right ) \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a+b}+\frac {4 \sqrt {a} \sqrt {b} (a-b) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{a+b}}{a+b}+\frac {b (3 a-b) \tanh (c+d x)}{(a+b) \left (a+b \tanh ^2(c+d x)\right )}\right )}{2 (a+b)}}{4 (a+b)}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 (a+b) \left (1-\tanh ^2(c+d x)\right )^2 \left (a+b \tanh ^2(c+d x)\right )}-\frac {\frac {(5 a-b) \tanh (c+d x)}{2 (a+b) \left (1-\tanh ^2(c+d x)\right ) \left (a+b \tanh ^2(c+d x)\right )}-\frac {3 \left (\frac {\frac {\left (a^2-6 a b+b^2\right ) \text {arctanh}(\tanh (c+d x))}{a+b}+\frac {4 \sqrt {a} \sqrt {b} (a-b) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{a+b}}{a+b}+\frac {b (3 a-b) \tanh (c+d x)}{(a+b) \left (a+b \tanh ^2(c+d x)\right )}\right )}{2 (a+b)}}{4 (a+b)}}{d}\) |
Input:
Int[Sinh[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^2,x]
Output:
(Tanh[c + d*x]/(4*(a + b)*(1 - Tanh[c + d*x]^2)^2*(a + b*Tanh[c + d*x]^2)) - (((5*a - b)*Tanh[c + d*x])/(2*(a + b)*(1 - Tanh[c + d*x]^2)*(a + b*Tanh [c + d*x]^2)) - (3*(((4*Sqrt[a]*(a - b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d *x])/Sqrt[a]])/(a + b) + ((a^2 - 6*a*b + b^2)*ArcTanh[Tanh[c + d*x]])/(a + b))/(a + b) + ((3*a - b)*b*Tanh[c + d*x])/((a + b)*(a + b*Tanh[c + d*x]^2 ))))/(2*(a + b)))/(4*(a + b)))/d
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. \(511\) vs. \(2(174)=348\).
Time = 71.35 (sec) , antiderivative size = 512, normalized size of antiderivative = 2.67
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 a b \left (\frac {\left (-\frac {a}{2}-\frac {b}{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-\frac {a}{2}-\frac {b}{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\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}+\frac {\left (3 a -3 b \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 )}{2}\right )}{\left (a +b \right )^{4}}+\frac {1}{4 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {a -7 b}{8 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {3 a -5 b}{8 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-3 a^{2}+18 a b -3 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 \left (a +b \right )^{4}}-\frac {1}{4 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-a +7 b}{8 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {3 a -5 b}{8 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (3 a^{2}-18 a b +3 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 \left (a +b \right )^{4}}}{d}\) | \(512\) |
| default | \(\frac {-\frac {2 a b \left (\frac {\left (-\frac {a}{2}-\frac {b}{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-\frac {a}{2}-\frac {b}{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\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}+\frac {\left (3 a -3 b \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 )}{2}\right )}{\left (a +b \right )^{4}}+\frac {1}{4 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {a -7 b}{8 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {3 a -5 b}{8 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-3 a^{2}+18 a b -3 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 \left (a +b \right )^{4}}-\frac {1}{4 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-a +7 b}{8 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {3 a -5 b}{8 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (3 a^{2}-18 a b +3 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 \left (a +b \right )^{4}}}{d}\) | \(512\) |
| risch | \(\frac {3 x \,a^{2}}{8 \left (a +b \right )^{2} \left (a^{2}+2 a b +b^{2}\right )}-\frac {9 x a b}{4 \left (a +b \right )^{2} \left (a^{2}+2 a b +b^{2}\right )}+\frac {3 x \,b^{2}}{8 \left (a +b \right )^{2} \left (a^{2}+2 a b +b^{2}\right )}+\frac {{\mathrm e}^{4 d x +4 c}}{64 \left (a^{2}+2 a b +b^{2}\right ) d}-\frac {{\mathrm e}^{2 d x +2 c} a}{8 \left (a +b \right ) \left (a^{2}+2 a b +b^{2}\right ) d}+\frac {{\mathrm e}^{2 d x +2 c} b}{8 \left (a +b \right ) \left (a^{2}+2 a b +b^{2}\right ) d}+\frac {{\mathrm e}^{-2 d x -2 c} a}{8 \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right ) d}-\frac {{\mathrm e}^{-2 d x -2 c} b}{8 \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right ) d}-\frac {{\mathrm e}^{-4 d x -4 c}}{64 \left (a^{2}+2 a b +b^{2}\right ) d}-\frac {a b \left ({\mathrm e}^{2 d x +2 c} a -{\mathrm e}^{2 d x +2 c} b +a +b \right )}{d \left (a +b \right )^{2} \left (a^{2}+2 a b +b^{2}\right ) \left ({\mathrm e}^{4 d x +4 c} a +b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 \,{\mathrm e}^{2 d x +2 c} b +a +b \right )}+\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right ) a}{4 \left (a +b \right )^{4} d}-\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right ) b}{4 \left (a +b \right )^{4} d}-\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right ) a}{4 \left (a +b \right )^{4} d}+\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right ) b}{4 \left (a +b \right )^{4} d}\) | \(557\) |
Input:
int(sinh(d*x+c)^4/(a+tanh(d*x+c)^2*b)^2,x,method=_RETURNVERBOSE)
Output:
1/d*(-2*a*b/(a+b)^4*(((-1/2*a-1/2*b)*tanh(1/2*d*x+1/2*c)^3+(-1/2*a-1/2*b)* 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* b*tanh(1/2*d*x+1/2*c)^2+a)+1/2*(3*a-3*b)*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))))+1/4/(a+b)^2/(tanh(1/2*d*x+1/2* c)-1)^4+1/2/(a+b)^2/(tanh(1/2*d*x+1/2*c)-1)^3-1/8*(a-7*b)/(a+b)^3/(tanh(1/ 2*d*x+1/2*c)-1)^2-1/8*(3*a-5*b)/(a+b)^3/(tanh(1/2*d*x+1/2*c)-1)+1/8/(a+b)^ 4*(-3*a^2+18*a*b-3*b^2)*ln(tanh(1/2*d*x+1/2*c)-1)-1/4/(a+b)^2/(tanh(1/2*d* x+1/2*c)+1)^4+1/2/(a+b)^2/(tanh(1/2*d*x+1/2*c)+1)^3-1/8*(-a+7*b)/(a+b)^3/( tanh(1/2*d*x+1/2*c)+1)^2-1/8*(3*a-5*b)/(a+b)^3/(tanh(1/2*d*x+1/2*c)+1)+1/8 /(a+b)^4*(3*a^2-18*a*b+3*b^2)*ln(tanh(1/2*d*x+1/2*c)+1))
Leaf count of result is larger than twice the leaf count of optimal. 3531 vs. \(2 (174) = 348\).
Time = 0.22 (sec) , antiderivative size = 7366, normalized size of antiderivative = 38.36 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(sinh(d*x+c)^4/(a+b*tanh(d*x+c)^2)^2,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\text {Timed out} \] Input:
integrate(sinh(d*x+c)**4/(a+b*tanh(d*x+c)**2)**2,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 1690 vs. \(2 (174) = 348\).
Time = 0.34 (sec) , antiderivative size = 1690, normalized size of antiderivative = 8.80 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(sinh(d*x+c)^4/(a+b*tanh(d*x+c)^2)^2,x, algorithm="maxima")
Output:
-1/4*(a*b - 2*b^2)*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) - 1/2*b*log((a + b)*e^(4*d*x + 4*c) + 2*(a - b)*e^(2*d*x + 2*c) + a + b)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) + 1/4*(a*b - 2*b^2)*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) + 1/2*b*log(2*(a - b)*e^(-2*d*x - 2*c) + (a + b)*e^(-4*d*x - 4*c) + a + b)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) + 1/32*(3*a^3*b - 33*a^2*b^2 + 13*a*b^3 + b^4)*arctan(1/2*((a + b)*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))/(( a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*sqrt(a*b)*d) + 1/8*(3*a^2*b - 6*a*b^2 - b^3)*arctan(1/2*((a + b)*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))/ ((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*sqrt(a*b)*d) - 1/32*(3*a^3*b - 33*a^2 *b^2 + 13*a*b^3 + b^4)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sqrt( a*b))/((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*sqrt(a*b)*d) - 1/8* (3*a^2*b - 6*a*b^2 - b^3)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sq rt(a*b))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*sqrt(a*b)*d) - 3/16*(3*a*b + b^2)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sqrt(a*b))/((a^3 + 2*a ^2*b + a*b^2)*sqrt(a*b)*d) - 1/16*(a^3*b - 5*a^2*b^2 - 5*a*b^3 + b^4 + (a^ 3*b - 15*a^2*b^2 + 15*a*b^3 - b^4)*e^(2*d*x + 2*c))/((a^6 + 5*a^5*b + 10*a ^4*b^2 + 10*a^3*b^3 + 5*a^2*b^4 + a*b^5 + (a^6 + 5*a^5*b + 10*a^4*b^2 + 10 *a^3*b^3 + 5*a^2*b^4 + a*b^5)*e^(4*d*x + 4*c) + 2*(a^6 + 3*a^5*b + 2*a^...
Leaf count of result is larger than twice the leaf count of optimal. 476 vs. \(2 (174) = 348\).
Time = 1.29 (sec) , antiderivative size = 476, normalized size of antiderivative = 2.48 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {\frac {24 \, {\left (a^{2} - 6 \, a b + b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} - \frac {{\left (18 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 108 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 18 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 8 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a^{2} + 2 \, a b + b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {96 \, {\left (a^{2} b - a b^{2}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \sqrt {a b}} + \frac {a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a b e^{\left (4 \, d x + 4 \, c\right )} + b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 8 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )}}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} - \frac {64 \, {\left (a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a^{2} b + a b^{2}\right )}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}}}{64 \, d} \] Input:
integrate(sinh(d*x+c)^4/(a+b*tanh(d*x+c)^2)^2,x, algorithm="giac")
Output:
1/64*(24*(a^2 - 6*a*b + b^2)*(d*x + c)/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^ 3 + b^4) - (18*a^2*e^(4*d*x + 4*c) - 108*a*b*e^(4*d*x + 4*c) + 18*b^2*e^(4 *d*x + 4*c) - 8*a^2*e^(2*d*x + 2*c) + 8*b^2*e^(2*d*x + 2*c) + a^2 + 2*a*b + b^2)*e^(-4*d*x - 4*c)/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) + 96*( a^2*b - a*b^2)*arctan(1/2*(a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) + a - b)/ sqrt(a*b))/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*sqrt(a*b)) + (a^2* e^(4*d*x + 4*c) + 2*a*b*e^(4*d*x + 4*c) + b^2*e^(4*d*x + 4*c) - 8*a^2*e^(2 *d*x + 2*c) + 8*b^2*e^(2*d*x + 2*c))/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) - 64*(a^2*b*e^(2*d*x + 2*c) - a*b^2*e^(2*d*x + 2*c) + a^2*b + a*b^2 )/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*(a*e^(4*d*x + 4*c) + b*e^(4 *d*x + 4*c) + 2*a*e^(2*d*x + 2*c) - 2*b*e^(2*d*x + 2*c) + a + b)))/d
Timed out. \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^4}{{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \] Input:
int(sinh(c + d*x)^4/(a + b*tanh(c + d*x)^2)^2,x)
Output:
int(sinh(c + d*x)^4/(a + b*tanh(c + d*x)^2)^2, x)
Time = 0.31 (sec) , antiderivative size = 1361, normalized size of antiderivative = 7.09 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:
int(sinh(d*x+c)^4/(a+b*tanh(d*x+c)^2)^2,x)
Output:
(96*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt (b))/sqrt(a))*a**2 - 96*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x )*sqrt(a + b) - sqrt(b))/sqrt(a))*b**2 + 192*e**(6*c + 6*d*x)*sqrt(b)*sqrt (a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**2 - 384*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a) )*a*b + 192*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b ) - sqrt(b))/sqrt(a))*b**2 + 96*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a)*atan((e** (c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**2 - 96*e**(4*c + 4*d*x)*sqrt( b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*b**2 - 96*e* *(8*c + 8*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) + sqrt(b))/s qrt(a))*a**2 + 96*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt (a + b) + sqrt(b))/sqrt(a))*b**2 - 192*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a)*at an((e**(c + d*x)*sqrt(a + b) + sqrt(b))/sqrt(a))*a**2 + 384*e**(6*c + 6*d* x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) + sqrt(b))/sqrt(a))*a*b - 192*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) + sq rt(b))/sqrt(a))*b**2 - 96*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d *x)*sqrt(a + b) + sqrt(b))/sqrt(a))*a**2 + 96*e**(4*c + 4*d*x)*sqrt(b)*sqr t(a)*atan((e**(c + d*x)*sqrt(a + b) + sqrt(b))/sqrt(a))*b**2 + e**(12*c + 12*d*x)*a**3 + 3*e**(12*c + 12*d*x)*a**2*b + 3*e**(12*c + 12*d*x)*a*b**2 + e**(12*c + 12*d*x)*b**3 - 6*e**(10*c + 10*d*x)*a**3 - 6*e**(10*c + 10*...