Integrand size = 23, antiderivative size = 151 \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {5 \sqrt {b} (3 a+7 b) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{9/2} d}+\frac {(a+3 b) \coth (c+d x)}{a^4 d}-\frac {\coth ^3(c+d x)}{3 a^3 d}+\frac {b (a+b) \tanh (c+d x)}{4 a^3 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {b (7 a+11 b) \tanh (c+d x)}{8 a^4 d \left (a+b \tanh ^2(c+d x)\right )} \] Output:
5/8*b^(1/2)*(3*a+7*b)*arctan(b^(1/2)*tanh(d*x+c)/a^(1/2))/a^(9/2)/d+(a+3*b )*coth(d*x+c)/a^4/d-1/3*coth(d*x+c)^3/a^3/d+1/4*b*(a+b)*tanh(d*x+c)/a^3/d/ (a+b*tanh(d*x+c)^2)^2+1/8*b*(7*a+11*b)*tanh(d*x+c)/a^4/d/(a+b*tanh(d*x+c)^ 2)
Time = 1.28 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.99 \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {15 \sqrt {b} (3 a+7 b) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )-8 \sqrt {a} \coth (c+d x) \left (-2 a-9 b+a \text {csch}^2(c+d x)\right )+\frac {3 \sqrt {a} b \left (9 a^2+6 a b-11 b^2+\left (9 a^2+20 a b+11 b^2\right ) \cosh (2 (c+d x))\right ) \sinh (2 (c+d x))}{(a-b+(a+b) \cosh (2 (c+d x)))^2}}{24 a^{9/2} d} \] Input:
Integrate[Csch[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^3,x]
Output:
(15*Sqrt[b]*(3*a + 7*b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]] - 8*Sqrt[a ]*Coth[c + d*x]*(-2*a - 9*b + a*Csch[c + d*x]^2) + (3*Sqrt[a]*b*(9*a^2 + 6 *a*b - 11*b^2 + (9*a^2 + 20*a*b + 11*b^2)*Cosh[2*(c + d*x)])*Sinh[2*(c + d *x)])/(a - b + (a + b)*Cosh[2*(c + d*x)])^2)/(24*a^(9/2)*d)
Time = 0.79 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 4146, 361, 25, 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.
| \(\displaystyle \int \frac {\text {csch}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (i c+i d x)^4 \left (a-b \tan (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 4146 |
| \(\displaystyle \frac {\int \frac {\coth ^4(c+d x) \left (1-\tanh ^2(c+d x)\right )}{\left (b \tanh ^2(c+d x)+a\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 361 |
| \(\displaystyle \frac {\frac {b (a+b) \tanh (c+d x)}{4 a^3 \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {1}{4} b \int -\frac {\coth ^4(c+d x) \left (\frac {3 (a+b) \tanh ^4(c+d x)}{a^3}-\frac {4 (a+b) \tanh ^2(c+d x)}{a^2 b}+\frac {4}{a b}\right )}{\left (b \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{4} b \int \frac {\coth ^4(c+d x) \left (\frac {3 (a+b) \tanh ^4(c+d x)}{a^3}-\frac {4 (a+b) \tanh ^2(c+d x)}{a^2 b}+\frac {4}{a b}\right )}{\left (b \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)+\frac {b (a+b) \tanh (c+d x)}{4 a^3 \left (a+b \tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 1582 |
| \(\displaystyle \frac {\frac {1}{4} b \left (\frac {\int \frac {\coth ^4(c+d x) \left (\frac {b^2 (7 a+11 b) \tanh ^4(c+d x)}{a}-8 b (a+2 b) \tanh ^2(c+d x)+8 a b\right )}{b \tanh ^2(c+d x)+a}d\tanh (c+d x)}{2 a^3 b^2}+\frac {(7 a+11 b) \tanh (c+d x)}{2 a^4 \left (a+b \tanh ^2(c+d x)\right )}\right )+\frac {b (a+b) \tanh (c+d x)}{4 a^3 \left (a+b \tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 1584 |
| \(\displaystyle \frac {\frac {1}{4} b \left (\frac {\int \left (8 b \coth ^4(c+d x)-\frac {8 b (a+3 b) \coth ^2(c+d x)}{a}+\frac {5 b^2 (3 a+7 b)}{a \left (b \tanh ^2(c+d x)+a\right )}\right )d\tanh (c+d x)}{2 a^3 b^2}+\frac {(7 a+11 b) \tanh (c+d x)}{2 a^4 \left (a+b \tanh ^2(c+d x)\right )}\right )+\frac {b (a+b) \tanh (c+d x)}{4 a^3 \left (a+b \tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {b (a+b) \tanh (c+d x)}{4 a^3 \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {1}{4} b \left (\frac {(7 a+11 b) \tanh (c+d x)}{2 a^4 \left (a+b \tanh ^2(c+d x)\right )}+\frac {\frac {5 b^{3/2} (3 a+7 b) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{3/2}}+\frac {8 b (a+3 b) \coth (c+d x)}{a}-\frac {8}{3} b \coth ^3(c+d x)}{2 a^3 b^2}\right )}{d}\) |
Input:
Int[Csch[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^3,x]
Output:
((b*(a + b)*Tanh[c + d*x])/(4*a^3*(a + b*Tanh[c + d*x]^2)^2) + (b*(((5*b^( 3/2)*(3*a + 7*b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/a^(3/2) + (8*b*( a + 3*b)*Coth[c + d*x])/a - (8*b*Coth[c + d*x]^3)/3)/(2*a^3*b^2) + ((7*a + 11*b)*Tanh[c + d*x])/(2*a^4*(a + b*Tanh[c + d*x]^2))))/4)/d
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_.)*((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. \(396\) vs. \(2(135)=270\).
Time = 23.36 (sec) , antiderivative size = 397, normalized size of antiderivative = 2.63
| method | result | size |
| derivativedivides | \(\frac {-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a}{3}-3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a -12 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{4}}-\frac {2 b \left (\frac {-\frac {a \left (9 a +13 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8}+\left (-\frac {27}{8} a^{2}-\frac {67}{8} a b -\frac {11}{2} b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {27}{8} a^{2}-\frac {67}{8} a b -\frac {11}{2} b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-\frac {9}{8} a^{2}-\frac {13}{8} a b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a \right )^{2}}+\frac {\left (15 a +35 b \right ) a \left (\frac {\left (a +\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (-a +\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{8}\right )}{a^{4}}-\frac {1}{24 a^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-3 a -12 b}{8 a^{4} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(397\) |
| default | \(\frac {-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a}{3}-3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a -12 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{4}}-\frac {2 b \left (\frac {-\frac {a \left (9 a +13 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8}+\left (-\frac {27}{8} a^{2}-\frac {67}{8} a b -\frac {11}{2} b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {27}{8} a^{2}-\frac {67}{8} a b -\frac {11}{2} b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-\frac {9}{8} a^{2}-\frac {13}{8} a b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a \right )^{2}}+\frac {\left (15 a +35 b \right ) a \left (\frac {\left (a +\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (-a +\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{8}\right )}{a^{4}}-\frac {1}{24 a^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-3 a -12 b}{8 a^{4} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(397\) |
| risch | \(-\frac {-105 b^{4}+270 a^{2} b^{2} {\mathrm e}^{10 d x +10 c}-325 a \,b^{3}-351 a^{2} b^{2}-260 a^{2} b^{2} {\mathrm e}^{6 d x +6 c}-16 a^{4}-147 a^{3} b -64 \,{\mathrm e}^{2 d x +2 c} a^{3} b +502 \,{\mathrm e}^{2 d x +2 c} a^{2} b^{2}+19 \,{\mathrm e}^{4 d x +4 c} a^{2} b^{2}+313 \,{\mathrm e}^{4 d x +4 c} a^{3} b +15 \,{\mathrm e}^{8 d x +8 c} a^{2} b^{2}+135 \,{\mathrm e}^{8 d x +8 c} a^{3} b +224 \,{\mathrm e}^{6 d x +6 c} a^{4}+176 \,{\mathrm e}^{8 d x +8 c} a^{4}+96 \,{\mathrm e}^{4 d x +4 c} a^{4}+320 b \,a^{3} {\mathrm e}^{6 d x +6 c}+630 \,{\mathrm e}^{10 d x +10 c} b^{4}-105 \,{\mathrm e}^{12 d x +12 c} b^{4}+630 \,{\mathrm e}^{2 d x +2 c} b^{4}-1575 \,{\mathrm e}^{4 d x +4 c} b^{4}+2100 \,{\mathrm e}^{6 d x +6 c} b^{4}-1575 \,{\mathrm e}^{8 d x +8 c} b^{4}+1600 \,{\mathrm e}^{6 d x +6 c} a \,b^{3}-1375 \,{\mathrm e}^{8 d x +8 c} a \,b^{3}+900 \,{\mathrm e}^{10 d x +10 c} a \,b^{3}-255 \,{\mathrm e}^{12 d x +12 c} a \,b^{3}-1725 \,{\mathrm e}^{4 d x +4 c} a \,b^{3}-195 \,{\mathrm e}^{12 d x +12 c} a^{2} b^{2}+1180 \,{\mathrm e}^{2 d x +2 c} a \,b^{3}-45 \,{\mathrm e}^{12 d x +12 c} a^{3} b -16 \,{\mathrm e}^{2 d x +2 c} a^{4}+48 \,{\mathrm e}^{10 d x +10 c} a^{4}}{12 \left ({\mathrm e}^{4 d x +4 c} a +b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 \,{\mathrm e}^{2 d x +2 c} b +a +b \right )^{2} \left (a +b \right ) \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3} d \,a^{4}}+\frac {15 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right )}{16 a^{4} d}+\frac {35 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right ) b}{16 a^{5} d}-\frac {15 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{16 a^{4} d}-\frac {35 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right ) b}{16 a^{5} d}\) | \(711\) |
Input:
int(csch(d*x+c)^4/(a+tanh(d*x+c)^2*b)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(-1/8/a^4*(1/3*tanh(1/2*d*x+1/2*c)^3*a-3*tanh(1/2*d*x+1/2*c)*a-12*b*ta nh(1/2*d*x+1/2*c))-2*b/a^4*((-1/8*a*(9*a+13*b)*tanh(1/2*d*x+1/2*c)^7+(-27/ 8*a^2-67/8*a*b-11/2*b^2)*tanh(1/2*d*x+1/2*c)^5+(-27/8*a^2-67/8*a*b-11/2*b^ 2)*tanh(1/2*d*x+1/2*c)^3+(-9/8*a^2-13/8*a*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)^2+ 1/8*(15*a+35*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))))-1/24/a^3/tanh(1/2*d*x+1/2*c)^3-1/8/a^4*(-3*a-12*b)/tanh (1/2*d*x+1/2*c))
Leaf count of result is larger than twice the leaf count of optimal. 7006 vs. \(2 (135) = 270\).
Time = 0.28 (sec) , antiderivative size = 14334, normalized size of antiderivative = 94.93 \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)^4/(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int \frac {\operatorname {csch}^{4}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3}}\, dx \] Input:
integrate(csch(d*x+c)**4/(a+b*tanh(d*x+c)**2)**3,x)
Output:
Integral(csch(c + d*x)**4/(a + b*tanh(c + d*x)**2)**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 615 vs. \(2 (135) = 270\).
Time = 0.36 (sec) , antiderivative size = 615, normalized size of antiderivative = 4.07 \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {16 \, a^{4} + 147 \, a^{3} b + 351 \, a^{2} b^{2} + 325 \, a b^{3} + 105 \, b^{4} + 2 \, {\left (8 \, a^{4} + 32 \, a^{3} b - 251 \, a^{2} b^{2} - 590 \, a b^{3} - 315 \, b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - {\left (96 \, a^{4} + 313 \, a^{3} b + 19 \, a^{2} b^{2} - 1725 \, a b^{3} - 1575 \, b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - 4 \, {\left (56 \, a^{4} + 80 \, a^{3} b - 65 \, a^{2} b^{2} + 400 \, a b^{3} + 525 \, b^{4}\right )} e^{\left (-6 \, d x - 6 \, c\right )} - {\left (176 \, a^{4} + 135 \, a^{3} b + 15 \, a^{2} b^{2} - 1375 \, a b^{3} - 1575 \, b^{4}\right )} e^{\left (-8 \, d x - 8 \, c\right )} - 6 \, {\left (8 \, a^{4} + 45 \, a^{2} b^{2} + 150 \, a b^{3} + 105 \, b^{4}\right )} e^{\left (-10 \, d x - 10 \, c\right )} + 15 \, {\left (3 \, a^{3} b + 13 \, a^{2} b^{2} + 17 \, a b^{3} + 7 \, b^{4}\right )} e^{\left (-12 \, d x - 12 \, c\right )}}{12 \, {\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3} + {\left (a^{7} - 5 \, a^{6} b - 13 \, a^{5} b^{2} - 7 \, a^{4} b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - {\left (3 \, a^{7} + a^{6} b - 23 \, a^{5} b^{2} - 21 \, a^{4} b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - {\left (3 \, a^{7} - 7 \, a^{6} b + 25 \, a^{5} b^{2} + 35 \, a^{4} b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (3 \, a^{7} - 7 \, a^{6} b + 25 \, a^{5} b^{2} + 35 \, a^{4} b^{3}\right )} e^{\left (-8 \, d x - 8 \, c\right )} + {\left (3 \, a^{7} + a^{6} b - 23 \, a^{5} b^{2} - 21 \, a^{4} b^{3}\right )} e^{\left (-10 \, d x - 10 \, c\right )} - {\left (a^{7} - 5 \, a^{6} b - 13 \, a^{5} b^{2} - 7 \, a^{4} b^{3}\right )} e^{\left (-12 \, d x - 12 \, c\right )} - {\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right )} e^{\left (-14 \, d x - 14 \, c\right )}\right )} d} - \frac {5 \, {\left (3 \, a b + 7 \, b^{2}\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{4} d} \] Input:
integrate(csch(d*x+c)^4/(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")
Output:
1/12*(16*a^4 + 147*a^3*b + 351*a^2*b^2 + 325*a*b^3 + 105*b^4 + 2*(8*a^4 + 32*a^3*b - 251*a^2*b^2 - 590*a*b^3 - 315*b^4)*e^(-2*d*x - 2*c) - (96*a^4 + 313*a^3*b + 19*a^2*b^2 - 1725*a*b^3 - 1575*b^4)*e^(-4*d*x - 4*c) - 4*(56* a^4 + 80*a^3*b - 65*a^2*b^2 + 400*a*b^3 + 525*b^4)*e^(-6*d*x - 6*c) - (176 *a^4 + 135*a^3*b + 15*a^2*b^2 - 1375*a*b^3 - 1575*b^4)*e^(-8*d*x - 8*c) - 6*(8*a^4 + 45*a^2*b^2 + 150*a*b^3 + 105*b^4)*e^(-10*d*x - 10*c) + 15*(3*a^ 3*b + 13*a^2*b^2 + 17*a*b^3 + 7*b^4)*e^(-12*d*x - 12*c))/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3 + (a^7 - 5*a^6*b - 13*a^5*b^2 - 7*a^4*b^3)*e^(-2*d*x - 2*c) - (3*a^7 + a^6*b - 23*a^5*b^2 - 21*a^4*b^3)*e^(-4*d*x - 4*c) - (3*a ^7 - 7*a^6*b + 25*a^5*b^2 + 35*a^4*b^3)*e^(-6*d*x - 6*c) + (3*a^7 - 7*a^6* b + 25*a^5*b^2 + 35*a^4*b^3)*e^(-8*d*x - 8*c) + (3*a^7 + a^6*b - 23*a^5*b^ 2 - 21*a^4*b^3)*e^(-10*d*x - 10*c) - (a^7 - 5*a^6*b - 13*a^5*b^2 - 7*a^4*b ^3)*e^(-12*d*x - 12*c) - (a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*e^(-14*d*x - 14*c))*d) - 5/8*(3*a*b + 7*b^2)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sqrt(a*b))/(sqrt(a*b)*a^4*d)
Leaf count of result is larger than twice the leaf count of optimal. 395 vs. \(2 (135) = 270\).
Time = 0.61 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.62 \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=-\frac {\frac {6 \, {\left (9 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} + 7 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 13 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 11 \, b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 27 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} + 15 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 5 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 33 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 27 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} + 37 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 23 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 33 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a^{3} b + 29 \, a^{2} b^{2} + 31 \, a b^{3} + 11 \, b^{4}\right )}}{{\left (a^{5} + a^{4} b\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}^{2}} - \frac {15 \, {\left (3 \, a b + 7 \, b^{2}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{\sqrt {a b} a^{4}} - \frac {16 \, {\left (9 \, b e^{\left (4 \, d x + 4 \, c\right )} - 6 \, a e^{\left (2 \, d x + 2 \, c\right )} - 18 \, b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a + 9 \, b\right )}}{a^{4} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{24 \, d} \] Input:
integrate(csch(d*x+c)^4/(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")
Output:
-1/24*(6*(9*a^3*b*e^(6*d*x + 6*c) + 7*a^2*b^2*e^(6*d*x + 6*c) - 13*a*b^3*e ^(6*d*x + 6*c) - 11*b^4*e^(6*d*x + 6*c) + 27*a^3*b*e^(4*d*x + 4*c) + 15*a^ 2*b^2*e^(4*d*x + 4*c) + 5*a*b^3*e^(4*d*x + 4*c) + 33*b^4*e^(4*d*x + 4*c) + 27*a^3*b*e^(2*d*x + 2*c) + 37*a^2*b^2*e^(2*d*x + 2*c) - 23*a*b^3*e^(2*d*x + 2*c) - 33*b^4*e^(2*d*x + 2*c) + 9*a^3*b + 29*a^2*b^2 + 31*a*b^3 + 11*b^ 4)/((a^5 + a^4*b)*(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)^2) - 15*(3*a*b + 7*b^2)*arctan(1/2*(a* e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))/(sqrt(a*b)*a^4) - 16*(9*b*e^(4*d*x + 4*c) - 6*a*e^(2*d*x + 2*c) - 18*b*e^(2*d*x + 2*c) + 2*a + 9*b)/(a^4*(e^(2*d*x + 2*c) - 1)^3))/d
Timed out. \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int \frac {1}{{\mathrm {sinh}\left (c+d\,x\right )}^4\,{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \] Input:
int(1/(sinh(c + d*x)^4*(a + b*tanh(c + d*x)^2)^3),x)
Output:
int(1/(sinh(c + d*x)^4*(a + b*tanh(c + d*x)^2)^3), x)
Time = 0.58 (sec) , antiderivative size = 5176, normalized size of antiderivative = 34.28 \[ \int \frac {\text {csch}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:
int(csch(d*x+c)^4/(a+b*tanh(d*x+c)^2)^3,x)
Output:
(45*e**(14*c + 14*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sq rt(b))/sqrt(a))*a**5 - 75*e**(14*c + 14*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**4*b - 1230*e**(14*c + 14*d*x)*sqr t(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**3*b**2 - 2790*e**(14*c + 14*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**2*b**3 - 2415*e**(14*c + 14*d*x)*sqrt(b)*sqrt(a)*ata n((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a*b**4 - 735*e**(14*c + 14 *d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*b **5 + 45*e**(12*c + 12*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**5 - 435*e**(12*c + 12*d*x)*sqrt(b)*sqrt(a)*atan((e **(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**4*b - 270*e**(12*c + 12*d*x )*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**3* b**2 + 6090*e**(12*c + 12*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**2*b**3 + 11025*e**(12*c + 12*d*x)*sqrt(b)*sqrt( a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a*b**4 + 5145*e**(12 *c + 12*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqr t(a))*b**5 - 135*e**(10*c + 10*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqr t(a + b) - sqrt(b))/sqrt(a))*a**5 + 585*e**(10*c + 10*d*x)*sqrt(b)*sqrt(a) *atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**4*b + 3450*e**(10*c + 10*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sq...