Integrand size = 23, antiderivative size = 70 \[ \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {\sqrt {b} (a+b) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {(a+b) \coth (c+d x)}{a^2 d}-\frac {\coth ^3(c+d x)}{3 a d} \] Output:
b^(1/2)*(a+b)*arctan(b^(1/2)*tanh(d*x+c)/a^(1/2))/a^(5/2)/d+(a+b)*coth(d*x +c)/a^2/d-1/3*coth(d*x+c)^3/a/d
Time = 0.44 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.01 \[ \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {3 \sqrt {b} (a+b) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )+\sqrt {a} \coth (c+d x) \left (2 a+3 b-a \text {csch}^2(c+d x)\right )}{3 a^{5/2} d} \] Input:
Integrate[Csch[c + d*x]^4/(a + b*Tanh[c + d*x]^2),x]
Output:
(3*Sqrt[b]*(a + b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]] + Sqrt[a]*Coth[ c + d*x]*(2*a + 3*b - a*Csch[c + d*x]^2))/(3*a^(5/2)*d)
Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 4146, 359, 264, 218}
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.
| \(\displaystyle \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (i c+i d x)^4 \left (a-b \tan (i c+i d x)^2\right )}dx\) |
| \(\Big \downarrow \) 4146 |
| \(\displaystyle \frac {\int \frac {\coth ^4(c+d x) \left (1-\tanh ^2(c+d x)\right )}{b \tanh ^2(c+d x)+a}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle \frac {-\frac {(a+b) \int \frac {\coth ^2(c+d x)}{b \tanh ^2(c+d x)+a}d\tanh (c+d x)}{a}-\frac {\coth ^3(c+d x)}{3 a}}{d}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {-\frac {(a+b) \left (-\frac {b \int \frac {1}{b \tanh ^2(c+d x)+a}d\tanh (c+d x)}{a}-\frac {\coth (c+d x)}{a}\right )}{a}-\frac {\coth ^3(c+d x)}{3 a}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {-\frac {(a+b) \left (-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\coth (c+d x)}{a}\right )}{a}-\frac {\coth ^3(c+d x)}{3 a}}{d}\) |
Input:
Int[Csch[c + d*x]^4/(a + b*Tanh[c + d*x]^2),x]
Output:
(-1/3*Coth[c + d*x]^3/a - ((a + b)*(-((Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d* x])/Sqrt[a]])/a^(3/2)) - Coth[c + d*x]/a))/a)/d
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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[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. \(245\) vs. \(2(60)=120\).
Time = 3.60 (sec) , antiderivative size = 246, normalized size of antiderivative = 3.51
| method | result | size |
| risch | \(-\frac {2 \left (-3 b \,{\mathrm e}^{4 d x +4 c}+6 \,{\mathrm e}^{2 d x +2 c} a +6 \,{\mathrm e}^{2 d x +2 c} b -2 a -3 b \right )}{3 a^{2} d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}+\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right )}{2 a^{2} d}+\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right ) b}{2 a^{3} d}-\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{2 a^{2} d}-\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right ) b}{2 a^{3} d}\) | \(246\) |
| derivativedivides | \(\frac {-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a}{3}-3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a -4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{2}}-\frac {1}{24 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-3 a -4 b}{8 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b \left (a +b \right ) \left (\frac {\left (a +\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (-a +\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{a}}{d}\) | \(247\) |
| default | \(\frac {-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a}{3}-3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a -4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{2}}-\frac {1}{24 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-3 a -4 b}{8 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b \left (a +b \right ) \left (\frac {\left (a +\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (-a +\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{a}}{d}\) | \(247\) |
Input:
int(csch(d*x+c)^4/(a+tanh(d*x+c)^2*b),x,method=_RETURNVERBOSE)
Output:
-2/3*(-3*b*exp(4*d*x+4*c)+6*exp(2*d*x+2*c)*a+6*exp(2*d*x+2*c)*b-2*a-3*b)/a ^2/d/(exp(2*d*x+2*c)-1)^3+1/2/a^2*(-a*b)^(1/2)/d*ln(exp(2*d*x+2*c)+(2*(-a* b)^(1/2)+a-b)/(a+b))+1/2/a^3*(-a*b)^(1/2)/d*ln(exp(2*d*x+2*c)+(2*(-a*b)^(1 /2)+a-b)/(a+b))*b-1/2/a^2*(-a*b)^(1/2)/d*ln(exp(2*d*x+2*c)-(2*(-a*b)^(1/2) -a+b)/(a+b))-1/2/a^3*(-a*b)^(1/2)/d*ln(exp(2*d*x+2*c)-(2*(-a*b)^(1/2)-a+b) /(a+b))*b
Leaf count of result is larger than twice the leaf count of optimal. 653 vs. \(2 (60) = 120\).
Time = 0.12 (sec) , antiderivative size = 1628, normalized size of antiderivative = 23.26 \[ \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)^4/(a+b*tanh(d*x+c)^2),x, algorithm="fricas")
Output:
[1/6*(12*b*cosh(d*x + c)^4 + 48*b*cosh(d*x + c)*sinh(d*x + c)^3 + 12*b*sin h(d*x + c)^4 - 24*(a + b)*cosh(d*x + c)^2 + 24*(3*b*cosh(d*x + c)^2 - a - b)*sinh(d*x + c)^2 + 3*((a + b)*cosh(d*x + c)^6 + 6*(a + b)*cosh(d*x + c)* sinh(d*x + c)^5 + (a + b)*sinh(d*x + c)^6 - 3*(a + b)*cosh(d*x + c)^4 + 3* (5*(a + b)*cosh(d*x + c)^2 - a - b)*sinh(d*x + c)^4 + 4*(5*(a + b)*cosh(d* x + c)^3 - 3*(a + b)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(a + b)*cosh(d*x + c)^2 + 3*(5*(a + b)*cosh(d*x + c)^4 - 6*(a + b)*cosh(d*x + c)^2 + a + b)* sinh(d*x + c)^2 + 6*((a + b)*cosh(d*x + c)^5 - 2*(a + b)*cosh(d*x + c)^3 + (a + b)*cosh(d*x + c))*sinh(d*x + c) - a - b)*sqrt(-b/a)*log(((a^2 + 2*a* b + b^2)*cosh(d*x + c)^4 + 4*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^4 + 2*(a^2 - b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 - b^2)*sinh(d*x + c)^2 + a^2 - 6*a*b + b^2 + 4*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + (a^2 - b^2)* cosh(d*x + c))*sinh(d*x + c) + 4*((a^2 + a*b)*cosh(d*x + c)^2 + 2*(a^2 + a *b)*cosh(d*x + c)*sinh(d*x + c) + (a^2 + a*b)*sinh(d*x + c)^2 + a^2 - a*b) *sqrt(-b/a))/((a + b)*cosh(d*x + c)^4 + 4*(a + b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a + b)*sinh(d*x + c)^4 + 2*(a - b)*cosh(d*x + c)^2 + 2*(3*(a + b) *cosh(d*x + c)^2 + a - b)*sinh(d*x + c)^2 + 4*((a + b)*cosh(d*x + c)^3 + ( a - b)*cosh(d*x + c))*sinh(d*x + c) + a + b)) + 48*(b*cosh(d*x + c)^3 - (a + b)*cosh(d*x + c))*sinh(d*x + c) + 8*a + 12*b)/(a^2*d*cosh(d*x + c)^6...
\[ \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int \frac {\operatorname {csch}^{4}{\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \] Input:
integrate(csch(d*x+c)**4/(a+b*tanh(d*x+c)**2),x)
Output:
Integral(csch(c + d*x)**4/(a + b*tanh(c + d*x)**2), x)
Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (60) = 120\).
Time = 0.15 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.91 \[ \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {2 \, {\left (6 \, {\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, b e^{\left (-4 \, d x - 4 \, c\right )} - 2 \, a - 3 \, b\right )}}{3 \, {\left (3 \, a^{2} e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, a^{2} e^{\left (-4 \, d x - 4 \, c\right )} + a^{2} e^{\left (-6 \, d x - 6 \, c\right )} - a^{2}\right )} d} - \frac {{\left (a b + b^{2}\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{\sqrt {a b} a^{2} d} \] Input:
integrate(csch(d*x+c)^4/(a+b*tanh(d*x+c)^2),x, algorithm="maxima")
Output:
2/3*(6*(a + b)*e^(-2*d*x - 2*c) - 3*b*e^(-4*d*x - 4*c) - 2*a - 3*b)/((3*a^ 2*e^(-2*d*x - 2*c) - 3*a^2*e^(-4*d*x - 4*c) + a^2*e^(-6*d*x - 6*c) - a^2)* d) - (a*b + b^2)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sqrt(a*b))/ (sqrt(a*b)*a^2*d)
Time = 0.31 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.70 \[ \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {\frac {3 \, {\left (a b + b^{2}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {2 \, {\left (3 \, b e^{\left (4 \, d x + 4 \, c\right )} - 6 \, a e^{\left (2 \, d x + 2 \, c\right )} - 6 \, b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a + 3 \, b\right )}}{a^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{3 \, d} \] Input:
integrate(csch(d*x+c)^4/(a+b*tanh(d*x+c)^2),x, algorithm="giac")
Output:
1/3*(3*(a*b + 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^2) + 2*(3*b*e^(4*d*x + 4*c) - 6*a*e^(2*d*x + 2 *c) - 6*b*e^(2*d*x + 2*c) + 2*a + 3*b)/(a^2*(e^(2*d*x + 2*c) - 1)^3))/d
Time = 1.65 (sec) , antiderivative size = 254, normalized size of antiderivative = 3.63 \[ \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {2\,b}{a^2\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {8}{3\,a\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {4}{a\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {\sqrt {-b}\,\ln \left (-\frac {4\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{a^2}-\frac {2\,\sqrt {-b}\,\left (a\,d+b\,d+a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}-b\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^{5/2}\,d}\right )\,\left (a+b\right )}{2\,a^{5/2}\,d}-\frac {\sqrt {-b}\,\ln \left (\frac {2\,\sqrt {-b}\,\left (a\,d+b\,d+a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}-b\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^{5/2}\,d}-\frac {4\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{a^2}\right )\,\left (a+b\right )}{2\,a^{5/2}\,d} \] Input:
int(1/(sinh(c + d*x)^4*(a + b*tanh(c + d*x)^2)),x)
Output:
(2*b)/(a^2*d*(exp(2*c + 2*d*x) - 1)) - 8/(3*a*d*(3*exp(2*c + 2*d*x) - 3*ex p(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1)) - 4/(a*d*(exp(4*c + 4*d*x) - 2*exp (2*c + 2*d*x) + 1)) + ((-b)^(1/2)*log(- (4*b*exp(2*c + 2*d*x))/a^2 - (2*(- b)^(1/2)*(a*d + b*d + a*d*exp(2*c + 2*d*x) - b*d*exp(2*c + 2*d*x)))/(a^(5/ 2)*d))*(a + b))/(2*a^(5/2)*d) - ((-b)^(1/2)*log((2*(-b)^(1/2)*(a*d + b*d + a*d*exp(2*c + 2*d*x) - b*d*exp(2*c + 2*d*x)))/(a^(5/2)*d) - (4*b*exp(2*c + 2*d*x))/a^2)*(a + b))/(2*a^(5/2)*d)
Time = 0.30 (sec) , antiderivative size = 683, normalized size of antiderivative = 9.76 \[ \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {3 e^{6 d x +6 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}-\sqrt {b}}{\sqrt {a}}\right ) a +3 e^{6 d x +6 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}-\sqrt {b}}{\sqrt {a}}\right ) b -9 e^{4 d x +4 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}-\sqrt {b}}{\sqrt {a}}\right ) a -9 e^{4 d x +4 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}-\sqrt {b}}{\sqrt {a}}\right ) b +9 e^{2 d x +2 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}-\sqrt {b}}{\sqrt {a}}\right ) a +9 e^{2 d x +2 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}-\sqrt {b}}{\sqrt {a}}\right ) b -3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}-\sqrt {b}}{\sqrt {a}}\right ) a -3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}-\sqrt {b}}{\sqrt {a}}\right ) b -3 e^{6 d x +6 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}+\sqrt {b}}{\sqrt {a}}\right ) a -3 e^{6 d x +6 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}+\sqrt {b}}{\sqrt {a}}\right ) b +9 e^{4 d x +4 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}+\sqrt {b}}{\sqrt {a}}\right ) a +9 e^{4 d x +4 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}+\sqrt {b}}{\sqrt {a}}\right ) b -9 e^{2 d x +2 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}+\sqrt {b}}{\sqrt {a}}\right ) a -9 e^{2 d x +2 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}+\sqrt {b}}{\sqrt {a}}\right ) b +3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}+\sqrt {b}}{\sqrt {a}}\right ) a +3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}+\sqrt {b}}{\sqrt {a}}\right ) b +2 e^{6 d x +6 c} a b -12 e^{2 d x +2 c} a^{2}-6 e^{2 d x +2 c} a b +4 a^{2}+4 a b}{3 a^{3} d \left (e^{6 d x +6 c}-3 e^{4 d x +4 c}+3 e^{2 d x +2 c}-1\right )} \] Input:
int(csch(d*x+c)^4/(a+b*tanh(d*x+c)^2),x)
Output:
(3*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt( b))/sqrt(a))*a + 3*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqr t(a + b) - sqrt(b))/sqrt(a))*b - 9*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a)*atan(( e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a - 9*e**(4*c + 4*d*x)*sqrt(b )*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*b + 9*e**(2*c + 2*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a ))*a + 9*e**(2*c + 2*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*b - 3*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a - 3*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - s qrt(b))/sqrt(a))*b - 3*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x) *sqrt(a + b) + sqrt(b))/sqrt(a))*a - 3*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a)*at an((e**(c + d*x)*sqrt(a + b) + sqrt(b))/sqrt(a))*b + 9*e**(4*c + 4*d*x)*sq rt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) + sqrt(b))/sqrt(a))*a + 9*e** (4*c + 4*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) + sqrt(b))/sq rt(a))*b - 9*e**(2*c + 2*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) + sqrt(b))/sqrt(a))*a - 9*e**(2*c + 2*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) + sqrt(b))/sqrt(a))*b + 3*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) + sqrt(b))/sqrt(a))*a + 3*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) + sqrt(b))/sqrt(a))*b + 2*e**(6*c + 6*d*x)*a*b - 12*e**(2 *c + 2*d*x)*a**2 - 6*e**(2*c + 2*d*x)*a*b + 4*a**2 + 4*a*b)/(3*a**3*d*(...