Integrand size = 23, antiderivative size = 85 \[ \int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {(a+2 b) \text {arctanh}(\cosh (c+d x))}{2 a^2 d}-\frac {\sqrt {b} \sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d} \] Output:
1/2*(a+2*b)*arctanh(cosh(d*x+c))/a^2/d-b^(1/2)*(a+b)^(1/2)*arctanh(b^(1/2) *sech(d*x+c)/(a+b)^(1/2))/a^2/d-1/2*coth(d*x+c)*csch(d*x+c)/a/d
Result contains complex when optimal does not.
Time = 0.84 (sec) , antiderivative size = 198, normalized size of antiderivative = 2.33 \[ \int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=-\frac {8 i \sqrt {b} \sqrt {a+b} \arctan \left (\frac {-i \sqrt {a+b}-\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )+8 i \sqrt {b} \sqrt {a+b} \arctan \left (\frac {-i \sqrt {a+b}+\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )+a \text {csch}^2\left (\frac {1}{2} (c+d x)\right )-4 a \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )-8 b \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+4 a \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+8 b \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+a \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 a^2 d} \] Input:
Integrate[Csch[c + d*x]^3/(a + b*Tanh[c + d*x]^2),x]
Output:
-1/8*((8*I)*Sqrt[b]*Sqrt[a + b]*ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]] + (8*I)*Sqrt[b]*Sqrt[a + b]*ArcTan[((-I)*Sqrt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]] + a*Csch[(c + d*x)/2]^2 - 4*a*Log[Co sh[(c + d*x)/2]] - 8*b*Log[Cosh[(c + d*x)/2]] + 4*a*Log[Sinh[(c + d*x)/2]] + 8*b*Log[Sinh[(c + d*x)/2]] + a*Sech[(c + d*x)/2]^2)/(a^2*d)
Time = 0.31 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 26, 4147, 373, 397, 219, 221}
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.
| \(\displaystyle \int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i}{\sin (i c+i d x)^3 \left (a-b \tan (i c+i d x)^2\right )}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {1}{\sin (i c+i d x)^3 \left (a-b \tan (i c+i d x)^2\right )}dx\) |
| \(\Big \downarrow \) 4147 |
| \(\displaystyle -\frac {\int \frac {\text {sech}^2(c+d x)}{\left (1-\text {sech}^2(c+d x)\right )^2 \left (-b \text {sech}^2(c+d x)+a+b\right )}d\text {sech}(c+d x)}{d}\) |
| \(\Big \downarrow \) 373 |
| \(\displaystyle -\frac {\frac {\text {sech}(c+d x)}{2 a \left (1-\text {sech}^2(c+d x)\right )}-\frac {\int \frac {b \text {sech}^2(c+d x)+a+b}{\left (1-\text {sech}^2(c+d x)\right ) \left (-b \text {sech}^2(c+d x)+a+b\right )}d\text {sech}(c+d x)}{2 a}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle -\frac {\frac {\text {sech}(c+d x)}{2 a \left (1-\text {sech}^2(c+d x)\right )}-\frac {\frac {(a+2 b) \int \frac {1}{1-\text {sech}^2(c+d x)}d\text {sech}(c+d x)}{a}-\frac {2 b (a+b) \int \frac {1}{-b \text {sech}^2(c+d x)+a+b}d\text {sech}(c+d x)}{a}}{2 a}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {\text {sech}(c+d x)}{2 a \left (1-\text {sech}^2(c+d x)\right )}-\frac {\frac {(a+2 b) \text {arctanh}(\text {sech}(c+d x))}{a}-\frac {2 b (a+b) \int \frac {1}{-b \text {sech}^2(c+d x)+a+b}d\text {sech}(c+d x)}{a}}{2 a}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\text {sech}(c+d x)}{2 a \left (1-\text {sech}^2(c+d x)\right )}-\frac {\frac {(a+2 b) \text {arctanh}(\text {sech}(c+d x))}{a}-\frac {2 \sqrt {b} \sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{a}}{2 a}}{d}\) |
Input:
Int[Csch[c + d*x]^3/(a + b*Tanh[c + d*x]^2),x]
Output:
-((-1/2*(((a + 2*b)*ArcTanh[Sech[c + d*x]])/a - (2*Sqrt[b]*Sqrt[a + b]*Arc Tanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/a)/a + Sech[c + d*x]/(2*a*(1 - Sech[c + d*x]^2)))/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[((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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*(b*c - a*d)*(p + 1))), x] - Simp[e^2/(2*(b*c - a*d)*(p + 1)) Int[(e *x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(m - 1) + d*(m + 2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 1] && LeQ[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[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]
Time = 1.88 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.31
| method | result | size |
| derivativedivides | \(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-2 a -4 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2}}-\frac {b \left (a +b \right ) \operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{a^{2} \sqrt {a b +b^{2}}}}{d}\) | \(111\) |
| default | \(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-2 a -4 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2}}-\frac {b \left (a +b \right ) \operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{a^{2} \sqrt {a b +b^{2}}}}{d}\) | \(111\) |
| risch | \(-\frac {{\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}+1\right )}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{2 a d}-\frac {b \ln \left ({\mathrm e}^{d x +c}-1\right )}{d \,a^{2}}+\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{2 a d}+\frac {b \ln \left ({\mathrm e}^{d x +c}+1\right )}{d \,a^{2}}+\frac {\sqrt {a b +b^{2}}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {a b +b^{2}}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{2 d \,a^{2}}-\frac {\sqrt {a b +b^{2}}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {a b +b^{2}}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{2 d \,a^{2}}\) | \(211\) |
Input:
int(csch(d*x+c)^3/(a+tanh(d*x+c)^2*b),x,method=_RETURNVERBOSE)
Output:
1/d*(1/8*tanh(1/2*d*x+1/2*c)^2/a-1/8/a/tanh(1/2*d*x+1/2*c)^2+1/4/a^2*(-2*a -4*b)*ln(tanh(1/2*d*x+1/2*c))-b*(a+b)/a^2/(a*b+b^2)^(1/2)*arctanh(1/4*(2*t anh(1/2*d*x+1/2*c)^2*a+2*a+4*b)/(a*b+b^2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 865 vs. \(2 (73) = 146\).
Time = 0.14 (sec) , antiderivative size = 1797, normalized size of antiderivative = 21.14 \[ \int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)^3/(a+b*tanh(d*x+c)^2),x, algorithm="fricas")
Output:
[-1/2*(2*a*cosh(d*x + c)^3 + 6*a*cosh(d*x + c)*sinh(d*x + c)^2 + 2*a*sinh( d*x + c)^3 - (cosh(d*x + c)^4 + 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4 + 2*(3*cosh(d*x + c)^2 - 1)*sinh(d*x + c)^2 - 2*cosh(d*x + c)^2 + 4*(cosh(d*x + c)^3 - cosh(d*x + c))*sinh(d*x + c) + 1)*sqrt(a*b + b^2)*log (((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 + 3*b)*cosh(d*x + c)^2 + 2*(3*(a + b)*cosh(d*x + c)^2 + a + 3*b)*sinh(d*x + c)^2 + 4*((a + b)*cosh(d*x + c)^3 + (a + 3*b) *cosh(d*x + c))*sinh(d*x + c) - 4*(cosh(d*x + c)^3 + 3*cosh(d*x + c)*sinh( d*x + c)^2 + sinh(d*x + c)^3 + (3*cosh(d*x + c)^2 + 1)*sinh(d*x + c) + cos h(d*x + c))*sqrt(a*b + b^2) + a + b)/((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)) + 2* a*cosh(d*x + c) - ((a + 2*b)*cosh(d*x + c)^4 + 4*(a + 2*b)*cosh(d*x + c)*s inh(d*x + c)^3 + (a + 2*b)*sinh(d*x + c)^4 - 2*(a + 2*b)*cosh(d*x + c)^2 + 2*(3*(a + 2*b)*cosh(d*x + c)^2 - a - 2*b)*sinh(d*x + c)^2 + 4*((a + 2*b)* cosh(d*x + c)^3 - (a + 2*b)*cosh(d*x + c))*sinh(d*x + c) + a + 2*b)*log(co sh(d*x + c) + sinh(d*x + c) + 1) + ((a + 2*b)*cosh(d*x + c)^4 + 4*(a + 2*b )*cosh(d*x + c)*sinh(d*x + c)^3 + (a + 2*b)*sinh(d*x + c)^4 - 2*(a + 2*b)* cosh(d*x + c)^2 + 2*(3*(a + 2*b)*cosh(d*x + c)^2 - a - 2*b)*sinh(d*x + ...
\[ \int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int \frac {\operatorname {csch}^{3}{\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \] Input:
integrate(csch(d*x+c)**3/(a+b*tanh(d*x+c)**2),x)
Output:
Integral(csch(c + d*x)**3/(a + b*tanh(c + d*x)**2), x)
\[ \int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{3}}{b \tanh \left (d x + c\right )^{2} + a} \,d x } \] Input:
integrate(csch(d*x+c)^3/(a+b*tanh(d*x+c)^2),x, algorithm="maxima")
Output:
-(e^(3*d*x + 3*c) + e^(d*x + c))/(a*d*e^(4*d*x + 4*c) - 2*a*d*e^(2*d*x + 2 *c) + a*d) + 1/2*(a + 2*b)*log((e^(d*x + c) + 1)*e^(-c))/(a^2*d) - 1/2*(a + 2*b)*log((e^(d*x + c) - 1)*e^(-c))/(a^2*d) + 8*integrate(1/4*((a*b*e^(3* c) + b^2*e^(3*c))*e^(3*d*x) - (a*b*e^c + b^2*e^c)*e^(d*x))/(a^3 + a^2*b + (a^3*e^(4*c) + a^2*b*e^(4*c))*e^(4*d*x) + 2*(a^3*e^(2*c) - a^2*b*e^(2*c))* e^(2*d*x)), x)
Exception generated. \[ \int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(csch(d*x+c)^3/(a+b*tanh(d*x+c)^2),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
Time = 1.93 (sec) , antiderivative size = 787, normalized size of antiderivative = 9.26 \[ \int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (18\,b^7\,\sqrt {-a^4\,d^2}+48\,a^2\,b^5\,\sqrt {-a^4\,d^2}+27\,a^3\,b^4\,\sqrt {-a^4\,d^2}+8\,a^4\,b^3\,\sqrt {-a^4\,d^2}+a^5\,b^2\,\sqrt {-a^4\,d^2}+45\,a\,b^6\,\sqrt {-a^4\,d^2}\right )}{9\,a^2\,b^6\,d\,\sqrt {a^2+4\,a\,b+4\,b^2}+18\,a^3\,b^5\,d\,\sqrt {a^2+4\,a\,b+4\,b^2}+15\,a^4\,b^4\,d\,\sqrt {a^2+4\,a\,b+4\,b^2}+6\,a^5\,b^3\,d\,\sqrt {a^2+4\,a\,b+4\,b^2}+a^6\,b^2\,d\,\sqrt {a^2+4\,a\,b+4\,b^2}}\right )\,\sqrt {a^2+4\,a\,b+4\,b^2}}{\sqrt {-a^4\,d^2}}-\frac {\left (2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a+b\right )\,\sqrt {-a^4\,d^2}}{2\,a^2\,d\,\sqrt {b\,\left (a+b\right )}}\right )+2\,\mathrm {atan}\left (\frac {\left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {64\,\left (2\,a^4\,b\,d\,\sqrt {b^2+a\,b}+6\,a^2\,b^3\,d\,\sqrt {b^2+a\,b}+6\,a^3\,b^2\,d\,\sqrt {b^2+a\,b}\right )}{a^9\,d^2\,{\left (a+b\right )}^2\,\left (a^2+2\,a\,b+b^2\right )}-\frac {32\,\left (3\,b^4\,\sqrt {-a^4\,d^2}+4\,a^2\,b^2\,\sqrt {-a^4\,d^2}+6\,a\,b^3\,\sqrt {-a^4\,d^2}+a^3\,b\,\sqrt {-a^4\,d^2}\right )}{a^7\,d\,\left (a+b\right )\,\sqrt {-a^4\,d^2}\,\sqrt {b\,\left (a+b\right )}\,\left (a^2+2\,a\,b+b^2\right )}\right )-\frac {32\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\left (3\,b^4\,\sqrt {-a^4\,d^2}+4\,a^2\,b^2\,\sqrt {-a^4\,d^2}+6\,a\,b^3\,\sqrt {-a^4\,d^2}+a^3\,b\,\sqrt {-a^4\,d^2}\right )}{a^7\,d\,\left (a+b\right )\,\sqrt {-a^4\,d^2}\,\sqrt {b\,\left (a+b\right )}\,\left (a^2+2\,a\,b+b^2\right )}\right )\,\left (a^8\,\sqrt {-a^4\,d^2}+a^5\,b^3\,\sqrt {-a^4\,d^2}+3\,a^6\,b^2\,\sqrt {-a^4\,d^2}+3\,a^7\,b\,\sqrt {-a^4\,d^2}\right )}{64\,a^2\,b+192\,a\,b^2+192\,b^3}\right )\right )\,\sqrt {b^2+a\,b}}{2\,\sqrt {-a^4\,d^2}}-\frac {{\mathrm {e}}^{c+d\,x}}{a\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}}{a\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \] Input:
int(1/(sinh(c + d*x)^3*(a + b*tanh(c + d*x)^2)),x)
Output:
(atan((exp(d*x)*exp(c)*(18*b^7*(-a^4*d^2)^(1/2) + 48*a^2*b^5*(-a^4*d^2)^(1 /2) + 27*a^3*b^4*(-a^4*d^2)^(1/2) + 8*a^4*b^3*(-a^4*d^2)^(1/2) + a^5*b^2*( -a^4*d^2)^(1/2) + 45*a*b^6*(-a^4*d^2)^(1/2)))/(9*a^2*b^6*d*(4*a*b + a^2 + 4*b^2)^(1/2) + 18*a^3*b^5*d*(4*a*b + a^2 + 4*b^2)^(1/2) + 15*a^4*b^4*d*(4* a*b + a^2 + 4*b^2)^(1/2) + 6*a^5*b^3*d*(4*a*b + a^2 + 4*b^2)^(1/2) + a^6*b ^2*d*(4*a*b + a^2 + 4*b^2)^(1/2)))*(4*a*b + a^2 + 4*b^2)^(1/2))/(-a^4*d^2) ^(1/2) - ((2*atan((exp(d*x)*exp(c)*(a + b)*(-a^4*d^2)^(1/2))/(2*a^2*d*(b*( a + b))^(1/2))) + 2*atan(((exp(d*x)*exp(c)*((64*(2*a^4*b*d*(a*b + b^2)^(1/ 2) + 6*a^2*b^3*d*(a*b + b^2)^(1/2) + 6*a^3*b^2*d*(a*b + b^2)^(1/2)))/(a^9* d^2*(a + b)^2*(2*a*b + a^2 + b^2)) - (32*(3*b^4*(-a^4*d^2)^(1/2) + 4*a^2*b ^2*(-a^4*d^2)^(1/2) + 6*a*b^3*(-a^4*d^2)^(1/2) + a^3*b*(-a^4*d^2)^(1/2)))/ (a^7*d*(a + b)*(-a^4*d^2)^(1/2)*(b*(a + b))^(1/2)*(2*a*b + a^2 + b^2))) - (32*exp(3*c)*exp(3*d*x)*(3*b^4*(-a^4*d^2)^(1/2) + 4*a^2*b^2*(-a^4*d^2)^(1/ 2) + 6*a*b^3*(-a^4*d^2)^(1/2) + a^3*b*(-a^4*d^2)^(1/2)))/(a^7*d*(a + b)*(- a^4*d^2)^(1/2)*(b*(a + b))^(1/2)*(2*a*b + a^2 + b^2)))*(a^8*(-a^4*d^2)^(1/ 2) + a^5*b^3*(-a^4*d^2)^(1/2) + 3*a^6*b^2*(-a^4*d^2)^(1/2) + 3*a^7*b*(-a^4 *d^2)^(1/2)))/(192*a*b^2 + 64*a^2*b + 192*b^3)))*(a*b + b^2)^(1/2))/(2*(-a ^4*d^2)^(1/2)) - exp(c + d*x)/(a*d*(exp(2*c + 2*d*x) - 1)) - (2*exp(c + d* x))/(a*d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1))
Time = 0.28 (sec) , antiderivative size = 570, normalized size of antiderivative = 6.71 \[ \int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {e^{4 d x +4 c} \sqrt {b}\, \sqrt {a +b}\, \mathrm {log}\left (e^{2 d x +2 c} \sqrt {a +b}+\sqrt {a +b}-2 e^{d x +c} \sqrt {b}\right )-e^{4 d x +4 c} \sqrt {b}\, \sqrt {a +b}\, \mathrm {log}\left (e^{2 d x +2 c} \sqrt {a +b}+\sqrt {a +b}+2 e^{d x +c} \sqrt {b}\right )-2 e^{2 d x +2 c} \sqrt {b}\, \sqrt {a +b}\, \mathrm {log}\left (e^{2 d x +2 c} \sqrt {a +b}+\sqrt {a +b}-2 e^{d x +c} \sqrt {b}\right )+2 e^{2 d x +2 c} \sqrt {b}\, \sqrt {a +b}\, \mathrm {log}\left (e^{2 d x +2 c} \sqrt {a +b}+\sqrt {a +b}+2 e^{d x +c} \sqrt {b}\right )+\sqrt {b}\, \sqrt {a +b}\, \mathrm {log}\left (e^{2 d x +2 c} \sqrt {a +b}+\sqrt {a +b}-2 e^{d x +c} \sqrt {b}\right )-\sqrt {b}\, \sqrt {a +b}\, \mathrm {log}\left (e^{2 d x +2 c} \sqrt {a +b}+\sqrt {a +b}+2 e^{d x +c} \sqrt {b}\right )-e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) a -2 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) b +e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) a +2 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) b -2 e^{3 d x +3 c} a +2 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) a +4 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) b -2 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) a -4 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) b -2 e^{d x +c} a -\mathrm {log}\left (e^{d x +c}-1\right ) a -2 \,\mathrm {log}\left (e^{d x +c}-1\right ) b +\mathrm {log}\left (e^{d x +c}+1\right ) a +2 \,\mathrm {log}\left (e^{d x +c}+1\right ) b}{2 a^{2} d \left (e^{4 d x +4 c}-2 e^{2 d x +2 c}+1\right )} \] Input:
int(csch(d*x+c)^3/(a+b*tanh(d*x+c)^2),x)
Output:
(e**(4*c + 4*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + s qrt(a + b) - 2*e**(c + d*x)*sqrt(b)) - e**(4*c + 4*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)) - 2*e**(2*c + 2*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)) + 2*e**(2*c + 2*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)*sq rt(b)) + 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)) - sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqr t(a + b) + sqrt(a + b) + 2*e**(c + d*x)*sqrt(b)) - e**(4*c + 4*d*x)*log(e* *(c + d*x) - 1)*a - 2*e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*b + e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)*a + 2*e**(4*c + 4*d*x)*log(e**(c + d*x) + 1) *b - 2*e**(3*c + 3*d*x)*a + 2*e**(2*c + 2*d*x)*log(e**(c + d*x) - 1)*a + 4 *e**(2*c + 2*d*x)*log(e**(c + d*x) - 1)*b - 2*e**(2*c + 2*d*x)*log(e**(c + d*x) + 1)*a - 4*e**(2*c + 2*d*x)*log(e**(c + d*x) + 1)*b - 2*e**(c + d*x) *a - log(e**(c + d*x) - 1)*a - 2*log(e**(c + d*x) - 1)*b + log(e**(c + d*x ) + 1)*a + 2*log(e**(c + d*x) + 1)*b)/(2*a**2*d*(e**(4*c + 4*d*x) - 2*e**( 2*c + 2*d*x) + 1))