Integrand size = 27, antiderivative size = 121 \[ \int \frac {\sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a^2 x}{b \left (a^2+b^2\right )}+\frac {b x}{a^2+b^2}+\frac {2 a^3 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}+\frac {a \text {sech}(c+d x)}{\left (a^2+b^2\right ) d}-\frac {b \tanh (c+d x)}{\left (a^2+b^2\right ) d} \] Output:
a^2*x/b/(a^2+b^2)+b*x/(a^2+b^2)+2*a^3*arctanh((b-a*tanh(1/2*d*x+1/2*c))/(a ^2+b^2)^(1/2))/b/(a^2+b^2)^(3/2)/d+a*sech(d*x+c)/(a^2+b^2)/d-b*tanh(d*x+c) /(a^2+b^2)/d
Time = 0.76 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.79 \[ \int \frac {\sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {c+d x}{b}+\frac {2 a^3 \arctan \left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{b \left (-a^2-b^2\right )^{3/2}}+\frac {\text {sech}(c+d x) (a-b \sinh (c+d x))}{a^2+b^2}}{d} \] Input:
Integrate[(Sinh[c + d*x]*Tanh[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
Output:
((c + d*x)/b + (2*a^3*ArcTan[(b - a*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]])/ (b*(-a^2 - b^2)^(3/2)) + (Sech[c + d*x]*(a - b*Sinh[c + d*x]))/(a^2 + b^2) )/d
Result contains complex when optimal does not.
Time = 0.83 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.05, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {3042, 26, 3381, 25, 26, 3042, 25, 26, 3086, 24, 3214, 3042, 3139, 1083, 217, 3954, 24}
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.
| \(\displaystyle \int \frac {\sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \sin (i c+i d x)^3}{\cos (i c+i d x)^2 (a-i b \sin (i c+i d x))}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\sin (i c+i d x)^3}{\cos (i c+i d x)^2 (a-i b \sin (i c+i d x))}dx\) |
| \(\Big \downarrow \) 3381 |
| \(\displaystyle i \left (-\frac {a^2 \int \frac {i \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {i b \int -\tanh ^2(c+d x)dx}{a^2+b^2}+\frac {a \int i \text {sech}(c+d x) \tanh (c+d x)dx}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (-\frac {a^2 \int \frac {i \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{a^2+b^2}-\frac {i b \int \tanh ^2(c+d x)dx}{a^2+b^2}+\frac {a \int i \text {sech}(c+d x) \tanh (c+d x)dx}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\frac {i a^2 \int \frac {\sinh (c+d x)}{a+b \sinh (c+d x)}dx}{a^2+b^2}-\frac {i b \int \tanh ^2(c+d x)dx}{a^2+b^2}+\frac {i a \int \text {sech}(c+d x) \tanh (c+d x)dx}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-\frac {i a^2 \int -\frac {i \sin (i c+i d x)}{a-i b \sin (i c+i d x)}dx}{a^2+b^2}-\frac {i b \int -\tan (i c+i d x)^2dx}{a^2+b^2}+\frac {i a \int -i \sec (i c+i d x) \tan (i c+i d x)dx}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (-\frac {i a^2 \int -\frac {i \sin (i c+i d x)}{a-i b \sin (i c+i d x)}dx}{a^2+b^2}+\frac {i b \int \tan (i c+i d x)^2dx}{a^2+b^2}+\frac {i a \int -i \sec (i c+i d x) \tan (i c+i d x)dx}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\frac {a^2 \int \frac {\sin (i c+i d x)}{a-i b \sin (i c+i d x)}dx}{a^2+b^2}+\frac {i b \int \tan (i c+i d x)^2dx}{a^2+b^2}+\frac {a \int \sec (i c+i d x) \tan (i c+i d x)dx}{a^2+b^2}\right )\) |
| \(\Big \downarrow \) 3086 |
| \(\displaystyle i \left (-\frac {a^2 \int \frac {\sin (i c+i d x)}{a-i b \sin (i c+i d x)}dx}{a^2+b^2}+\frac {i b \int \tan (i c+i d x)^2dx}{a^2+b^2}-\frac {i a \int 1d\text {sech}(c+d x)}{d \left (a^2+b^2\right )}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle i \left (-\frac {a^2 \int \frac {\sin (i c+i d x)}{a-i b \sin (i c+i d x)}dx}{a^2+b^2}+\frac {i b \int \tan (i c+i d x)^2dx}{a^2+b^2}-\frac {i a \text {sech}(c+d x)}{d \left (a^2+b^2\right )}\right )\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle i \left (\frac {i b \int \tan (i c+i d x)^2dx}{a^2+b^2}-\frac {a^2 \left (\frac {i x}{b}-\frac {i a \int \frac {1}{a+b \sinh (c+d x)}dx}{b}\right )}{a^2+b^2}-\frac {i a \text {sech}(c+d x)}{d \left (a^2+b^2\right )}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-\frac {a^2 \left (\frac {i x}{b}-\frac {i a \int \frac {1}{a-i b \sin (i c+i d x)}dx}{b}\right )}{a^2+b^2}+\frac {i b \int \tan (i c+i d x)^2dx}{a^2+b^2}-\frac {i a \text {sech}(c+d x)}{d \left (a^2+b^2\right )}\right )\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle i \left (\frac {i b \int \tan (i c+i d x)^2dx}{a^2+b^2}-\frac {a^2 \left (\frac {i x}{b}-\frac {2 a \int \frac {1}{-a \tanh ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tanh \left (\frac {1}{2} (c+d x)\right )+a}d\left (i \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{b d}\right )}{a^2+b^2}-\frac {i a \text {sech}(c+d x)}{d \left (a^2+b^2\right )}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle i \left (\frac {i b \int \tan (i c+i d x)^2dx}{a^2+b^2}-\frac {a^2 \left (\frac {4 a \int \frac {1}{\tanh ^2\left (\frac {1}{2} (c+d x)\right )-4 \left (a^2+b^2\right )}d\left (2 i a \tanh \left (\frac {1}{2} (c+d x)\right )-2 i b\right )}{b d}+\frac {i x}{b}\right )}{a^2+b^2}-\frac {i a \text {sech}(c+d x)}{d \left (a^2+b^2\right )}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle i \left (\frac {i b \int \tan (i c+i d x)^2dx}{a^2+b^2}-\frac {a^2 \left (\frac {i x}{b}-\frac {2 i a \text {arctanh}\left (\frac {\tanh \left (\frac {1}{2} (c+d x)\right )}{2 \sqrt {a^2+b^2}}\right )}{b d \sqrt {a^2+b^2}}\right )}{a^2+b^2}-\frac {i a \text {sech}(c+d x)}{d \left (a^2+b^2\right )}\right )\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle i \left (\frac {i b \left (\frac {\tanh (c+d x)}{d}-\int 1dx\right )}{a^2+b^2}-\frac {a^2 \left (\frac {i x}{b}-\frac {2 i a \text {arctanh}\left (\frac {\tanh \left (\frac {1}{2} (c+d x)\right )}{2 \sqrt {a^2+b^2}}\right )}{b d \sqrt {a^2+b^2}}\right )}{a^2+b^2}-\frac {i a \text {sech}(c+d x)}{d \left (a^2+b^2\right )}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle i \left (-\frac {a^2 \left (\frac {i x}{b}-\frac {2 i a \text {arctanh}\left (\frac {\tanh \left (\frac {1}{2} (c+d x)\right )}{2 \sqrt {a^2+b^2}}\right )}{b d \sqrt {a^2+b^2}}\right )}{a^2+b^2}+\frac {i b \left (\frac {\tanh (c+d x)}{d}-x\right )}{a^2+b^2}-\frac {i a \text {sech}(c+d x)}{d \left (a^2+b^2\right )}\right )\) |
Input:
Int[(Sinh[c + d*x]*Tanh[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
Output:
I*(-((a^2*((I*x)/b - ((2*I)*a*ArcTanh[Tanh[(c + d*x)/2]/(2*Sqrt[a^2 + b^2] )])/(b*Sqrt[a^2 + b^2]*d)))/(a^2 + b^2)) - (I*a*Sech[c + d*x])/((a^2 + b^2 )*d) + (I*b*(-x + Tanh[c + d*x]/d))/(a^2 + b^2))
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[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a*(d^2/(a^2 - b^2)) Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 2), x], x] + (-Simp[ b*(d/(a^2 - b^2)) Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 1), x], x] - Simp[a^2*(d^2/(g^2*(a^2 - b^2))) Int[(g*Cos[e + f*x])^(p + 2)*((d*Sin[ e + f*x])^(n - 2)/(a + b*Sin[e + f*x])), x], x]) /; FreeQ[{a, b, d, e, f, g }, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[p, -1] && GtQ[n, 1 ]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Time = 0.42 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.03
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{\left (a^{2}+b^{2}\right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}-\frac {2 a^{3} \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b}}{d}\) | \(125\) |
| default | \(\frac {-\frac {2 \left (b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{\left (a^{2}+b^{2}\right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}-\frac {2 a^{3} \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b}}{d}\) | \(125\) |
| risch | \(\frac {x}{b}+\frac {2 a \,{\mathrm e}^{d x +c}+2 b}{d \left (a^{2}+b^{2}\right ) \left (1+{\mathrm e}^{2 d x +2 c}\right )}+\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d b}-\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d b}\) | \(181\) |
Input:
int(sinh(d*x+c)*tanh(d*x+c)^2/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(-2/(a^2+b^2)*(b*tanh(1/2*d*x+1/2*c)-a)/(1+tanh(1/2*d*x+1/2*c)^2)-2/b* a^3/(a^2+b^2)^(3/2)*arctanh(1/2*(2*a*tanh(1/2*d*x+1/2*c)-2*b)/(a^2+b^2)^(1 /2))+1/b*ln(1+tanh(1/2*d*x+1/2*c))-1/b*ln(tanh(1/2*d*x+1/2*c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 459 vs. \(2 (118) = 236\).
Time = 0.09 (sec) , antiderivative size = 459, normalized size of antiderivative = 3.79 \[ \int \frac {\sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d x \cosh \left (d x + c\right )^{2} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d x \sinh \left (d x + c\right )^{2} + 2 \, a^{2} b^{2} + 2 \, b^{4} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d x + {\left (a^{3} \cosh \left (d x + c\right )^{2} + 2 \, a^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{3} \sinh \left (d x + c\right )^{2} + a^{3}\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) + a\right )}}{b \cosh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) + 2 \, {\left (b \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right ) - b}\right ) + 2 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right ) + 2 \, {\left (a^{3} b + a b^{3} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d x \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d \sinh \left (d x + c\right )^{2} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d} \] Input:
integrate(sinh(d*x+c)*tanh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas ")
Output:
((a^4 + 2*a^2*b^2 + b^4)*d*x*cosh(d*x + c)^2 + (a^4 + 2*a^2*b^2 + b^4)*d*x *sinh(d*x + c)^2 + 2*a^2*b^2 + 2*b^4 + (a^4 + 2*a^2*b^2 + b^4)*d*x + (a^3* cosh(d*x + c)^2 + 2*a^3*cosh(d*x + c)*sinh(d*x + c) + a^3*sinh(d*x + c)^2 + a^3)*sqrt(a^2 + b^2)*log((b^2*cosh(d*x + c)^2 + b^2*sinh(d*x + c)^2 + 2* a*b*cosh(d*x + c) + 2*a^2 + b^2 + 2*(b^2*cosh(d*x + c) + a*b)*sinh(d*x + c ) + 2*sqrt(a^2 + b^2)*(b*cosh(d*x + c) + b*sinh(d*x + c) + a))/(b*cosh(d*x + c)^2 + b*sinh(d*x + c)^2 + 2*a*cosh(d*x + c) + 2*(b*cosh(d*x + c) + a)* sinh(d*x + c) - b)) + 2*(a^3*b + a*b^3)*cosh(d*x + c) + 2*(a^3*b + a*b^3 + (a^4 + 2*a^2*b^2 + b^4)*d*x*cosh(d*x + c))*sinh(d*x + c))/((a^4*b + 2*a^2 *b^3 + b^5)*d*cosh(d*x + c)^2 + 2*(a^4*b + 2*a^2*b^3 + b^5)*d*cosh(d*x + c )*sinh(d*x + c) + (a^4*b + 2*a^2*b^3 + b^5)*d*sinh(d*x + c)^2 + (a^4*b + 2 *a^2*b^3 + b^5)*d)
\[ \int \frac {\sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\sinh {\left (c + d x \right )} \tanh ^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:
integrate(sinh(d*x+c)*tanh(d*x+c)**2/(a+b*sinh(d*x+c)),x)
Output:
Integral(sinh(c + d*x)*tanh(c + d*x)**2/(a + b*sinh(c + d*x)), x)
Time = 0.12 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.17 \[ \int \frac {\sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a^{3} \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} b + b^{3}\right )} \sqrt {a^{2} + b^{2}} d} + \frac {2 \, {\left (a e^{\left (-d x - c\right )} - b\right )}}{{\left (a^{2} + b^{2} + {\left (a^{2} + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} d} + \frac {d x + c}{b d} \] Input:
integrate(sinh(d*x+c)*tanh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="maxima ")
Output:
-a^3*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt (a^2 + b^2)))/((a^2*b + b^3)*sqrt(a^2 + b^2)*d) + 2*(a*e^(-d*x - c) - b)/( (a^2 + b^2 + (a^2 + b^2)*e^(-2*d*x - 2*c))*d) + (d*x + c)/(b*d)
Time = 0.15 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.06 \[ \int \frac {\sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {a^{3} \log \left (\frac {{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{2} b + b^{3}\right )} \sqrt {a^{2} + b^{2}}} - \frac {d x + c}{b} - \frac {2 \, {\left (a e^{\left (d x + c\right )} + b\right )}}{{\left (a^{2} + b^{2}\right )} {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}}}{d} \] Input:
integrate(sinh(d*x+c)*tanh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="giac")
Output:
-(a^3*log(abs(2*b*e^(d*x + c) + 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*e^(d*x + c) + 2*a + 2*sqrt(a^2 + b^2)))/((a^2*b + b^3)*sqrt(a^2 + b^2)) - (d*x + c) /b - 2*(a*e^(d*x + c) + b)/((a^2 + b^2)*(e^(2*d*x + 2*c) + 1)))/d
Time = 3.15 (sec) , antiderivative size = 468, normalized size of antiderivative = 3.87 \[ \int \frac {\sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {2\,b}{d\,\left (a^2+b^2\right )}+\frac {2\,a\,{\mathrm {e}}^{c+d\,x}}{d\,\left (a^2+b^2\right )}}{{\mathrm {e}}^{2\,c+2\,d\,x}+1}+\frac {x}{b}+\frac {2\,\mathrm {atan}\left (\left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {2\,a^3}{b^3\,d\,\left (a^2\,b+b^3\right )\,\sqrt {a^6}\,\left (a^2+b^2\right )}+\frac {2\,\left (a\,b^3\,d\,\sqrt {a^6}+a^3\,b\,d\,\sqrt {a^6}\right )}{a^2\,b^2\,\left (a^2\,b+b^3\right )\,\sqrt {-b^2\,d^2\,{\left (a^2+b^2\right )}^3}\,\sqrt {-a^6\,b^2\,d^2-3\,a^4\,b^4\,d^2-3\,a^2\,b^6\,d^2-b^8\,d^2}}\right )-\frac {2\,\left (b^4\,d\,\sqrt {a^6}+a^2\,b^2\,d\,\sqrt {a^6}\right )}{a^2\,b^2\,\left (a^2\,b+b^3\right )\,\sqrt {-b^2\,d^2\,{\left (a^2+b^2\right )}^3}\,\sqrt {-a^6\,b^2\,d^2-3\,a^4\,b^4\,d^2-3\,a^2\,b^6\,d^2-b^8\,d^2}}\right )\,\left (\frac {b^4\,\sqrt {-a^6\,b^2\,d^2-3\,a^4\,b^4\,d^2-3\,a^2\,b^6\,d^2-b^8\,d^2}}{2}+\frac {a^2\,b^2\,\sqrt {-a^6\,b^2\,d^2-3\,a^4\,b^4\,d^2-3\,a^2\,b^6\,d^2-b^8\,d^2}}{2}\right )\right )\,\sqrt {a^6}}{\sqrt {-a^6\,b^2\,d^2-3\,a^4\,b^4\,d^2-3\,a^2\,b^6\,d^2-b^8\,d^2}} \] Input:
int((sinh(c + d*x)*tanh(c + d*x)^2)/(a + b*sinh(c + d*x)),x)
Output:
((2*b)/(d*(a^2 + b^2)) + (2*a*exp(c + d*x))/(d*(a^2 + b^2)))/(exp(2*c + 2* d*x) + 1) + x/b + (2*atan((exp(d*x)*exp(c)*((2*a^3)/(b^3*d*(a^2*b + b^3)*( a^6)^(1/2)*(a^2 + b^2)) + (2*(a*b^3*d*(a^6)^(1/2) + a^3*b*d*(a^6)^(1/2)))/ (a^2*b^2*(a^2*b + b^3)*(-b^2*d^2*(a^2 + b^2)^3)^(1/2)*(- b^8*d^2 - 3*a^2*b ^6*d^2 - 3*a^4*b^4*d^2 - a^6*b^2*d^2)^(1/2))) - (2*(b^4*d*(a^6)^(1/2) + a^ 2*b^2*d*(a^6)^(1/2)))/(a^2*b^2*(a^2*b + b^3)*(-b^2*d^2*(a^2 + b^2)^3)^(1/2 )*(- b^8*d^2 - 3*a^2*b^6*d^2 - 3*a^4*b^4*d^2 - a^6*b^2*d^2)^(1/2)))*((b^4* (- b^8*d^2 - 3*a^2*b^6*d^2 - 3*a^4*b^4*d^2 - a^6*b^2*d^2)^(1/2))/2 + (a^2* b^2*(- b^8*d^2 - 3*a^2*b^6*d^2 - 3*a^4*b^4*d^2 - a^6*b^2*d^2)^(1/2))/2))*( a^6)^(1/2))/(- b^8*d^2 - 3*a^2*b^6*d^2 - 3*a^4*b^4*d^2 - a^6*b^2*d^2)^(1/2 )
Time = 0.20 (sec) , antiderivative size = 396, normalized size of antiderivative = 3.27 \[ \int \frac {\sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-2 e^{2 d x +2 c} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{d x +c} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{3} i -2 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{d x +c} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{3} i -e^{2 d x +2 c} \tanh \left (d x +c \right ) a^{4}-2 e^{2 d x +2 c} \tanh \left (d x +c \right ) a^{2} b^{2}-e^{2 d x +2 c} \tanh \left (d x +c \right ) b^{4}+e^{2 d x +2 c} a^{4} d x +2 e^{2 d x +2 c} a^{4}+2 e^{2 d x +2 c} a^{2} b^{2} d x +2 e^{2 d x +2 c} a^{2} b^{2}+e^{2 d x +2 c} b^{4} d x +2 e^{d x +c} a^{3} b +2 e^{d x +c} a \,b^{3}-\tanh \left (d x +c \right ) a^{4}-2 \tanh \left (d x +c \right ) a^{2} b^{2}-\tanh \left (d x +c \right ) b^{4}+a^{4} d x +2 a^{2} b^{2} d x +b^{4} d x}{b d \left (e^{2 d x +2 c} a^{4}+2 e^{2 d x +2 c} a^{2} b^{2}+e^{2 d x +2 c} b^{4}+a^{4}+2 a^{2} b^{2}+b^{4}\right )} \] Input:
int(sinh(d*x+c)*tanh(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
( - 2*e**(2*c + 2*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqr t(a**2 + b**2))*a**3*i - 2*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i) /sqrt(a**2 + b**2))*a**3*i - e**(2*c + 2*d*x)*tanh(c + d*x)*a**4 - 2*e**(2 *c + 2*d*x)*tanh(c + d*x)*a**2*b**2 - e**(2*c + 2*d*x)*tanh(c + d*x)*b**4 + e**(2*c + 2*d*x)*a**4*d*x + 2*e**(2*c + 2*d*x)*a**4 + 2*e**(2*c + 2*d*x) *a**2*b**2*d*x + 2*e**(2*c + 2*d*x)*a**2*b**2 + e**(2*c + 2*d*x)*b**4*d*x + 2*e**(c + d*x)*a**3*b + 2*e**(c + d*x)*a*b**3 - tanh(c + d*x)*a**4 - 2*t anh(c + d*x)*a**2*b**2 - tanh(c + d*x)*b**4 + a**4*d*x + 2*a**2*b**2*d*x + b**4*d*x)/(b*d*(e**(2*c + 2*d*x)*a**4 + 2*e**(2*c + 2*d*x)*a**2*b**2 + e* *(2*c + 2*d*x)*b**4 + a**4 + 2*a**2*b**2 + b**4))