Integrand size = 21, antiderivative size = 92 \[ \int \frac {\sinh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=-\frac {3 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{2 (a+b)^{5/2} d}+\frac {3 \cosh (c+d x)}{2 (a+b)^2 d}-\frac {\cosh (c+d x)}{2 (a+b) d \left (a+b-b \text {sech}^2(c+d x)\right )} \]
3/2*cosh(d*x+c)/(a+b)^2/d-1/2*cosh(d*x+c)/(a+b)/d/(a+b-b*sech(d*x+c)^2)-3/ 2*arctanh(sech(d*x+c)*b^(1/2)/(a+b)^(1/2))*b^(1/2)/(a+b)^(5/2)/d
Result contains complex when optimal does not.
Time = 1.47 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.45 \[ \int \frac {\sinh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {-\frac {3 i \sqrt {b} \left (\arctan \left (\frac {-i \sqrt {a+b}-\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )+\arctan \left (\frac {-i \sqrt {a+b}+\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )\right )}{(a+b)^{5/2}}+\frac {2 \cosh (c+d x) \left (1-\frac {b}{a-b+(a+b) \cosh (2 (c+d x))}\right )}{(a+b)^2}}{2 d} \]
(((-3*I)*Sqrt[b]*(ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sq rt[b]] + ArcTan[((-I)*Sqrt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]]))/ (a + b)^(5/2) + (2*Cosh[c + d*x]*(1 - b/(a - b + (a + b)*Cosh[2*(c + d*x)] )))/(a + b)^2)/(2*d)
Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 26, 4147, 253, 264, 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.
| \(\displaystyle \int \frac {\sinh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \sin (i c+i d x)}{\left (a-b \tan (i c+i d x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\sin (i c+i d x)}{\left (a-b \tan (i c+i d x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 4147 |
| \(\displaystyle -\frac {\int \frac {\cosh ^2(c+d x)}{\left (-b \text {sech}^2(c+d x)+a+b\right )^2}d\text {sech}(c+d x)}{d}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle -\frac {\frac {3 \int \frac {\cosh ^2(c+d x)}{-b \text {sech}^2(c+d x)+a+b}d\text {sech}(c+d x)}{2 (a+b)}+\frac {\cosh (c+d x)}{2 (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )}}{d}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle -\frac {\frac {3 \left (\frac {b \int \frac {1}{-b \text {sech}^2(c+d x)+a+b}d\text {sech}(c+d x)}{a+b}-\frac {\cosh (c+d x)}{a+b}\right )}{2 (a+b)}+\frac {\cosh (c+d x)}{2 (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {3 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{3/2}}-\frac {\cosh (c+d x)}{a+b}\right )}{2 (a+b)}+\frac {\cosh (c+d x)}{2 (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )}}{d}\) |
-(((3*((Sqrt[b]*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/(a + b)^(3/2 ) - Cosh[c + d*x]/(a + b)))/(2*(a + b)) + Cosh[c + d*x]/(2*(a + b)*(a + b - b*Sech[c + d*x]^2)))/d)
3.1.36.3.1 Defintions of rubi rules used
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/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
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[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]
Leaf count of result is larger than twice the leaf count of optimal. \(166\) vs. \(2(78)=156\).
Time = 1.72 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.82
| method | result | size |
| derivativedivides | \(\frac {\frac {2 b \left (\frac {-\frac {\left (a +2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a}-\frac {1}{2}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a}-\frac {3 \,\operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{4 \sqrt {a b +b^{2}}}\right )}{\left (a +b \right )^{2}}-\frac {1}{\left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {1}{\left (a +b \right )^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{d}\) | \(167\) |
| default | \(\frac {\frac {2 b \left (\frac {-\frac {\left (a +2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a}-\frac {1}{2}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a}-\frac {3 \,\operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{4 \sqrt {a b +b^{2}}}\right )}{\left (a +b \right )^{2}}-\frac {1}{\left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {1}{\left (a +b \right )^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{d}\) | \(167\) |
| risch | \(\frac {{\mathrm e}^{d x +c}}{2 d \left (a^{2}+2 a b +b^{2}\right )}+\frac {{\mathrm e}^{-d x -c}}{2 d \left (a^{2}+2 a b +b^{2}\right )}-\frac {b \,{\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}+1\right )}{d \left (a +b \right )^{2} \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )}+\frac {3 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{4 \left (a +b \right )^{3} d}-\frac {3 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{4 \left (a +b \right )^{3} d}\) | \(230\) |
1/d*(2*b/(a+b)^2*((-1/2*(a+2*b)/a*tanh(1/2*d*x+1/2*c)^2-1/2)/(tanh(1/2*d*x +1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)-3/4/(a* b+b^2)^(1/2)*arctanh(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a+2*a+4*b)/(a*b+b^2)^(1/ 2)))-1/(a+b)^2/(tanh(1/2*d*x+1/2*c)-1)+1/(a+b)^2/(1+tanh(1/2*d*x+1/2*c)))
Leaf count of result is larger than twice the leaf count of optimal. 1114 vs. \(2 (81) = 162\).
Time = 0.31 (sec) , antiderivative size = 2252, normalized size of antiderivative = 24.48 \[ \int \frac {\sinh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \]
[1/4*(2*(a + b)*cosh(d*x + c)^6 + 12*(a + b)*cosh(d*x + c)*sinh(d*x + c)^5 + 2*(a + b)*sinh(d*x + c)^6 + 6*(a - b)*cosh(d*x + c)^4 + 6*(5*(a + b)*co sh(d*x + c)^2 + a - b)*sinh(d*x + c)^4 + 8*(5*(a + b)*cosh(d*x + c)^3 + 3* (a - b)*cosh(d*x + c))*sinh(d*x + c)^3 + 6*(a - b)*cosh(d*x + c)^2 + 6*(5* (a + b)*cosh(d*x + c)^4 + 6*(a - b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c) ^2 + 3*((a + b)*cosh(d*x + c)^5 + 5*(a + b)*cosh(d*x + c)*sinh(d*x + c)^4 + (a + b)*sinh(d*x + c)^5 + 2*(a - b)*cosh(d*x + c)^3 + 2*(5*(a + b)*cosh( d*x + c)^2 + a - b)*sinh(d*x + c)^3 + 2*(5*(a + b)*cosh(d*x + c)^3 + 3*(a - b)*cosh(d*x + c))*sinh(d*x + c)^2 + (a + b)*cosh(d*x + c) + (5*(a + b)*c osh(d*x + c)^4 + 6*(a - b)*cosh(d*x + c)^2 + a + b)*sinh(d*x + c))*sqrt(b/ (a + b))*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*((a + b)*cosh(d*x + c)^3 + 3 *(a + b)*cosh(d*x + c)*sinh(d*x + c)^2 + (a + b)*sinh(d*x + c)^3 + (a + b) *cosh(d*x + c) + (3*(a + b)*cosh(d*x + c)^2 + a + b)*sinh(d*x + c))*sqrt(b /(a + b)) + 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)) + 12*((a + b)*cosh(...
\[ \int \frac {\sinh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int \frac {\sinh {\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \]
\[ \int \frac {\sinh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int { \frac {\sinh \left (d x + c\right )}{{\left (b \tanh \left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
1/2*((a*e^(6*c) + b*e^(6*c))*e^(6*d*x) + 3*(a*e^(4*c) - b*e^(4*c))*e^(4*d* x) + 3*(a*e^(2*c) - b*e^(2*c))*e^(2*d*x) + a + b)/((a^3*d*e^(5*c) + 3*a^2* b*d*e^(5*c) + 3*a*b^2*d*e^(5*c) + b^3*d*e^(5*c))*e^(5*d*x) + 2*(a^3*d*e^(3 *c) + a^2*b*d*e^(3*c) - a*b^2*d*e^(3*c) - b^3*d*e^(3*c))*e^(3*d*x) + (a^3* d*e^c + 3*a^2*b*d*e^c + 3*a*b^2*d*e^c + b^3*d*e^c)*e^(d*x)) + 1/2*integrat e(6*(b*e^(3*d*x + 3*c) - b*e^(d*x + c))/(a^3 + 3*a^2*b + 3*a*b^2 + b^3 + ( a^3*e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c) + b^3*e^(4*c))*e^(4*d*x) + 2*(a^3*e^(2*c) + a^2*b*e^(2*c) - a*b^2*e^(2*c) - b^3*e^(2*c))*e^(2*d*x)), x)
\[ \int \frac {\sinh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int { \frac {\sinh \left (d x + c\right )}{{\left (b \tanh \left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\sinh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int \frac {\mathrm {sinh}\left (c+d\,x\right )}{{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \]