Integrand size = 21, antiderivative size = 53 \[ \int \frac {\sinh (c+d x)}{a+b \tanh ^2(c+d x)} \, dx=-\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{3/2} d}+\frac {\cosh (c+d x)}{(a+b) d} \]
Result contains complex when optimal does not.
Time = 0.66 (sec) , antiderivative size = 107, normalized size of antiderivative = 2.02 \[ \int \frac {\sinh (c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {-i \sqrt {b} \left (\arctan \left (\frac {-i \sqrt {a+b}-\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )+\arctan \left (\frac {-i \sqrt {a+b}+\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )\right )+\sqrt {a+b} \cosh (c+d x)}{(a+b)^{3/2} d} \]
((-I)*Sqrt[b]*(ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[ b]] + ArcTan[((-I)*Sqrt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]]) + Sq rt[a + b]*Cosh[c + d*x])/((a + b)^(3/2)*d)
Time = 0.25 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 26, 4147, 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.
| \(\displaystyle \int \frac {\sinh (c+d x)}{a+b \tanh ^2(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \sin (i c+i d x)}{a-b \tan (i c+i d x)^2}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\sin (i c+i d x)}{a-b \tan (i c+i d x)^2}dx\) |
| \(\Big \downarrow \) 4147 |
| \(\displaystyle -\frac {\int \frac {\cosh ^2(c+d x)}{-b \text {sech}^2(c+d x)+a+b}d\text {sech}(c+d x)}{d}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle -\frac {\frac {b \int \frac {1}{-b \text {sech}^2(c+d x)+a+b}d\text {sech}(c+d x)}{a+b}-\frac {\cosh (c+d x)}{a+b}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{3/2}}-\frac {\cosh (c+d x)}{a+b}}{d}\) |
-(((Sqrt[b]*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/(a + b)^(3/2) - Cosh[c + d*x]/(a + b))/d)
3.1.28.3.1 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)/(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. \(103\) vs. \(2(45)=90\).
Time = 0.48 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.96
| method | result | size |
| derivativedivides | \(\frac {-\frac {b \,\operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{\left (a +b \right ) \sqrt {a b +b^{2}}}+\frac {4}{\left (4 a +4 b \right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {4}{\left (4 a +4 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(104\) |
| default | \(\frac {-\frac {b \,\operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{\left (a +b \right ) \sqrt {a b +b^{2}}}+\frac {4}{\left (4 a +4 b \right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {4}{\left (4 a +4 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(104\) |
| risch | \(\frac {{\mathrm e}^{d x +c}}{2 d \left (a +b \right )}+\frac {{\mathrm e}^{-d x -c}}{2 d \left (a +b \right )}+\frac {\sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{2 \left (a +b \right )^{2} d}-\frac {\sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{2 \left (a +b \right )^{2} d}\) | \(135\) |
1/d*(-b/(a+b)/(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))+4/(4*a+4*b)/(1+tanh(1/2*d*x+1/2*c))-4/(4*a+4*b)/(tanh (1/2*d*x+1/2*c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (45) = 90\).
Time = 0.31 (sec) , antiderivative size = 666, normalized size of antiderivative = 12.57 \[ \int \frac {\sinh (c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\left [\frac {\sqrt {\frac {b}{a + b}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )} \log \left (\frac {{\left (a + b\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a + b\right )} \sinh \left (d x + c\right )^{4} + 2 \, {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + a + 3 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} + {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 4 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (a + b\right )} \sinh \left (d x + c\right )^{3} + {\left (a + b\right )} \cosh \left (d x + c\right ) + {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + a + b\right )} \sinh \left (d x + c\right )\right )} \sqrt {\frac {b}{a + b}} + a + b}{{\left (a + b\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a + b\right )} \sinh \left (d x + c\right )^{4} + 2 \, {\left (a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} + {\left (a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a + b}\right ) + \cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1}{2 \, {\left ({\left (a + b\right )} d \cosh \left (d x + c\right ) + {\left (a + b\right )} d \sinh \left (d x + c\right )\right )}}, -\frac {2 \, \sqrt {-\frac {b}{a + b}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )} \arctan \left (\frac {{\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (a + b\right )} \sinh \left (d x + c\right )^{3} + {\left (a - 3 \, b\right )} \cosh \left (d x + c\right ) + {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + a - 3 \, b\right )} \sinh \left (d x + c\right )\right )} \sqrt {-\frac {b}{a + b}}}{2 \, b}\right ) - 2 \, \sqrt {-\frac {b}{a + b}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )} \arctan \left (\frac {{\left ({\left (a + b\right )} \cosh \left (d x + c\right ) + {\left (a + b\right )} \sinh \left (d x + c\right )\right )} \sqrt {-\frac {b}{a + b}}}{2 \, b}\right ) - \cosh \left (d x + c\right )^{2} - 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) - \sinh \left (d x + c\right )^{2} - 1}{2 \, {\left ({\left (a + b\right )} d \cosh \left (d x + c\right ) + {\left (a + b\right )} d \sinh \left (d x + c\right )\right )}}\right ] \]
[1/2*(sqrt(b/(a + b))*(cosh(d*x + c) + sinh(d*x + c))*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))*s inh(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)*c osh(d*x + c)^2 + a + b)*sinh(d*x + c))*sqrt(b/(a + b)) + a + b)/((a + b)*c osh(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))*s inh(d*x + c) + a + b)) + cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)/((a + b)*d*cosh(d*x + c) + (a + b)*d*sinh(d*x + c)), -1/2*(2*sqrt(-b/(a + b))*(cosh(d*x + c) + sinh(d*x + c))*arctan(1/2*((a + b)*cosh(d*x + c)^3 + 3*(a + b)*cosh(d*x + c)*sinh(d*x + c)^2 + (a + b)*si nh(d*x + c)^3 + (a - 3*b)*cosh(d*x + c) + (3*(a + b)*cosh(d*x + c)^2 + a - 3*b)*sinh(d*x + c))*sqrt(-b/(a + b))/b) - 2*sqrt(-b/(a + b))*(cosh(d*x + c) + sinh(d*x + c))*arctan(1/2*((a + b)*cosh(d*x + c) + (a + b)*sinh(d*x + c))*sqrt(-b/(a + b))/b) - cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) - sinh(d*x + c)^2 - 1)/((a + b)*d*cosh(d*x + c) + (a + b)*d*sinh(d*x + c) )]
\[ \int \frac {\sinh (c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int \frac {\sinh {\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {\sinh (c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int { \frac {\sinh \left (d x + c\right )}{b \tanh \left (d x + c\right )^{2} + a} \,d x } \]
1/2*(e^(2*d*x + 2*c) + 1)*e^(-d*x)/(a*d*e^c + b*d*e^c) + 1/2*integrate(4*( b*e^(3*d*x + 3*c) - b*e^(d*x + c))/(a^2 + 2*a*b + b^2 + (a^2*e^(4*c) + 2*a *b*e^(4*c) + b^2*e^(4*c))*e^(4*d*x) + 2*(a^2*e^(2*c) - b^2*e^(2*c))*e^(2*d *x)), x)
\[ \int \frac {\sinh (c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int { \frac {\sinh \left (d x + c\right )}{b \tanh \left (d x + c\right )^{2} + a} \,d x } \]
Time = 2.69 (sec) , antiderivative size = 520, normalized size of antiderivative = 9.81 \[ \int \frac {\sinh (c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {{\mathrm {e}}^{c+d\,x}}{2\,d\,\left (a+b\right )}+\frac {{\mathrm {e}}^{-c-d\,x}}{2\,d\,\left (a+b\right )}-\frac {\sqrt {b}\,\left (2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2\,{\left (a+b\right )}^3}}{2\,\sqrt {b}\,d\,\left (a+b\right )}\right )-2\,\mathrm {atan}\left (\frac {\left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {2\,a\,\sqrt {b}}{d\,{\left (a+b\right )}^2\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}+\frac {4\,\left (2\,a^2\,b^{3/2}\,d+2\,a\,b^{5/2}\,d\right )}{\left (a+b\right )\,\sqrt {-d^2\,{\left (a+b\right )}^3}\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}\right )+\frac {2\,a\,\sqrt {b}\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}}{d\,{\left (a+b\right )}^2\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}\right )\,\left (a^4\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}+b^4\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}+4\,a\,b^3\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}+4\,a^3\,b\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}+6\,a^2\,b^2\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}\right )}{4\,a\,b}\right )\right )}{2\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}} \]
exp(c + d*x)/(2*d*(a + b)) + exp(- c - d*x)/(2*d*(a + b)) - (b^(1/2)*(2*at an((exp(d*x)*exp(c)*(-d^2*(a + b)^3)^(1/2))/(2*b^(1/2)*d*(a + b))) - 2*ata n(((exp(d*x)*exp(c)*((2*a*b^(1/2))/(d*(a + b)^2*(3*a*b^2 + 3*a^2*b + a^3 + b^3)) + (4*(2*a^2*b^(3/2)*d + 2*a*b^(5/2)*d))/((a + b)*(-d^2*(a + b)^3)^( 1/2)*(- a^3*d^2 - b^3*d^2 - 3*a*b^2*d^2 - 3*a^2*b*d^2)^(1/2)*(3*a*b^2 + 3* a^2*b + a^3 + b^3))) + (2*a*b^(1/2)*exp(3*c)*exp(3*d*x))/(d*(a + b)^2*(3*a *b^2 + 3*a^2*b + a^3 + b^3)))*(a^4*(- a^3*d^2 - b^3*d^2 - 3*a*b^2*d^2 - 3* a^2*b*d^2)^(1/2) + b^4*(- a^3*d^2 - b^3*d^2 - 3*a*b^2*d^2 - 3*a^2*b*d^2)^( 1/2) + 4*a*b^3*(- a^3*d^2 - b^3*d^2 - 3*a*b^2*d^2 - 3*a^2*b*d^2)^(1/2) + 4 *a^3*b*(- a^3*d^2 - b^3*d^2 - 3*a*b^2*d^2 - 3*a^2*b*d^2)^(1/2) + 6*a^2*b^2 *(- a^3*d^2 - b^3*d^2 - 3*a*b^2*d^2 - 3*a^2*b*d^2)^(1/2)))/(4*a*b))))/(2*( - a^3*d^2 - b^3*d^2 - 3*a*b^2*d^2 - 3*a^2*b*d^2)^(1/2))