Integrand size = 15, antiderivative size = 89 \[ \int \frac {A+B \tanh (x)}{a+b \sinh (x)} \, dx=\frac {b B \arctan (\sinh (x))}{a^2+b^2}-\frac {2 A \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}}+\frac {a B \log (\cosh (x))}{a^2+b^2}-\frac {a B \log (a+b \sinh (x))}{a^2+b^2} \]
b*B*arctan(sinh(x))/(a^2+b^2)+a*B*ln(cosh(x))/(a^2+b^2)-a*B*ln(a+b*sinh(x) )/(a^2+b^2)-2*A*arctanh((b-a*tanh(1/2*x))/(a^2+b^2)^(1/2))/(a^2+b^2)^(1/2)
Result contains complex when optimal does not.
Time = 0.65 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.67 \[ \int \frac {A+B \tanh (x)}{a+b \sinh (x)} \, dx=\frac {\cosh (x) \left (2 b \sqrt {-a^2-b^2} B \arctan \left (\tanh \left (\frac {x}{2}\right )\right )+2 A \left (a^2+b^2\right ) \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )+a \sqrt {-a^2-b^2} B (\log (\cosh (x))-\log (a+b \sinh (x)))\right ) (A+B \tanh (x))}{(a-i b) (a+i b) \sqrt {-a^2-b^2} (A \cosh (x)+B \sinh (x))} \]
(Cosh[x]*(2*b*Sqrt[-a^2 - b^2]*B*ArcTan[Tanh[x/2]] + 2*A*(a^2 + b^2)*ArcTa n[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]] + a*Sqrt[-a^2 - b^2]*B*(Log[Cosh[x]] - Log[a + b*Sinh[x]]))*(A + B*Tanh[x]))/((a - I*b)*(a + I*b)*Sqrt[-a^2 - b^2]*(A*Cosh[x] + B*Sinh[x]))
Time = 0.38 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3042, 4901, 2009}
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.
| \(\displaystyle \int \frac {A+B \tanh (x)}{a+b \sinh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A-i B \tan (i x)}{a-i b \sin (i x)}dx\) |
| \(\Big \downarrow \) 4901 |
| \(\displaystyle \int \left (\frac {A}{a+b \sinh (x)}+\frac {B \tanh (x)}{a+b \sinh (x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 A \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}}+\frac {b B \arctan (\sinh (x))}{a^2+b^2}-\frac {a B \log (a+b \sinh (x))}{a^2+b^2}+\frac {a B \log (\cosh (x))}{a^2+b^2}\) |
(b*B*ArcTan[Sinh[x]])/(a^2 + b^2) - (2*A*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^ 2 + b^2]])/Sqrt[a^2 + b^2] + (a*B*Log[Cosh[x]])/(a^2 + b^2) - (a*B*Log[a + b*Sinh[x]])/(a^2 + b^2)
3.3.49.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = ExpandTrig[u, x]}, Int[v, x] /; SumQ[v]] /; !InertTrigFreeQ[u]
Time = 0.58 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.31
| method | result | size |
| default | \(\frac {-B a \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a \right )-\frac {2 \left (-a^{2} A -A \,b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}}{a^{2}+b^{2}}+\frac {2 B \left (\frac {a \ln \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )}{2}+b \arctan \left (\tanh \left (\frac {x}{2}\right )\right )\right )}{a^{2}+b^{2}}\) | \(117\) |
| risch | \(-\frac {2 x B a}{a^{2}+b^{2}}-\frac {2 x \,a^{3} B}{-a^{4}-2 a^{2} b^{2}-b^{4}}-\frac {2 x B a \,b^{2}}{-a^{4}-2 a^{2} b^{2}-b^{4}}+\frac {i B \ln \left ({\mathrm e}^{x}+i\right ) b}{a^{2}+b^{2}}+\frac {B \ln \left ({\mathrm e}^{x}+i\right ) a}{a^{2}+b^{2}}-\frac {i B \ln \left ({\mathrm e}^{x}-i\right ) b}{a^{2}+b^{2}}+\frac {B \ln \left ({\mathrm e}^{x}-i\right ) a}{a^{2}+b^{2}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {A a -\sqrt {A^{2} a^{2}+A^{2} b^{2}}}{A b}\right ) a B}{a^{2}+b^{2}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {A a -\sqrt {A^{2} a^{2}+A^{2} b^{2}}}{A b}\right ) \sqrt {A^{2} a^{2}+A^{2} b^{2}}}{a^{2}+b^{2}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {A a +\sqrt {A^{2} a^{2}+A^{2} b^{2}}}{A b}\right ) a B}{a^{2}+b^{2}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {A a +\sqrt {A^{2} a^{2}+A^{2} b^{2}}}{A b}\right ) \sqrt {A^{2} a^{2}+A^{2} b^{2}}}{a^{2}+b^{2}}\) | \(362\) |
2/(a^2+b^2)*(-1/2*B*a*ln(tanh(1/2*x)^2*a-2*b*tanh(1/2*x)-a)-(-A*a^2-A*b^2) /(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tanh(1/2*x)-2*b)/(a^2+b^2)^(1/2)))+2*B/( a^2+b^2)*(1/2*a*ln(1+tanh(1/2*x)^2)+b*arctan(tanh(1/2*x)))
Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (85) = 170\).
Time = 1.19 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.93 \[ \int \frac {A+B \tanh (x)}{a+b \sinh (x)} \, dx=\frac {2 \, B b \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - B a \log \left (\frac {2 \, {\left (b \sinh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + B a \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + \sqrt {a^{2} + b^{2}} A \log \left (\frac {b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) - b}\right )}{a^{2} + b^{2}} \]
(2*B*b*arctan(cosh(x) + sinh(x)) - B*a*log(2*(b*sinh(x) + a)/(cosh(x) - si nh(x))) + B*a*log(2*cosh(x)/(cosh(x) - sinh(x))) + sqrt(a^2 + b^2)*A*log(( b^2*cosh(x)^2 + b^2*sinh(x)^2 + 2*a*b*cosh(x) + 2*a^2 + b^2 + 2*(b^2*cosh( x) + a*b)*sinh(x) - 2*sqrt(a^2 + b^2)*(b*cosh(x) + b*sinh(x) + a))/(b*cosh (x)^2 + b*sinh(x)^2 + 2*a*cosh(x) + 2*(b*cosh(x) + a)*sinh(x) - b)))/(a^2 + b^2)
\[ \int \frac {A+B \tanh (x)}{a+b \sinh (x)} \, dx=\int \frac {A + B \tanh {\left (x \right )}}{a + b \sinh {\left (x \right )}}\, dx \]
Time = 0.33 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.40 \[ \int \frac {A+B \tanh (x)}{a+b \sinh (x)} \, dx=-B {\left (\frac {2 \, b \arctan \left (e^{\left (-x\right )}\right )}{a^{2} + b^{2}} + \frac {a \log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{a^{2} + b^{2}} - \frac {a \log \left (e^{\left (-2 \, x\right )} + 1\right )}{a^{2} + b^{2}}\right )} + \frac {A \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}}} \]
-B*(2*b*arctan(e^(-x))/(a^2 + b^2) + a*log(-2*a*e^(-x) + b*e^(-2*x) - b)/( a^2 + b^2) - a*log(e^(-2*x) + 1)/(a^2 + b^2)) + A*log((b*e^(-x) - a - sqrt (a^2 + b^2))/(b*e^(-x) - a + sqrt(a^2 + b^2)))/sqrt(a^2 + b^2)
Time = 0.29 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.38 \[ \int \frac {A+B \tanh (x)}{a+b \sinh (x)} \, dx=\frac {2 \, B b \arctan \left (e^{x}\right )}{a^{2} + b^{2}} + \frac {B a \log \left (e^{\left (2 \, x\right )} + 1\right )}{a^{2} + b^{2}} - \frac {B a \log \left ({\left | b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b \right |}\right )}{a^{2} + b^{2}} + \frac {A \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}}} \]
2*B*b*arctan(e^x)/(a^2 + b^2) + B*a*log(e^(2*x) + 1)/(a^2 + b^2) - B*a*log (abs(b*e^(2*x) + 2*a*e^x - b))/(a^2 + b^2) + A*log(abs(2*b*e^x + 2*a - 2*s qrt(a^2 + b^2))/abs(2*b*e^x + 2*a + 2*sqrt(a^2 + b^2)))/sqrt(a^2 + b^2)
Time = 9.69 (sec) , antiderivative size = 914, normalized size of antiderivative = 10.27 \[ \int \frac {A+B \tanh (x)}{a+b \sinh (x)} \, dx=-\frac {\ln \left (\frac {32\,B\,\left (A^2\,a\,b+{\mathrm {e}}^x\,A^2\,b^2-4\,{\mathrm {e}}^x\,A\,B\,a^2+2\,A\,B\,a\,b-{\mathrm {e}}^x\,A\,B\,b^2-4\,{\mathrm {e}}^x\,B^2\,a^2+B^2\,a\,b\right )}{b^5}-\frac {\left (\frac {32\,\left (-A^2\,a^2\,b-2\,{\mathrm {e}}^x\,A^2\,a\,b^2+A^2\,b^3+8\,{\mathrm {e}}^x\,A\,B\,a^3-4\,A\,B\,a^2\,b+2\,{\mathrm {e}}^x\,A\,B\,a\,b^2+4\,{\mathrm {e}}^x\,B^2\,a^3-3\,B^2\,a^2\,b-5\,{\mathrm {e}}^x\,B^2\,a\,b^2+B^2\,b^3\right )}{b^5}-\frac {\left (A\,\sqrt {{\left (a^2+b^2\right )}^3}+B\,a^3+B\,a\,b^2\right )\,\left (a\,b^5\,\left (64\,A-128\,B\right )+a^5\,b\,\left (64\,A-128\,B\right )+96\,b^6\,{\mathrm {e}}^x\,\left (A-3\,B\right )+a^3\,b^3\,\left (128\,A-256\,B\right )-128\,{\mathrm {e}}^x\,\left (A-2\,B\right )\,{\left (a^2+b^2\right )}^3+192\,a^2\,b^4\,{\mathrm {e}}^x\,\left (A-3\,B\right )+96\,a^4\,b^2\,{\mathrm {e}}^x\,\left (A-3\,B\right )-96\,A\,a^2\,b\,\sqrt {{\left (a^2+b^2\right )}^3}+128\,A\,a^3\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}+32\,A\,a\,b^2\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}\right )}{b^5\,{\left (a^2+b^2\right )}^3}\right )\,\left (A\,\sqrt {{\left (a^2+b^2\right )}^3}+B\,a^3+B\,a\,b^2\right )}{{\left (a^2+b^2\right )}^2}\right )\,\left (A\,\sqrt {{\left (a^2+b^2\right )}^3}+B\,a^3+B\,a\,b^2\right )}{a^4+2\,a^2\,b^2+b^4}-\frac {\ln \left (\frac {32\,B\,\left (A^2\,a\,b+{\mathrm {e}}^x\,A^2\,b^2-4\,{\mathrm {e}}^x\,A\,B\,a^2+2\,A\,B\,a\,b-{\mathrm {e}}^x\,A\,B\,b^2-4\,{\mathrm {e}}^x\,B^2\,a^2+B^2\,a\,b\right )}{b^5}-\frac {\left (\frac {32\,\left (-A^2\,a^2\,b-2\,{\mathrm {e}}^x\,A^2\,a\,b^2+A^2\,b^3+8\,{\mathrm {e}}^x\,A\,B\,a^3-4\,A\,B\,a^2\,b+2\,{\mathrm {e}}^x\,A\,B\,a\,b^2+4\,{\mathrm {e}}^x\,B^2\,a^3-3\,B^2\,a^2\,b-5\,{\mathrm {e}}^x\,B^2\,a\,b^2+B^2\,b^3\right )}{b^5}-\frac {\left (B\,a^3-A\,\sqrt {{\left (a^2+b^2\right )}^3}+B\,a\,b^2\right )\,\left (a\,b^5\,\left (64\,A-128\,B\right )+a^5\,b\,\left (64\,A-128\,B\right )+96\,b^6\,{\mathrm {e}}^x\,\left (A-3\,B\right )+a^3\,b^3\,\left (128\,A-256\,B\right )-128\,{\mathrm {e}}^x\,\left (A-2\,B\right )\,{\left (a^2+b^2\right )}^3+192\,a^2\,b^4\,{\mathrm {e}}^x\,\left (A-3\,B\right )+96\,a^4\,b^2\,{\mathrm {e}}^x\,\left (A-3\,B\right )+96\,A\,a^2\,b\,\sqrt {{\left (a^2+b^2\right )}^3}-128\,A\,a^3\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}-32\,A\,a\,b^2\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}\right )}{b^5\,{\left (a^2+b^2\right )}^3}\right )\,\left (B\,a^3-A\,\sqrt {{\left (a^2+b^2\right )}^3}+B\,a\,b^2\right )}{{\left (a^2+b^2\right )}^2}\right )\,\left (B\,a^3-A\,\sqrt {{\left (a^2+b^2\right )}^3}+B\,a\,b^2\right )}{a^4+2\,a^2\,b^2+b^4}+\frac {B\,\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}{a-b\,1{}\mathrm {i}}+\frac {B\,\ln \left ({\mathrm {e}}^x-\mathrm {i}\right )\,1{}\mathrm {i}}{-b+a\,1{}\mathrm {i}} \]
(B*log(exp(x) + 1i))/(a - b*1i) - (log((32*B*(A^2*b^2*exp(x) - 4*B^2*a^2*e xp(x) + A^2*a*b + B^2*a*b - 4*A*B*a^2*exp(x) - A*B*b^2*exp(x) + 2*A*B*a*b) )/b^5 - (((32*(A^2*b^3 + B^2*b^3 - A^2*a^2*b - 3*B^2*a^2*b + 4*B^2*a^3*exp (x) - 5*B^2*a*b^2*exp(x) - 4*A*B*a^2*b + 8*A*B*a^3*exp(x) - 2*A^2*a*b^2*ex p(x) + 2*A*B*a*b^2*exp(x)))/b^5 - ((B*a^3 - A*((a^2 + b^2)^3)^(1/2) + B*a* b^2)*(a*b^5*(64*A - 128*B) + a^5*b*(64*A - 128*B) + 96*b^6*exp(x)*(A - 3*B ) + a^3*b^3*(128*A - 256*B) - 128*exp(x)*(A - 2*B)*(a^2 + b^2)^3 + 192*a^2 *b^4*exp(x)*(A - 3*B) + 96*a^4*b^2*exp(x)*(A - 3*B) + 96*A*a^2*b*((a^2 + b ^2)^3)^(1/2) - 128*A*a^3*exp(x)*((a^2 + b^2)^3)^(1/2) - 32*A*a*b^2*exp(x)* ((a^2 + b^2)^3)^(1/2)))/(b^5*(a^2 + b^2)^3))*(B*a^3 - A*((a^2 + b^2)^3)^(1 /2) + B*a*b^2))/(a^2 + b^2)^2)*(B*a^3 - A*((a^2 + b^2)^3)^(1/2) + B*a*b^2) )/(a^4 + b^4 + 2*a^2*b^2) - (log((32*B*(A^2*b^2*exp(x) - 4*B^2*a^2*exp(x) + A^2*a*b + B^2*a*b - 4*A*B*a^2*exp(x) - A*B*b^2*exp(x) + 2*A*B*a*b))/b^5 - (((32*(A^2*b^3 + B^2*b^3 - A^2*a^2*b - 3*B^2*a^2*b + 4*B^2*a^3*exp(x) - 5*B^2*a*b^2*exp(x) - 4*A*B*a^2*b + 8*A*B*a^3*exp(x) - 2*A^2*a*b^2*exp(x) + 2*A*B*a*b^2*exp(x)))/b^5 - ((A*((a^2 + b^2)^3)^(1/2) + B*a^3 + B*a*b^2)*( a*b^5*(64*A - 128*B) + a^5*b*(64*A - 128*B) + 96*b^6*exp(x)*(A - 3*B) + a^ 3*b^3*(128*A - 256*B) - 128*exp(x)*(A - 2*B)*(a^2 + b^2)^3 + 192*a^2*b^4*e xp(x)*(A - 3*B) + 96*a^4*b^2*exp(x)*(A - 3*B) - 96*A*a^2*b*((a^2 + b^2)^3) ^(1/2) + 128*A*a^3*exp(x)*((a^2 + b^2)^3)^(1/2) + 32*A*a*b^2*exp(x)*((a...