Integrand size = 16, antiderivative size = 195 \[ \int \frac {\sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=\frac {3 a^2 b \arctan \left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac {\left (2 a^2+b^2\right ) \cosh (x)}{-a^2 b^2+b^4}+\frac {a \left (a^2+2 b^2\right ) \sinh (x)}{b^3 \left (a^2-b^2\right )}-\frac {a^3}{b^3 (a+b)^2 \left (1-\tanh \left (\frac {x}{2}\right )\right )}+\frac {a^3}{(a-b)^2 b^3 \left (1+\tanh \left (\frac {x}{2}\right )\right )}+\frac {2 a^2 \left (a+b \tanh \left (\frac {x}{2}\right )\right )}{\left (a^2-b^2\right )^2 \left (a+2 b \tanh \left (\frac {x}{2}\right )+a \tanh ^2\left (\frac {x}{2}\right )\right )} \]
3*a^2*b*arctan((b*cosh(x)+a*sinh(x))/(a^2-b^2)^(1/2))/(a^2-b^2)^(5/2)+(2*a ^2+b^2)*cosh(x)/(-a^2*b^2+b^4)+a*(a^2+2*b^2)*sinh(x)/b^3/(a^2-b^2)-a^3/b^3 /(a+b)^2/(1-tanh(1/2*x))+a^3/(a-b)^2/b^3/(1+tanh(1/2*x))+2*a^2*(a+b*tanh(1 /2*x))/(a^2-b^2)^2/(a+2*b*tanh(1/2*x)+a*tanh(1/2*x)^2)
Time = 0.45 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.05 \[ \int \frac {\sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=\frac {a \sqrt {a-b} \left (a^3+a^2 b+a b^2+b^3\right ) \cosh ^2(x)-b \cosh (x) \left (-6 a^3 \sqrt {a+b} \arctan \left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right )+(a-b)^{3/2} (a+b)^2 \sinh (x)\right )+a \left (a^2 \sqrt {a-b} (a+b)+6 a b^2 \sqrt {a+b} \arctan \left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right ) \sinh (x)-2 \sqrt {a-b} b^2 (a+b) \sinh ^2(x)\right )}{(a-b)^{5/2} (a+b)^3 (a \cosh (x)+b \sinh (x))} \]
(a*Sqrt[a - b]*(a^3 + a^2*b + a*b^2 + b^3)*Cosh[x]^2 - b*Cosh[x]*(-6*a^3*S qrt[a + b]*ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])] + (a - b)^( 3/2)*(a + b)^2*Sinh[x]) + a*(a^2*Sqrt[a - b]*(a + b) + 6*a*b^2*Sqrt[a + b] *ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])]*Sinh[x] - 2*Sqrt[a - b]*b^2*(a + b)*Sinh[x]^2))/((a - b)^(5/2)*(a + b)^3*(a*Cosh[x] + b*Sinh[x] ))
Result contains complex when optimal does not.
Time = 1.34 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.68, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 26, 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.
\(\displaystyle \int \frac {\sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \sin (i x)^3}{(a \cos (i x)-i b \sin (i x))^2}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\sin (i x)^3}{(a \cos (i x)-i b \sin (i x))^2}dx\) |
\(\Big \downarrow \) 4901 |
\(\displaystyle i \int \left (-\frac {i a^3 \cosh ^3(x)}{b^3 (i a \cosh (x)+i b \sinh (x))^2}-\frac {3 i a^2 \cosh ^2(x)}{b^3 (a \cosh (x)+b \sinh (x))}+\frac {2 i a \cosh (x)}{b^3}-\frac {i \sinh (x)}{b^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle i \left (\frac {i a^3}{b^3 (a+b)^2 \left (1-\tanh \left (\frac {x}{2}\right )\right )}-\frac {i a^3}{b^3 (a-b)^2 \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {2 i a^2 \left (3 a^2-b^2\right ) \arctan \left (\frac {a \tanh \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2}}-\frac {2 i a^2 b \arctan \left (\frac {a \tanh \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac {3 i a^2 \arctan \left (\frac {a \sinh (x)+b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2}}+\frac {3 i a^2 \cosh (x)}{b^2 \left (a^2-b^2\right )}-\frac {2 i a^2 \left (a+b \tanh \left (\frac {x}{2}\right )\right )}{\left (a^2-b^2\right )^2 \left (a \tanh ^2\left (\frac {x}{2}\right )+a+2 b \tanh \left (\frac {x}{2}\right )\right )}-\frac {3 i a^3 \sinh (x)}{b^3 \left (a^2-b^2\right )}+\frac {2 i a \sinh (x)}{b^3}-\frac {i \cosh (x)}{b^2}\right )\) |
I*(((3*I)*a^2*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(b*(a^2 - b ^2)^(3/2)) - ((2*I)*a^2*b*ArcTan[(b + a*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - ((2*I)*a^2*(3*a^2 - b^2)*ArcTan[(b + a*Tanh[x/2])/Sqrt[a^2 - b^2]])/(b*(a^2 - b^2)^(5/2)) - (I*Cosh[x])/b^2 + ((3*I)*a^2*Cosh[x])/(b^ 2*(a^2 - b^2)) + ((2*I)*a*Sinh[x])/b^3 - ((3*I)*a^3*Sinh[x])/(b^3*(a^2 - b ^2)) + (I*a^3)/(b^3*(a + b)^2*(1 - Tanh[x/2])) - (I*a^3)/((a - b)^2*b^3*(1 + Tanh[x/2])) - ((2*I)*a^2*(a + b*Tanh[x/2]))/((a^2 - b^2)^2*(a + 2*b*Tan h[x/2] + a*Tanh[x/2]^2)))
3.7.98.3.1 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[u_, x_Symbol] :> With[{v = ExpandTrig[u, x]}, Int[v, x] /; SumQ[v]] /; !InertTrigFreeQ[u]
Time = 1.04 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.62
method | result | size |
default | \(-\frac {1}{\left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {1}{\left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {4 a^{2} \left (\frac {-\frac {b \tanh \left (\frac {x}{2}\right )}{2}-\frac {a}{2}}{\tanh \left (\frac {x}{2}\right )^{2} a +2 b \tanh \left (\frac {x}{2}\right )+a}-\frac {3 b \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a +b \right )^{2} \left (a -b \right )^{2}}\) | \(121\) |
risch | \(\frac {{\mathrm e}^{x}}{2 a^{2}+4 a b +2 b^{2}}+\frac {{\mathrm e}^{-x}}{2 a^{2}-4 a b +2 b^{2}}+\frac {2 a^{3} {\mathrm e}^{x}}{\left (a -b \right )^{2} \left (a^{2}+2 a b +b^{2}\right ) \left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{2 x}+a -b \right )}-\frac {3 b \,a^{2} \ln \left ({\mathrm e}^{x}-\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}+\frac {3 b \,a^{2} \ln \left ({\mathrm e}^{x}+\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}\) | \(185\) |
-1/(a+b)^2/(tanh(1/2*x)-1)+1/(a-b)^2/(tanh(1/2*x)+1)-4*a^2/(a+b)^2/(a-b)^2 *((-1/2*b*tanh(1/2*x)-1/2*a)/(tanh(1/2*x)^2*a+2*b*tanh(1/2*x)+a)-3/2*b/(a^ 2-b^2)^(1/2)*arctan(1/2*(2*a*tanh(1/2*x)+2*b)/(a^2-b^2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 789 vs. \(2 (180) = 360\).
Time = 0.29 (sec) , antiderivative size = 1633, normalized size of antiderivative = 8.37 \[ \int \frac {\sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=\text {Too large to display} \]
[1/2*(a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5 + (a^5 - a^4*b - 2 *a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^4 + 4*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)*sinh(x)^3 + (a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*sinh(x)^4 + 6*(a^5 - a*b^4)*cosh(x)^2 + 6*(a^5 - a*b^4 + (a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^2)*si nh(x)^2 - 6*((a^3*b + a^2*b^2)*cosh(x)^3 + 3*(a^3*b + a^2*b^2)*cosh(x)*sin h(x)^2 + (a^3*b + a^2*b^2)*sinh(x)^3 + (a^3*b - a^2*b^2)*cosh(x) + (a^3*b - a^2*b^2 + 3*(a^3*b + a^2*b^2)*cosh(x)^2)*sinh(x))*sqrt(-a^2 + b^2)*log(( (a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 - 2*sqrt (-a^2 + b^2)*(cosh(x) + sinh(x)) - a + b)/((a + b)*cosh(x)^2 + 2*(a + b)*c osh(x)*sinh(x) + (a + b)*sinh(x)^2 + a - b)) + 4*((a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^3 + 3*(a^5 - a*b^4)*cosh(x))*sinh(x))/ ((a^7 + a^6*b - 3*a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 + 3*a^2*b^5 - a*b^6 - b^ 7)*cosh(x)^3 + 3*(a^7 + a^6*b - 3*a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 + 3*a^2* b^5 - a*b^6 - b^7)*cosh(x)*sinh(x)^2 + (a^7 + a^6*b - 3*a^5*b^2 - 3*a^4*b^ 3 + 3*a^3*b^4 + 3*a^2*b^5 - a*b^6 - b^7)*sinh(x)^3 + (a^7 - a^6*b - 3*a^5* b^2 + 3*a^4*b^3 + 3*a^3*b^4 - 3*a^2*b^5 - a*b^6 + b^7)*cosh(x) + (a^7 - a^ 6*b - 3*a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 - 3*a^2*b^5 - a*b^6 + b^7 + 3*(a^7 + a^6*b - 3*a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 + 3*a^2*b^5 - a*b^6 - b^7)*co sh(x)^2)*sinh(x)), 1/2*(a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b...
Timed out. \[ \int \frac {\sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Time = 0.27 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.89 \[ \int \frac {\sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=\frac {6 \, a^{2} b \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {a^{2} - b^{2}}}\right )}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} - b^{2}}} + \frac {e^{x}}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} + \frac {5 \, a^{3} e^{\left (2 \, x\right )} + 3 \, a^{2} b e^{\left (2 \, x\right )} + 3 \, a b^{2} e^{\left (2 \, x\right )} + b^{3} e^{\left (2 \, x\right )} + a^{3} + a^{2} b - a b^{2} - b^{3}}{2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (a e^{\left (3 \, x\right )} + b e^{\left (3 \, x\right )} + a e^{x} - b e^{x}\right )}} \]
6*a^2*b*arctan((a*e^x + b*e^x)/sqrt(a^2 - b^2))/((a^4 - 2*a^2*b^2 + b^4)*s qrt(a^2 - b^2)) + 1/2*e^x/(a^2 + 2*a*b + b^2) + 1/2*(5*a^3*e^(2*x) + 3*a^2 *b*e^(2*x) + 3*a*b^2*e^(2*x) + b^3*e^(2*x) + a^3 + a^2*b - a*b^2 - b^3)/(( a^4 - 2*a^2*b^2 + b^4)*(a*e^(3*x) + b*e^(3*x) + a*e^x - b*e^x))
Time = 2.57 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.31 \[ \int \frac {\sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx=\frac {{\mathrm {e}}^{-x}}{2\,{\left (a-b\right )}^2}+\frac {{\mathrm {e}}^x}{2\,{\left (a+b\right )}^2}+\frac {6\,\mathrm {atan}\left (\frac {a^2\,b\,{\mathrm {e}}^x\,\sqrt {a^{10}-5\,a^8\,b^2+10\,a^6\,b^4-10\,a^4\,b^6+5\,a^2\,b^8-b^{10}}}{a^5\,\sqrt {a^4\,b^2}-b^5\,\sqrt {a^4\,b^2}+2\,a^2\,b^3\,\sqrt {a^4\,b^2}-2\,a^3\,b^2\,\sqrt {a^4\,b^2}+a\,b^4\,\sqrt {a^4\,b^2}-a^4\,b\,\sqrt {a^4\,b^2}}\right )\,\sqrt {a^4\,b^2}}{\sqrt {a^{10}-5\,a^8\,b^2+10\,a^6\,b^4-10\,a^4\,b^6+5\,a^2\,b^8-b^{10}}}+\frac {2\,a^3\,{\mathrm {e}}^x}{{\left (a+b\right )}^2\,{\left (a-b\right )}^2\,\left (a-b+{\mathrm {e}}^{2\,x}\,\left (a+b\right )\right )} \]
exp(-x)/(2*(a - b)^2) + exp(x)/(2*(a + b)^2) + (6*atan((a^2*b*exp(x)*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^8*b^2)^(1/2))/(a^5*(a^ 4*b^2)^(1/2) - b^5*(a^4*b^2)^(1/2) + 2*a^2*b^3*(a^4*b^2)^(1/2) - 2*a^3*b^2 *(a^4*b^2)^(1/2) + a*b^4*(a^4*b^2)^(1/2) - a^4*b*(a^4*b^2)^(1/2)))*(a^4*b^ 2)^(1/2))/(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^8*b^2)^ (1/2) + (2*a^3*exp(x))/((a + b)^2*(a - b)^2*(a - b + exp(2*x)*(a + b)))