Integrand size = 12, antiderivative size = 87 \[ \int \frac {x \sinh (x)}{(a+b \cosh (x))^3} \, dx=\frac {a \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} b (a+b)^{3/2}}-\frac {x}{2 b (a+b \cosh (x))^2}-\frac {\sinh (x)}{2 \left (a^2-b^2\right ) (a+b \cosh (x))} \] Output:
a*arctanh((a-b)^(1/2)*tanh(1/2*x)/(a+b)^(1/2))/(a-b)^(3/2)/b/(a+b)^(3/2)-1 /2*x/b/(a+b*cosh(x))^2-1/2*sinh(x)/(a^2-b^2)/(a+b*cosh(x))
Time = 0.17 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00 \[ \int \frac {x \sinh (x)}{(a+b \cosh (x))^3} \, dx=\frac {1}{2} \left (\frac {\frac {2 a \arctan \left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{3/2}}-\frac {x}{(a+b \cosh (x))^2}}{b}-\frac {\sinh (x)}{(a-b) (a+b) (a+b \cosh (x))}\right ) \] Input:
Integrate[(x*Sinh[x])/(a + b*Cosh[x])^3,x]
Output:
(((2*a*ArcTan[((a - b)*Tanh[x/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) - x/(a + b*Cosh[x])^2)/b - Sinh[x]/((a - b)*(a + b)*(a + b*Cosh[x])))/2
Time = 0.43 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.18, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5988, 3042, 3143, 25, 27, 3042, 3138, 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.
\(\displaystyle \int \frac {x \sinh (x)}{(a+b \cosh (x))^3} \, dx\) |
\(\Big \downarrow \) 5988 |
\(\displaystyle \frac {\int \frac {1}{(a+b \cosh (x))^2}dx}{2 b}-\frac {x}{2 b (a+b \cosh (x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {x}{2 b (a+b \cosh (x))^2}+\frac {\int \frac {1}{\left (a+b \sin \left (i x+\frac {\pi }{2}\right )\right )^2}dx}{2 b}\) |
\(\Big \downarrow \) 3143 |
\(\displaystyle \frac {-\frac {\int -\frac {a}{a+b \cosh (x)}dx}{a^2-b^2}-\frac {b \sinh (x)}{\left (a^2-b^2\right ) (a+b \cosh (x))}}{2 b}-\frac {x}{2 b (a+b \cosh (x))^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {\int \frac {a}{a+b \cosh (x)}dx}{a^2-b^2}-\frac {b \sinh (x)}{\left (a^2-b^2\right ) (a+b \cosh (x))}}{2 b}-\frac {x}{2 b (a+b \cosh (x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {a \int \frac {1}{a+b \cosh (x)}dx}{a^2-b^2}-\frac {b \sinh (x)}{\left (a^2-b^2\right ) (a+b \cosh (x))}}{2 b}-\frac {x}{2 b (a+b \cosh (x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {x}{2 b (a+b \cosh (x))^2}+\frac {-\frac {b \sinh (x)}{\left (a^2-b^2\right ) (a+b \cosh (x))}+\frac {a \int \frac {1}{a+b \sin \left (i x+\frac {\pi }{2}\right )}dx}{a^2-b^2}}{2 b}\) |
\(\Big \downarrow \) 3138 |
\(\displaystyle \frac {\frac {2 a \int \frac {1}{-\left ((a-b) \tanh ^2\left (\frac {x}{2}\right )\right )+a+b}d\tanh \left (\frac {x}{2}\right )}{a^2-b^2}-\frac {b \sinh (x)}{\left (a^2-b^2\right ) (a+b \cosh (x))}}{2 b}-\frac {x}{2 b (a+b \cosh (x))^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {2 a \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right )}-\frac {b \sinh (x)}{\left (a^2-b^2\right ) (a+b \cosh (x))}}{2 b}-\frac {x}{2 b (a+b \cosh (x))^2}\) |
Input:
Int[(x*Sinh[x])/(a + b*Cosh[x])^3,x]
Output:
-1/2*x/(b*(a + b*Cosh[x])^2) + ((2*a*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[ a + b]])/(Sqrt[a - b]*Sqrt[a + b]*(a^2 - b^2)) - (b*Sinh[x])/((a^2 - b^2)* (a + b*Cosh[x])))/(2*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*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[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos [c + d*x]*((a + b*Sin[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 - b^2))), x] + Simp [1/((n + 1)*(a^2 - b^2)) Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n + 1) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
Int[(Cosh[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.)*((e_.) + (f_.)*(x_))^(m_. )*Sinh[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(e + f*x)^m*((a + b*Cosh[c + d*x])^(n + 1)/(b*d*(n + 1))), x] - Simp[f*(m/(b*d*(n + 1))) Int[(e + f*x) ^(m - 1)*(a + b*Cosh[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(230\) vs. \(2(73)=146\).
Time = 5.08 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.66
method | result | size |
risch | \(-\frac {2 a^{2} x \,{\mathrm e}^{2 x}-a b \,{\mathrm e}^{3 x}-2 b^{2} x \,{\mathrm e}^{2 x}-2 a^{2} {\mathrm e}^{2 x}-b^{2} {\mathrm e}^{2 x}-3 b \,{\mathrm e}^{x} a -b^{2}}{b \left ({\mathrm e}^{2 x} b +2 a \,{\mathrm e}^{x}+b \right )^{2} \left (a^{2}-b^{2}\right )}+\frac {a \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) b}-\frac {a \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) b}\) | \(231\) |
Input:
int(x*sinh(x)/(a+b*cosh(x))^3,x,method=_RETURNVERBOSE)
Output:
-1/b*(2*a^2*x*exp(x)^2-a*b*exp(x)^3-2*b^2*x*exp(x)^2-2*a^2*exp(x)^2-b^2*ex p(x)^2-3*b*exp(x)*a-b^2)/(exp(x)^2*b+2*a*exp(x)+b)^2/(a^2-b^2)+1/2/(a^2-b^ 2)^(1/2)*a/(a+b)/(a-b)/b*ln(exp(x)+(a*(a^2-b^2)^(1/2)-a^2+b^2)/b/(a^2-b^2) ^(1/2))-1/2/(a^2-b^2)^(1/2)*a/(a+b)/(a-b)/b*ln(exp(x)+(a*(a^2-b^2)^(1/2)+a ^2-b^2)/b/(a^2-b^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 808 vs. \(2 (73) = 146\).
Time = 0.10 (sec) , antiderivative size = 1692, normalized size of antiderivative = 19.45 \[ \int \frac {x \sinh (x)}{(a+b \cosh (x))^3} \, dx=\text {Too large to display} \] Input:
integrate(x*sinh(x)/(a+b*cosh(x))^3,x, algorithm="fricas")
Output:
[1/2*(2*a^2*b^2 - 2*b^4 + 2*(a^3*b - a*b^3)*cosh(x)^3 + 2*(a^3*b - a*b^3)* sinh(x)^3 + 2*(2*a^4 - a^2*b^2 - b^4 - 2*(a^4 - 2*a^2*b^2 + b^4)*x)*cosh(x )^2 + 2*(2*a^4 - a^2*b^2 - b^4 - 2*(a^4 - 2*a^2*b^2 + b^4)*x + 3*(a^3*b - a*b^3)*cosh(x))*sinh(x)^2 - (a*b^2*cosh(x)^4 + a*b^2*sinh(x)^4 + 4*a^2*b*c osh(x)^3 + 4*a^2*b*cosh(x) + 4*(a*b^2*cosh(x) + a^2*b)*sinh(x)^3 + a*b^2 + 2*(2*a^3 + a*b^2)*cosh(x)^2 + 2*(3*a*b^2*cosh(x)^2 + 6*a^2*b*cosh(x) + 2* a^3 + a*b^2)*sinh(x)^2 + 4*(a*b^2*cosh(x)^3 + 3*a^2*b*cosh(x)^2 + a^2*b + (2*a^3 + a*b^2)*cosh(x))*sinh(x))*sqrt(a^2 - b^2)*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)) + 6*(a^3*b - a*b^3)*cosh( x) + 2*(3*a^3*b - 3*a*b^3 + 3*(a^3*b - a*b^3)*cosh(x)^2 + 2*(2*a^4 - a^2*b ^2 - b^4 - 2*(a^4 - 2*a^2*b^2 + b^4)*x)*cosh(x))*sinh(x))/(a^4*b^3 - 2*a^2 *b^5 + b^7 + (a^4*b^3 - 2*a^2*b^5 + b^7)*cosh(x)^4 + (a^4*b^3 - 2*a^2*b^5 + b^7)*sinh(x)^4 + 4*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*cosh(x)^3 + 4*(a^5*b^2 - 2*a^3*b^4 + a*b^6 + (a^4*b^3 - 2*a^2*b^5 + b^7)*cosh(x))*sinh(x)^3 + 2*( 2*a^6*b - 3*a^4*b^3 + b^7)*cosh(x)^2 + 2*(2*a^6*b - 3*a^4*b^3 + b^7 + 3*(a ^4*b^3 - 2*a^2*b^5 + b^7)*cosh(x)^2 + 6*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*cosh (x))*sinh(x)^2 + 4*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*cosh(x) + 4*(a^5*b^2 - 2* a^3*b^4 + a*b^6 + (a^4*b^3 - 2*a^2*b^5 + b^7)*cosh(x)^3 + 3*(a^5*b^2 - ...
Timed out. \[ \int \frac {x \sinh (x)}{(a+b \cosh (x))^3} \, dx=\text {Timed out} \] Input:
integrate(x*sinh(x)/(a+b*cosh(x))**3,x)
Output:
Timed out
Exception generated. \[ \int \frac {x \sinh (x)}{(a+b \cosh (x))^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x*sinh(x)/(a+b*cosh(x))^3,x, algorithm="maxima")
Output:
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*a^2-4*b^2>0)', see `assume?` f or more de
\[ \int \frac {x \sinh (x)}{(a+b \cosh (x))^3} \, dx=\int { \frac {x \sinh \left (x\right )}{{\left (b \cosh \left (x\right ) + a\right )}^{3}} \,d x } \] Input:
integrate(x*sinh(x)/(a+b*cosh(x))^3,x, algorithm="giac")
Output:
integrate(x*sinh(x)/(b*cosh(x) + a)^3, x)
Timed out. \[ \int \frac {x \sinh (x)}{(a+b \cosh (x))^3} \, dx=\int \frac {x\,\mathrm {sinh}\left (x\right )}{{\left (a+b\,\mathrm {cosh}\left (x\right )\right )}^3} \,d x \] Input:
int((x*sinh(x))/(a + b*cosh(x))^3,x)
Output:
int((x*sinh(x))/(a + b*cosh(x))^3, x)
Time = 0.24 (sec) , antiderivative size = 924, normalized size of antiderivative = 10.62 \[ \int \frac {x \sinh (x)}{(a+b \cosh (x))^3} \, dx =\text {Too large to display} \] Input:
int(x*sinh(x)/(a+b*cosh(x))^3,x)
Output:
( - 2*e**(2*x)*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2) )*cosh(x)**2*a*b**3 - 4*e**x*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2))*cosh(x)**2*a**2*b**2 - 2*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2))*cosh(x)**2*a*b**3 - 4*e**(2*x)*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2))*cosh(x)*a**2*b**2 - 8*e**x*s qrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2))*cosh(x)*a**3*b - 4*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2))*cosh(x)* a**2*b**2 - 2*e**(2*x)*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2))*a**3*b - 4*e**x*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a **2 + b**2))*a**4 - 2*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2))*a**3*b - e**(2*x)*cosh(x)**2*a**2*b**3 + e**(2*x)*cosh(x)**2*b**5 + cosh(x)**2*a**2*b**3 - cosh(x)**2*b**5 - 2*e**(2*x)*cosh(x)*a**3*b**2 + 2*e**(2*x)*cosh(x)*a*b**4 + 2*cosh(x)*a**3*b**2 - 2*cosh(x)*a*b**4 - e**( 2*x)*a**4*b*x - e**(2*x)*a**4*b + 2*e**(2*x)*a**2*b**3*x + e**(2*x)*a**2*b **3 - e**(2*x)*b**5*x - 2*e**x*a**5*x + 4*e**x*a**3*b**2*x - 2*e**x*a*b**4 *x - a**4*b*x + a**4*b + 2*a**2*b**3*x - a**2*b**3 - b**5*x)/(2*b*(e**(2*x )*cosh(x)**2*a**4*b**3 - 2*e**(2*x)*cosh(x)**2*a**2*b**5 + e**(2*x)*cosh(x )**2*b**7 + 2*e**x*cosh(x)**2*a**5*b**2 - 4*e**x*cosh(x)**2*a**3*b**4 + 2* e**x*cosh(x)**2*a*b**6 + cosh(x)**2*a**4*b**3 - 2*cosh(x)**2*a**2*b**5 + c osh(x)**2*b**7 + 2*e**(2*x)*cosh(x)*a**5*b**2 - 4*e**(2*x)*cosh(x)*a**3...