Integrand size = 15, antiderivative size = 36 \[ \int \frac {\sinh ^3(x)}{a+b \cosh ^2(x)} \, dx=-\frac {(a+b) \arctan \left (\frac {\sqrt {b} \cosh (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}+\frac {\cosh (x)}{b} \] Output:
-(a+b)*arctan(b^(1/2)*cosh(x)/a^(1/2))/a^(1/2)/b^(3/2)+cosh(x)/b
Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.31 \[ \int \frac {\sinh ^3(x)}{a+b \cosh ^2(x)} \, dx=-\frac {(a+b) \left (\arctan \left (\frac {\sqrt {b}-i \sqrt {a+b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a}}\right )+\arctan \left (\frac {\sqrt {b}+i \sqrt {a+b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a}}\right )\right )}{\sqrt {a} b^{3/2}}+\frac {\cosh (x)}{b} \] Input:
Integrate[Sinh[x]^3/(a + b*Cosh[x]^2),x]
Output:
-(((a + b)*(ArcTan[(Sqrt[b] - I*Sqrt[a + b]*Tanh[x/2])/Sqrt[a]] + ArcTan[( Sqrt[b] + I*Sqrt[a + b]*Tanh[x/2])/Sqrt[a]]))/(Sqrt[a]*b^(3/2))) + Cosh[x] /b
Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 26, 3669, 299, 218}
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.
\(\displaystyle \int \frac {\sinh ^3(x)}{a+b \cosh ^2(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \cos \left (\frac {\pi }{2}+i x\right )^3}{a+b \sin \left (\frac {\pi }{2}+i x\right )^2}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\cos \left (i x+\frac {\pi }{2}\right )^3}{b \sin \left (i x+\frac {\pi }{2}\right )^2+a}dx\) |
\(\Big \downarrow \) 3669 |
\(\displaystyle -\int \frac {1-\cosh ^2(x)}{b \cosh ^2(x)+a}d\cosh (x)\) |
\(\Big \downarrow \) 299 |
\(\displaystyle \frac {\cosh (x)}{b}-\frac {(a+b) \int \frac {1}{b \cosh ^2(x)+a}d\cosh (x)}{b}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\cosh (x)}{b}-\frac {(a+b) \arctan \left (\frac {\sqrt {b} \cosh (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}\) |
Input:
Int[Sinh[x]^3/(a + b*Cosh[x]^2),x]
Output:
-(((a + b)*ArcTan[(Sqrt[b]*Cosh[x])/Sqrt[a]])/(Sqrt[a]*b^(3/2))) + Cosh[x] /b
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)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f S ubst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x] /ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Time = 8.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {\cosh \left (x \right )}{b}+\frac {\left (-a -b \right ) \arctan \left (\frac {b \cosh \left (x \right )}{\sqrt {a b}}\right )}{b \sqrt {a b}}\) | \(34\) |
default | \(\frac {\cosh \left (x \right )}{b}+\frac {\left (-a -b \right ) \arctan \left (\frac {b \cosh \left (x \right )}{\sqrt {a b}}\right )}{b \sqrt {a b}}\) | \(34\) |
risch | \(\frac {{\mathrm e}^{x}}{2 b}+\frac {{\mathrm e}^{-x}}{2 b}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{\sqrt {-a b}}+1\right ) a}{2 \sqrt {-a b}\, b}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{\sqrt {-a b}}+1\right )}{2 \sqrt {-a b}}+\frac {\ln \left ({\mathrm e}^{2 x}-\frac {2 a \,{\mathrm e}^{x}}{\sqrt {-a b}}+1\right ) a}{2 \sqrt {-a b}\, b}+\frac {\ln \left ({\mathrm e}^{2 x}-\frac {2 a \,{\mathrm e}^{x}}{\sqrt {-a b}}+1\right )}{2 \sqrt {-a b}}\) | \(130\) |
Input:
int(sinh(x)^3/(a+b*cosh(x)^2),x,method=_RETURNVERBOSE)
Output:
cosh(x)/b+(-a-b)/b/(a*b)^(1/2)*arctan(b*cosh(x)/(a*b)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (28) = 56\).
Time = 0.11 (sec) , antiderivative size = 419, normalized size of antiderivative = 11.64 \[ \int \frac {\sinh ^3(x)}{a+b \cosh ^2(x)} \, dx =\text {Too large to display} \] Input:
integrate(sinh(x)^3/(a+b*cosh(x)^2),x, algorithm="fricas")
Output:
[1/2*(a*b*cosh(x)^2 + 2*a*b*cosh(x)*sinh(x) + a*b*sinh(x)^2 - sqrt(-a*b)*( (a + b)*cosh(x) + (a + b)*sinh(x))*log((b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^ 3 + b*sinh(x)^4 - 2*(2*a - b)*cosh(x)^2 + 2*(3*b*cosh(x)^2 - 2*a + b)*sinh (x)^2 + 4*(b*cosh(x)^3 - (2*a - b)*cosh(x))*sinh(x) + 4*(cosh(x)^3 + 3*cos h(x)*sinh(x)^2 + sinh(x)^3 + (3*cosh(x)^2 + 1)*sinh(x) + cosh(x))*sqrt(-a* b) + b)/(b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 + 2*(2*a + b)*c osh(x)^2 + 2*(3*b*cosh(x)^2 + 2*a + b)*sinh(x)^2 + 4*(b*cosh(x)^3 + (2*a + b)*cosh(x))*sinh(x) + b)) + a*b)/(a*b^2*cosh(x) + a*b^2*sinh(x)), 1/2*(a* b*cosh(x)^2 + 2*a*b*cosh(x)*sinh(x) + a*b*sinh(x)^2 + 2*sqrt(a*b)*((a + b) *cosh(x) + (a + b)*sinh(x))*arctan(2*sqrt(a*b)/(b*cosh(x) + b*sinh(x))) + 2*sqrt(a*b)*((a + b)*cosh(x) + (a + b)*sinh(x))*arctan(1/2*(b*cosh(x)^3 + 3*b*cosh(x)*sinh(x)^2 + b*sinh(x)^3 + (4*a + b)*cosh(x) + (3*b*cosh(x)^2 + 4*a + b)*sinh(x))*sqrt(a*b)/(a*b)) + a*b)/(a*b^2*cosh(x) + a*b^2*sinh(x)) ]
Timed out. \[ \int \frac {\sinh ^3(x)}{a+b \cosh ^2(x)} \, dx=\text {Timed out} \] Input:
integrate(sinh(x)**3/(a+b*cosh(x)**2),x)
Output:
Timed out
\[ \int \frac {\sinh ^3(x)}{a+b \cosh ^2(x)} \, dx=\int { \frac {\sinh \left (x\right )^{3}}{b \cosh \left (x\right )^{2} + a} \,d x } \] Input:
integrate(sinh(x)^3/(a+b*cosh(x)^2),x, algorithm="maxima")
Output:
1/2*(e^(2*x) + 1)*e^(-x)/b - 1/8*integrate(16*((a + b)*e^(3*x) - (a + b)*e ^x)/(b^2*e^(4*x) + b^2 + 2*(2*a*b + b^2)*e^(2*x)), x)
Exception generated. \[ \int \frac {\sinh ^3(x)}{a+b \cosh ^2(x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(sinh(x)^3/(a+b*cosh(x)^2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Time = 2.66 (sec) , antiderivative size = 257, normalized size of antiderivative = 7.14 \[ \int \frac {\sinh ^3(x)}{a+b \cosh ^2(x)} \, dx=\frac {{\mathrm {e}}^{-x}}{2\,b}+\frac {{\mathrm {e}}^x}{2\,b}+\frac {\left (2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{3\,x}\,\left (a^3\,\sqrt {a\,b^3}+b^3\,\sqrt {a\,b^3}+3\,a\,b^2\,\sqrt {a\,b^3}+3\,a^2\,b\,\sqrt {a\,b^3}\right )}{2\,a\,b\,{\left ({\left (a+b\right )}^2\right )}^{3/2}}+\frac {a\,b^4\,{\mathrm {e}}^x\,\sqrt {a\,b^3}\,\left (\frac {8\,{\left (a^2+2\,a\,b+b^2\right )}^{3/2}}{a\,b^6\,{\left (a+b\right )}^3}+\frac {2\,\left (a^3\,\sqrt {a\,b^3}+b^3\,\sqrt {a\,b^3}+3\,a\,b^2\,\sqrt {a\,b^3}+3\,a^2\,b\,\sqrt {a\,b^3}\right )}{a^2\,b^5\,\sqrt {a\,b^3}\,{\left ({\left (a+b\right )}^2\right )}^{3/2}}\right )}{4}\right )-2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,{\left (a+b\right )}^3\,\sqrt {a\,b^3}}{2\,a\,b\,{\left ({\left (a+b\right )}^2\right )}^{3/2}}\right )\right )\,\sqrt {a^2+2\,a\,b+b^2}}{2\,\sqrt {a\,b^3}} \] Input:
int(sinh(x)^3/(a + b*cosh(x)^2),x)
Output:
exp(-x)/(2*b) + exp(x)/(2*b) + ((2*atan((exp(3*x)*(a^3*(a*b^3)^(1/2) + b^3 *(a*b^3)^(1/2) + 3*a*b^2*(a*b^3)^(1/2) + 3*a^2*b*(a*b^3)^(1/2)))/(2*a*b*(( a + b)^2)^(3/2)) + (a*b^4*exp(x)*(a*b^3)^(1/2)*((8*(2*a*b + a^2 + b^2)^(3/ 2))/(a*b^6*(a + b)^3) + (2*(a^3*(a*b^3)^(1/2) + b^3*(a*b^3)^(1/2) + 3*a*b^ 2*(a*b^3)^(1/2) + 3*a^2*b*(a*b^3)^(1/2)))/(a^2*b^5*(a*b^3)^(1/2)*((a + b)^ 2)^(3/2))))/4) - 2*atan((exp(x)*(a + b)^3*(a*b^3)^(1/2))/(2*a*b*((a + b)^2 )^(3/2))))*(2*a*b + a^2 + b^2)^(1/2))/(2*(a*b^3)^(1/2))
Time = 0.53 (sec) , antiderivative size = 654, normalized size of antiderivative = 18.17 \[ \int \frac {\sinh ^3(x)}{a+b \cosh ^2(x)} \, dx=\frac {-2 e^{x} \sqrt {b}\, \sqrt {a}\, \sqrt {a +b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}\, \mathit {atan} \left (\frac {e^{x} b}{\sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}}\right ) a -2 e^{x} \sqrt {b}\, \sqrt {a}\, \sqrt {a +b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}\, \mathit {atan} \left (\frac {e^{x} b}{\sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}}\right ) b +2 e^{x} \sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}\, \mathit {atan} \left (\frac {e^{x} b}{\sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}}\right ) a^{2}+2 e^{x} \sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}\, \mathit {atan} \left (\frac {e^{x} b}{\sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}}\right ) a b -e^{x} \sqrt {b}\, \sqrt {a}\, \sqrt {a +b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) a -e^{x} \sqrt {b}\, \sqrt {a}\, \sqrt {a +b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) b +e^{x} \sqrt {b}\, \sqrt {a}\, \sqrt {a +b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) a +e^{x} \sqrt {b}\, \sqrt {a}\, \sqrt {a +b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) b -e^{x} \sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) a^{2}-e^{x} \sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) a b +e^{x} \sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) a^{2}+e^{x} \sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) a b +e^{2 x} a \,b^{2}+a \,b^{2}}{2 e^{x} a \,b^{3}} \] Input:
int(sinh(x)^3/(a+b*cosh(x)^2),x)
Output:
( - 2*e**x*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)*atan((e**x*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)))*a - 2*e* *x*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)*atan( (e**x*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)))*b + 2*e**x*sqrt( b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)*atan((e**x*b)/(sqrt(b)*sqrt(2*sqr t(a)*sqrt(a + b) + 2*a + b)))*a**2 + 2*e**x*sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)*atan((e**x*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)))*a*b - e**x*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(a)*sqrt(a + b) - 2 *a - b)*log( - sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b) + e**x*sqrt(b))*a - e **x*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b)*log( - sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b) + e**x*sqrt(b))*b + e**x*sqrt(b)* sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b)*log(sqrt(2*sqrt( a)*sqrt(a + b) - 2*a - b) + e**x*sqrt(b))*a + e**x*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b)*log(sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b) + e**x*sqrt(b))*b - e**x*sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) - 2 *a - b)*log( - sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b) + e**x*sqrt(b))*a**2 - e**x*sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b)*log( - sqrt(2*sqrt(a) *sqrt(a + b) - 2*a - b) + e**x*sqrt(b))*a*b + e**x*sqrt(b)*sqrt(2*sqrt(a)* sqrt(a + b) - 2*a - b)*log(sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b) + e**x*sq rt(b))*a**2 + e**x*sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b)*log(sq...