Integrand size = 15, antiderivative size = 54 \[ \int \frac {\sinh ^5(x)}{a+b \cosh ^2(x)} \, dx=\frac {(a+b)^2 \arctan \left (\frac {\sqrt {b} \cosh (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2}}-\frac {(a+2 b) \cosh (x)}{b^2}+\frac {\cosh ^3(x)}{3 b} \] Output:
(a+b)^2*arctan(b^(1/2)*cosh(x)/a^(1/2))/a^(1/2)/b^(5/2)-(a+2*b)*cosh(x)/b^ 2+1/3*cosh(x)^3/b
Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.22 \[ \int \frac {\sinh ^5(x)}{a+b \cosh ^2(x)} \, dx=\frac {\frac {12 (a+b)^2 \arctan \left (\frac {\sqrt {b}-i \sqrt {a+b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {12 (a+b)^2 \arctan \left (\frac {\sqrt {b}+i \sqrt {a+b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a}}\right )}{\sqrt {a}}-3 \sqrt {b} (4 a+7 b) \cosh (x)+b^{3/2} \cosh (3 x)}{12 b^{5/2}} \] Input:
Integrate[Sinh[x]^5/(a + b*Cosh[x]^2),x]
Output:
((12*(a + b)^2*ArcTan[(Sqrt[b] - I*Sqrt[a + b]*Tanh[x/2])/Sqrt[a]])/Sqrt[a ] + (12*(a + b)^2*ArcTan[(Sqrt[b] + I*Sqrt[a + b]*Tanh[x/2])/Sqrt[a]])/Sqr t[a] - 3*Sqrt[b]*(4*a + 7*b)*Cosh[x] + b^(3/2)*Cosh[3*x])/(12*b^(5/2))
Time = 0.29 (sec) , antiderivative size = 54, 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, 300, 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.
\(\displaystyle \int \frac {\sinh ^5(x)}{a+b \cosh ^2(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \cos \left (\frac {\pi }{2}+i x\right )^5}{a+b \sin \left (\frac {\pi }{2}+i x\right )^2}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\cos \left (i x+\frac {\pi }{2}\right )^5}{b \sin \left (i x+\frac {\pi }{2}\right )^2+a}dx\) |
\(\Big \downarrow \) 3669 |
\(\displaystyle \int \frac {\left (1-\cosh ^2(x)\right )^2}{a+b \cosh ^2(x)}d\cosh (x)\) |
\(\Big \downarrow \) 300 |
\(\displaystyle \int \left (\frac {a^2+2 a b+b^2}{b^2 \left (a+b \cosh ^2(x)\right )}-\frac {a+2 b}{b^2}+\frac {\cosh ^2(x)}{b}\right )d\cosh (x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(a+b)^2 \arctan \left (\frac {\sqrt {b} \cosh (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2}}-\frac {(a+2 b) \cosh (x)}{b^2}+\frac {\cosh ^3(x)}{3 b}\) |
Input:
Int[Sinh[x]^5/(a + b*Cosh[x]^2),x]
Output:
((a + b)^2*ArcTan[(Sqrt[b]*Cosh[x])/Sqrt[a]])/(Sqrt[a]*b^(5/2)) - ((a + 2* b)*Cosh[x])/b^2 + Cosh[x]^3/(3*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)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int [PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c , d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
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 = 120.99 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(-\frac {-\frac {b \cosh \left (x \right )^{3}}{3}+\cosh \left (x \right ) a +2 b \cosh \left (x \right )}{b^{2}}+\frac {\left (a^{2}+2 a b +b^{2}\right ) \arctan \left (\frac {b \cosh \left (x \right )}{\sqrt {a b}}\right )}{b^{2} \sqrt {a b}}\) | \(54\) |
default | \(-\frac {-\frac {b \cosh \left (x \right )^{3}}{3}+\cosh \left (x \right ) a +2 b \cosh \left (x \right )}{b^{2}}+\frac {\left (a^{2}+2 a b +b^{2}\right ) \arctan \left (\frac {b \cosh \left (x \right )}{\sqrt {a b}}\right )}{b^{2} \sqrt {a b}}\) | \(54\) |
risch | \(\frac {{\mathrm e}^{3 x}}{24 b}-\frac {a \,{\mathrm e}^{x}}{2 b^{2}}-\frac {7 \,{\mathrm e}^{x}}{8 b}-\frac {{\mathrm e}^{-x} a}{2 b^{2}}-\frac {7 \,{\mathrm e}^{-x}}{8 b}+\frac {{\mathrm e}^{-3 x}}{24 b}-\frac {\ln \left ({\mathrm e}^{2 x}-\frac {2 a \,{\mathrm e}^{x}}{\sqrt {-a b}}+1\right ) a^{2}}{2 \sqrt {-a b}\, b^{2}}-\frac {\ln \left ({\mathrm e}^{2 x}-\frac {2 a \,{\mathrm e}^{x}}{\sqrt {-a b}}+1\right ) a}{\sqrt {-a b}\, b}-\frac {\ln \left ({\mathrm e}^{2 x}-\frac {2 a \,{\mathrm e}^{x}}{\sqrt {-a b}}+1\right )}{2 \sqrt {-a b}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{\sqrt {-a b}}+1\right ) a^{2}}{2 \sqrt {-a b}\, b^{2}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{\sqrt {-a b}}+1\right ) a}{\sqrt {-a b}\, b}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{\sqrt {-a b}}+1\right )}{2 \sqrt {-a b}}\) | \(229\) |
Input:
int(sinh(x)^5/(a+b*cosh(x)^2),x,method=_RETURNVERBOSE)
Output:
-1/b^2*(-1/3*b*cosh(x)^3+cosh(x)*a+2*b*cosh(x))+(a^2+2*a*b+b^2)/b^2/(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. 516 vs. \(2 (44) = 88\).
Time = 0.11 (sec) , antiderivative size = 1067, normalized size of antiderivative = 19.76 \[ \int \frac {\sinh ^5(x)}{a+b \cosh ^2(x)} \, dx=\text {Too large to display} \] Input:
integrate(sinh(x)^5/(a+b*cosh(x)^2),x, algorithm="fricas")
Output:
[1/24*(a*b^2*cosh(x)^6 + 6*a*b^2*cosh(x)*sinh(x)^5 + a*b^2*sinh(x)^6 - 3*( 4*a^2*b + 7*a*b^2)*cosh(x)^4 + 3*(5*a*b^2*cosh(x)^2 - 4*a^2*b - 7*a*b^2)*s inh(x)^4 + 4*(5*a*b^2*cosh(x)^3 - 3*(4*a^2*b + 7*a*b^2)*cosh(x))*sinh(x)^3 + a*b^2 - 3*(4*a^2*b + 7*a*b^2)*cosh(x)^2 + 3*(5*a*b^2*cosh(x)^4 - 4*a^2* b - 7*a*b^2 - 6*(4*a^2*b + 7*a*b^2)*cosh(x)^2)*sinh(x)^2 - 12*((a^2 + 2*a* b + b^2)*cosh(x)^3 + 3*(a^2 + 2*a*b + b^2)*cosh(x)^2*sinh(x) + 3*(a^2 + 2* a*b + b^2)*cosh(x)*sinh(x)^2 + (a^2 + 2*a*b + b^2)*sinh(x)^3)*sqrt(-a*b)*l og((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)*c osh(x))*sinh(x) - 4*(cosh(x)^3 + 3*cosh(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)*s inh(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) + b)) + 6*(a*b^ 2*cosh(x)^5 - 2*(4*a^2*b + 7*a*b^2)*cosh(x)^3 - (4*a^2*b + 7*a*b^2)*cosh(x ))*sinh(x))/(a*b^3*cosh(x)^3 + 3*a*b^3*cosh(x)^2*sinh(x) + 3*a*b^3*cosh(x) *sinh(x)^2 + a*b^3*sinh(x)^3), 1/24*(a*b^2*cosh(x)^6 + 6*a*b^2*cosh(x)*sin h(x)^5 + a*b^2*sinh(x)^6 - 3*(4*a^2*b + 7*a*b^2)*cosh(x)^4 + 3*(5*a*b^2*co sh(x)^2 - 4*a^2*b - 7*a*b^2)*sinh(x)^4 + 4*(5*a*b^2*cosh(x)^3 - 3*(4*a^2*b + 7*a*b^2)*cosh(x))*sinh(x)^3 + a*b^2 - 3*(4*a^2*b + 7*a*b^2)*cosh(x)^2 + 3*(5*a*b^2*cosh(x)^4 - 4*a^2*b - 7*a*b^2 - 6*(4*a^2*b + 7*a*b^2)*cosh(...
Timed out. \[ \int \frac {\sinh ^5(x)}{a+b \cosh ^2(x)} \, dx=\text {Timed out} \] Input:
integrate(sinh(x)**5/(a+b*cosh(x)**2),x)
Output:
Timed out
\[ \int \frac {\sinh ^5(x)}{a+b \cosh ^2(x)} \, dx=\int { \frac {\sinh \left (x\right )^{5}}{b \cosh \left (x\right )^{2} + a} \,d x } \] Input:
integrate(sinh(x)^5/(a+b*cosh(x)^2),x, algorithm="maxima")
Output:
1/24*(b*e^(6*x) - 3*(4*a + 7*b)*e^(4*x) - 3*(4*a + 7*b)*e^(2*x) + b)*e^(-3 *x)/b^2 + 1/32*integrate(64*((a^2 + 2*a*b + b^2)*e^(3*x) - (a^2 + 2*a*b + b^2)*e^x)/(b^3*e^(4*x) + b^3 + 2*(2*a*b^2 + b^3)*e^(2*x)), x)
Exception generated. \[ \int \frac {\sinh ^5(x)}{a+b \cosh ^2(x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(sinh(x)^5/(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 = 2.40 (sec) , antiderivative size = 548, normalized size of antiderivative = 10.15 \[ \int \frac {\sinh ^5(x)}{a+b \cosh ^2(x)} \, dx=\frac {{\mathrm {e}}^{-3\,x}}{24\,b}+\frac {{\mathrm {e}}^{3\,x}}{24\,b}-\frac {{\mathrm {e}}^{-x}\,\left (4\,a+7\,b\right )}{8\,b^2}+\frac {\left (2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,{\left (a+b\right )}^2\,\sqrt {a\,b^5}}{2\,a\,b^2\,\sqrt {{\left (a+b\right )}^4}}\right )-2\,\mathrm {atan}\left (\frac {a\,b^6\,{\mathrm {e}}^x\,\left (\frac {4\,\left (6\,a^2\,b^4\,\sqrt {a^4+4\,a^3\,b+6\,a^2\,b^2+4\,a\,b^3+b^4}+6\,a^3\,b^3\,\sqrt {a^4+4\,a^3\,b+6\,a^2\,b^2+4\,a\,b^3+b^4}+2\,a^4\,b^2\,\sqrt {a^4+4\,a^3\,b+6\,a^2\,b^2+4\,a\,b^3+b^4}+2\,a\,b^5\,\sqrt {a^4+4\,a^3\,b+6\,a^2\,b^2+4\,a\,b^3+b^4}\right )}{a^2\,b^{11}\,{\left (a+b\right )}^2}+\frac {2\,\left (a^5\,\sqrt {a\,b^5}+b^5\,\sqrt {a\,b^5}+5\,a\,b^4\,\sqrt {a\,b^5}+5\,a^4\,b\,\sqrt {a\,b^5}+10\,a^2\,b^3\,\sqrt {a\,b^5}+10\,a^3\,b^2\,\sqrt {a\,b^5}\right )}{a^2\,b^8\,\sqrt {a\,b^5}\,\sqrt {{\left (a+b\right )}^4}}\right )\,\sqrt {a\,b^5}}{4\,a^3+12\,a^2\,b+12\,a\,b^2+4\,b^3}+\frac {2\,{\mathrm {e}}^{3\,x}\,\left (a^5\,\sqrt {a\,b^5}+b^5\,\sqrt {a\,b^5}+5\,a\,b^4\,\sqrt {a\,b^5}+5\,a^4\,b\,\sqrt {a\,b^5}+10\,a^2\,b^3\,\sqrt {a\,b^5}+10\,a^3\,b^2\,\sqrt {a\,b^5}\right )}{a\,b^2\,\sqrt {{\left (a+b\right )}^4}\,\left (4\,a^3+12\,a^2\,b+12\,a\,b^2+4\,b^3\right )}\right )\right )\,\sqrt {a^4+4\,a^3\,b+6\,a^2\,b^2+4\,a\,b^3+b^4}}{2\,\sqrt {a\,b^5}}-\frac {{\mathrm {e}}^x\,\left (4\,a+7\,b\right )}{8\,b^2} \] Input:
int(sinh(x)^5/(a + b*cosh(x)^2),x)
Output:
exp(-3*x)/(24*b) + exp(3*x)/(24*b) - (exp(-x)*(4*a + 7*b))/(8*b^2) + ((2*a tan((exp(x)*(a + b)^2*(a*b^5)^(1/2))/(2*a*b^2*((a + b)^4)^(1/2))) - 2*atan ((a*b^6*exp(x)*((4*(6*a^2*b^4*(4*a*b^3 + 4*a^3*b + a^4 + b^4 + 6*a^2*b^2)^ (1/2) + 6*a^3*b^3*(4*a*b^3 + 4*a^3*b + a^4 + b^4 + 6*a^2*b^2)^(1/2) + 2*a^ 4*b^2*(4*a*b^3 + 4*a^3*b + a^4 + b^4 + 6*a^2*b^2)^(1/2) + 2*a*b^5*(4*a*b^3 + 4*a^3*b + a^4 + b^4 + 6*a^2*b^2)^(1/2)))/(a^2*b^11*(a + b)^2) + (2*(a^5 *(a*b^5)^(1/2) + b^5*(a*b^5)^(1/2) + 5*a*b^4*(a*b^5)^(1/2) + 5*a^4*b*(a*b^ 5)^(1/2) + 10*a^2*b^3*(a*b^5)^(1/2) + 10*a^3*b^2*(a*b^5)^(1/2)))/(a^2*b^8* (a*b^5)^(1/2)*((a + b)^4)^(1/2)))*(a*b^5)^(1/2))/(12*a*b^2 + 12*a^2*b + 4* a^3 + 4*b^3) + (2*exp(3*x)*(a^5*(a*b^5)^(1/2) + b^5*(a*b^5)^(1/2) + 5*a*b^ 4*(a*b^5)^(1/2) + 5*a^4*b*(a*b^5)^(1/2) + 10*a^2*b^3*(a*b^5)^(1/2) + 10*a^ 3*b^2*(a*b^5)^(1/2)))/(a*b^2*((a + b)^4)^(1/2)*(12*a*b^2 + 12*a^2*b + 4*a^ 3 + 4*b^3))))*(4*a*b^3 + 4*a^3*b + a^4 + b^4 + 6*a^2*b^2)^(1/2))/(2*(a*b^5 )^(1/2)) - (exp(x)*(4*a + 7*b))/(8*b^2)
Time = 0.37 (sec) , antiderivative size = 1084, normalized size of antiderivative = 20.07 \[ \int \frac {\sinh ^5(x)}{a+b \cosh ^2(x)} \, dx =\text {Too large to display} \] Input:
int(sinh(x)^5/(a+b*cosh(x)^2),x)
Output:
(24*e**(3*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 + 48*e**(3*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*b + 24*e**(3*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 - 24*e**(3*x)*sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)*atan((e**x*b)/(s qrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)))*a**3 - 48*e**(3*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**2*b - 24*e**(3*x)*sqrt(b)*sqrt(2*sqrt(a)*sqr t(a + b) + 2*a + b)*atan((e**x*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2* a + b)))*a*b**2 + 12*e**(3*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(a)*s qrt(a + b) - 2*a - b)*log( - sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b) + e**x* sqrt(b))*a**2 + 24*e**(3*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(a)*sqr t(a + b) - 2*a - b)*log( - sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b) + e**x*sq rt(b))*a*b + 12*e**(3*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**2 - 12*e**(3*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 **2 - 24*e**(3*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(a)*sqrt(a + b...