Integrand size = 15, antiderivative size = 59 \[ \int \frac {\text {sech}^3(x)}{a+b \cosh ^2(x)} \, dx=\frac {(a-2 b) \arctan (\sinh (x))}{2 a^2}+\frac {b^{3/2} \arctan \left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{a^2 \sqrt {a+b}}+\frac {\text {sech}(x) \tanh (x)}{2 a} \] Output:
1/2*(a-2*b)*arctan(sinh(x))/a^2+b^(3/2)*arctan(b^(1/2)*sinh(x)/(a+b)^(1/2) )/a^2/(a+b)^(1/2)+1/2*sech(x)*tanh(x)/a
Time = 0.14 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.98 \[ \int \frac {\text {sech}^3(x)}{a+b \cosh ^2(x)} \, dx=\frac {-\frac {2 b^{3/2} \arctan \left (\frac {\sqrt {a+b} \text {csch}(x)}{\sqrt {b}}\right )}{\sqrt {a+b}}+2 (a-2 b) \arctan \left (\tanh \left (\frac {x}{2}\right )\right )+a \text {sech}(x) \tanh (x)}{2 a^2} \] Input:
Integrate[Sech[x]^3/(a + b*Cosh[x]^2),x]
Output:
((-2*b^(3/2)*ArcTan[(Sqrt[a + b]*Csch[x])/Sqrt[b]])/Sqrt[a + b] + 2*(a - 2 *b)*ArcTan[Tanh[x/2]] + a*Sech[x]*Tanh[x])/(2*a^2)
Time = 0.29 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.20, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3042, 3665, 316, 25, 397, 216, 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.
| \(\displaystyle \int \frac {\text {sech}^3(x)}{a+b \cosh ^2(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin \left (\frac {\pi }{2}+i x\right )^3 \left (a+b \sin \left (\frac {\pi }{2}+i x\right )^2\right )}dx\) |
| \(\Big \downarrow \) 3665 |
| \(\displaystyle \int \frac {1}{\left (\sinh ^2(x)+1\right )^2 \left (a+b \sinh ^2(x)+b\right )}d\sinh (x)\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\sinh (x)}{2 a \left (\sinh ^2(x)+1\right )}-\frac {\int -\frac {b \sinh ^2(x)+a-b}{\left (\sinh ^2(x)+1\right ) \left (b \sinh ^2(x)+a+b\right )}d\sinh (x)}{2 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {b \sinh ^2(x)+a-b}{\left (\sinh ^2(x)+1\right ) \left (b \sinh ^2(x)+a+b\right )}d\sinh (x)}{2 a}+\frac {\sinh (x)}{2 a \left (\sinh ^2(x)+1\right )}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {2 b^2 \int \frac {1}{b \sinh ^2(x)+a+b}d\sinh (x)}{a}+\frac {(a-2 b) \int \frac {1}{\sinh ^2(x)+1}d\sinh (x)}{a}}{2 a}+\frac {\sinh (x)}{2 a \left (\sinh ^2(x)+1\right )}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {2 b^2 \int \frac {1}{b \sinh ^2(x)+a+b}d\sinh (x)}{a}+\frac {(a-2 b) \arctan (\sinh (x))}{a}}{2 a}+\frac {\sinh (x)}{2 a \left (\sinh ^2(x)+1\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {2 b^{3/2} \arctan \left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{a \sqrt {a+b}}+\frac {(a-2 b) \arctan (\sinh (x))}{a}}{2 a}+\frac {\sinh (x)}{2 a \left (\sinh ^2(x)+1\right )}\) |
Input:
Int[Sech[x]^3/(a + b*Cosh[x]^2),x]
Output:
(((a - 2*b)*ArcTan[Sinh[x]])/a + (2*b^(3/2)*ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[ a + b]])/(a*Sqrt[a + b]))/(2*a) + Sinh[x]/(2*a*(1 + Sinh[x]^2))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Leaf count of result is larger than twice the leaf count of optimal. \(122\) vs. \(2(47)=94\).
Time = 1.64 (sec) , antiderivative size = 123, normalized size of antiderivative = 2.08
| method | result | size |
| default | \(\frac {\frac {2 \left (-\frac {a \tanh \left (\frac {x}{2}\right )^{3}}{2}+\frac {a \tanh \left (\frac {x}{2}\right )}{2}\right )}{\left (\tanh \left (\frac {x}{2}\right )^{2}+1\right )^{2}}+\left (a -2 b \right ) \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{a^{2}}+\frac {2 b^{2} \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )+2 \sqrt {a}}{2 \sqrt {b}}\right )}{2 \sqrt {a +b}\, \sqrt {b}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )-2 \sqrt {a}}{2 \sqrt {b}}\right )}{2 \sqrt {a +b}\, \sqrt {b}}\right )}{a^{2}}\) | \(123\) |
| risch | \(\frac {{\mathrm e}^{x} \left ({\mathrm e}^{2 x}-1\right )}{\left ({\mathrm e}^{2 x}+1\right )^{2} a}+\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{2 a}-\frac {i b \ln \left ({\mathrm e}^{x}+i\right )}{a^{2}}-\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{2 a}+\frac {i b \ln \left ({\mathrm e}^{x}-i\right )}{a^{2}}+\frac {\sqrt {-\left (a +b \right ) b}\, b \ln \left ({\mathrm e}^{2 x}+\frac {2 \sqrt {-\left (a +b \right ) b}\, {\mathrm e}^{x}}{b}-1\right )}{2 \left (a +b \right ) a^{2}}-\frac {\sqrt {-\left (a +b \right ) b}\, b \ln \left ({\mathrm e}^{2 x}-\frac {2 \sqrt {-\left (a +b \right ) b}\, {\mathrm e}^{x}}{b}-1\right )}{2 \left (a +b \right ) a^{2}}\) | \(154\) |
Input:
int(sech(x)^3/(a+b*cosh(x)^2),x,method=_RETURNVERBOSE)
Output:
2/a^2*((-1/2*a*tanh(1/2*x)^3+1/2*a*tanh(1/2*x))/(tanh(1/2*x)^2+1)^2+1/2*(a -2*b)*arctan(tanh(1/2*x)))+2*b^2/a^2*(1/2/(a+b)^(1/2)/b^(1/2)*arctan(1/2*( 2*(a+b)^(1/2)*tanh(1/2*x)+2*a^(1/2))/b^(1/2))+1/2/(a+b)^(1/2)/b^(1/2)*arct an(1/2*(2*(a+b)^(1/2)*tanh(1/2*x)-2*a^(1/2))/b^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (47) = 94\).
Time = 0.14 (sec) , antiderivative size = 963, normalized size of antiderivative = 16.32 \[ \int \frac {\text {sech}^3(x)}{a+b \cosh ^2(x)} \, dx=\text {Too large to display} \] Input:
integrate(sech(x)^3/(a+b*cosh(x)^2),x, algorithm="fricas")
Output:
[1/2*(2*a*cosh(x)^3 + 6*a*cosh(x)*sinh(x)^2 + 2*a*sinh(x)^3 + (b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 + 2*b*cosh(x)^2 + 2*(3*b*cosh(x)^2 + b)*sinh(x)^2 + 4*(b*cosh(x)^3 + b*cosh(x))*sinh(x) + b)*sqrt(-b/(a + b)) *log((b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 - 2*(2*a + 3*b)*co sh(x)^2 + 2*(3*b*cosh(x)^2 - 2*a - 3*b)*sinh(x)^2 + 4*(b*cosh(x)^3 - (2*a + 3*b)*cosh(x))*sinh(x) + 4*((a + b)*cosh(x)^3 + 3*(a + b)*cosh(x)*sinh(x) ^2 + (a + b)*sinh(x)^3 - (a + b)*cosh(x) + (3*(a + b)*cosh(x)^2 - a - b)*s inh(x))*sqrt(-b/(a + b)) + b)/(b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sin h(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)) + 2*((a - 2*b)*cosh(x)^4 + 4*(a - 2*b)*cosh(x)*sinh(x)^3 + (a - 2*b)*sinh(x)^4 + 2*(a - 2*b)*cosh(x )^2 + 2*(3*(a - 2*b)*cosh(x)^2 + a - 2*b)*sinh(x)^2 + 4*((a - 2*b)*cosh(x) ^3 + (a - 2*b)*cosh(x))*sinh(x) + a - 2*b)*arctan(cosh(x) + sinh(x)) - 2*a *cosh(x) + 2*(3*a*cosh(x)^2 - a)*sinh(x))/(a^2*cosh(x)^4 + 4*a^2*cosh(x)*s inh(x)^3 + a^2*sinh(x)^4 + 2*a^2*cosh(x)^2 + 2*(3*a^2*cosh(x)^2 + a^2)*sin h(x)^2 + a^2 + 4*(a^2*cosh(x)^3 + a^2*cosh(x))*sinh(x)), (a*cosh(x)^3 + 3* a*cosh(x)*sinh(x)^2 + a*sinh(x)^3 + (b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 + 2*b*cosh(x)^2 + 2*(3*b*cosh(x)^2 + b)*sinh(x)^2 + 4*(b*cosh (x)^3 + b*cosh(x))*sinh(x) + b)*sqrt(b/(a + b))*arctan(1/2*sqrt(b/(a + b)) *(cosh(x) + sinh(x))) + (b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x...
\[ \int \frac {\text {sech}^3(x)}{a+b \cosh ^2(x)} \, dx=\int \frac {\operatorname {sech}^{3}{\left (x \right )}}{a + b \cosh ^{2}{\left (x \right )}}\, dx \] Input:
integrate(sech(x)**3/(a+b*cosh(x)**2),x)
Output:
Integral(sech(x)**3/(a + b*cosh(x)**2), x)
\[ \int \frac {\text {sech}^3(x)}{a+b \cosh ^2(x)} \, dx=\int { \frac {\operatorname {sech}\left (x\right )^{3}}{b \cosh \left (x\right )^{2} + a} \,d x } \] Input:
integrate(sech(x)^3/(a+b*cosh(x)^2),x, algorithm="maxima")
Output:
(e^(3*x) - e^x)/(a*e^(4*x) + 2*a*e^(2*x) + a) + (a - 2*b)*arctan(e^x)/a^2 + 8*integrate(1/4*(b^2*e^(3*x) + b^2*e^x)/(a^2*b*e^(4*x) + a^2*b + 2*(2*a^ 3 + a^2*b)*e^(2*x)), x)
Exception generated. \[ \int \frac {\text {sech}^3(x)}{a+b \cosh ^2(x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(sech(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 = 2.94 (sec) , antiderivative size = 447, normalized size of antiderivative = 7.58 \[ \int \frac {\text {sech}^3(x)}{a+b \cosh ^2(x)} \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\left (a^7\,{\left (a^4\right )}^{3/2}-12\,b^3\,{\left (a^4\right )}^{5/2}-18\,b^7\,{\left (a^4\right )}^{3/2}+36\,a^2\,b^5\,{\left (a^4\right )}^{3/2}-30\,a^3\,b^4\,{\left (a^4\right )}^{3/2}+21\,a^5\,b^2\,{\left (a^4\right )}^{3/2}+9\,a\,b^6\,{\left (a^4\right )}^{3/2}-8\,a^6\,b\,{\left (a^4\right )}^{3/2}\right )}{a^{12}\,\sqrt {a^2-4\,a\,b+4\,b^2}-6\,a^{11}\,b\,\sqrt {a^2-4\,a\,b+4\,b^2}+9\,a^6\,b^6\,\sqrt {a^2-4\,a\,b+4\,b^2}-18\,a^8\,b^4\,\sqrt {a^2-4\,a\,b+4\,b^2}+6\,a^9\,b^3\,\sqrt {a^2-4\,a\,b+4\,b^2}+9\,a^{10}\,b^2\,\sqrt {a^2-4\,a\,b+4\,b^2}}\right )\,\sqrt {a^2-4\,a\,b+4\,b^2}}{\sqrt {a^4}}-\frac {2\,{\mathrm {e}}^x}{a\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )}+\frac {{\mathrm {e}}^x}{a\,\left ({\mathrm {e}}^{2\,x}+1\right )}-\frac {{\left (-b\right )}^{3/2}\,\ln \left (\frac {64\,\left ({\mathrm {e}}^{2\,x}-1\right )\,\left (a^3-3\,a^2\,b+3\,b^3\right )}{a^5\,{\left (a+b\right )}^2}-\frac {128\,{\mathrm {e}}^x\,\left (a^3-3\,a^2\,b+3\,b^3\right )}{a^5\,\sqrt {-b}\,{\left (a+b\right )}^{3/2}}\right )}{2\,a^2\,\sqrt {a+b}}+\frac {{\left (-b\right )}^{3/2}\,\ln \left (\frac {64\,\left ({\mathrm {e}}^{2\,x}-1\right )\,\left (a^3-3\,a^2\,b+3\,b^3\right )}{a^5\,{\left (a+b\right )}^2}+\frac {128\,{\mathrm {e}}^x\,\left (a^3-3\,a^2\,b+3\,b^3\right )}{a^5\,\sqrt {-b}\,{\left (a+b\right )}^{3/2}}\right )}{2\,a^2\,\sqrt {a+b}} \] Input:
int(1/(cosh(x)^3*(a + b*cosh(x)^2)),x)
Output:
(atan((exp(x)*(a^7*(a^4)^(3/2) - 12*b^3*(a^4)^(5/2) - 18*b^7*(a^4)^(3/2) + 36*a^2*b^5*(a^4)^(3/2) - 30*a^3*b^4*(a^4)^(3/2) + 21*a^5*b^2*(a^4)^(3/2) + 9*a*b^6*(a^4)^(3/2) - 8*a^6*b*(a^4)^(3/2)))/(a^12*(a^2 - 4*a*b + 4*b^2)^ (1/2) - 6*a^11*b*(a^2 - 4*a*b + 4*b^2)^(1/2) + 9*a^6*b^6*(a^2 - 4*a*b + 4* b^2)^(1/2) - 18*a^8*b^4*(a^2 - 4*a*b + 4*b^2)^(1/2) + 6*a^9*b^3*(a^2 - 4*a *b + 4*b^2)^(1/2) + 9*a^10*b^2*(a^2 - 4*a*b + 4*b^2)^(1/2)))*(a^2 - 4*a*b + 4*b^2)^(1/2))/(a^4)^(1/2) - (2*exp(x))/(a*(2*exp(2*x) + exp(4*x) + 1)) + exp(x)/(a*(exp(2*x) + 1)) - ((-b)^(3/2)*log((64*(exp(2*x) - 1)*(a^3 - 3*a ^2*b + 3*b^3))/(a^5*(a + b)^2) - (128*exp(x)*(a^3 - 3*a^2*b + 3*b^3))/(a^5 *(-b)^(1/2)*(a + b)^(3/2))))/(2*a^2*(a + b)^(1/2)) + ((-b)^(3/2)*log((64*( exp(2*x) - 1)*(a^3 - 3*a^2*b + 3*b^3))/(a^5*(a + b)^2) + (128*exp(x)*(a^3 - 3*a^2*b + 3*b^3))/(a^5*(-b)^(1/2)*(a + b)^(3/2))))/(2*a^2*(a + b)^(1/2))
Time = 0.29 (sec) , antiderivative size = 1545, normalized size of antiderivative = 26.19 \[ \int \frac {\text {sech}^3(x)}{a+b \cosh ^2(x)} \, dx =\text {Too large to display} \] Input:
int(sech(x)^3/(a+b*cosh(x)^2),x)
Output:
(2*e**(4*x)*atan(e**x)*a**2 - 2*e**(4*x)*atan(e**x)*a*b - 4*e**(4*x)*atan( e**x)*b**2 + 4*e**(2*x)*atan(e**x)*a**2 - 4*e**(2*x)*atan(e**x)*a*b - 8*e* *(2*x)*atan(e**x)*b**2 + 2*atan(e**x)*a**2 - 2*atan(e**x)*a*b - 4*atan(e** x)*b**2 - 2*e**(4*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)) ) - 4*e**(2*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))) - 2* 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))) + 2*e**(4*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*e**(4*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)))*b + 4*e**(2*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 + 4*e**(2*x)*sqrt (b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)*atan((e**x*b)/(sqrt(b)*sqrt(2*sq rt(a)*sqrt(a + b) + 2*a + b)))*b + 2*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*sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)*atan((e**x*b)/(sqrt(b)*s qrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)))*b - e**(4*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b)*log( - sqrt(2*sqrt(a)*sqrt(...