Integrand size = 15, antiderivative size = 61 \[ \int \frac {\text {csch}^3(x)}{a+b \cosh ^2(x)} \, dx=\frac {b^{3/2} \arctan \left (\frac {\sqrt {b} \cosh (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^2}+\frac {(a+3 b) \text {arctanh}(\cosh (x))}{2 (a+b)^2}-\frac {\coth (x) \text {csch}(x)}{2 (a+b)} \]
1/2*(a+3*b)*arctanh(cosh(x))/(a+b)^2-1/2*coth(x)*csch(x)/(a+b)+b^(3/2)*arc tan(cosh(x)*b^(1/2)/a^(1/2))/(a+b)^2/a^(1/2)
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 154, normalized size of antiderivative = 2.52 \[ \int \frac {\text {csch}^3(x)}{a+b \cosh ^2(x)} \, dx=\frac {8 b^{3/2} \arctan \left (\frac {\sqrt {b}-i \sqrt {a+b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a}}\right )+8 b^{3/2} \arctan \left (\frac {\sqrt {b}+i \sqrt {a+b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a}}\right )-\sqrt {a} (a+b) \text {csch}^2\left (\frac {x}{2}\right )+4 \sqrt {a} (a+3 b) \left (\log \left (\cosh \left (\frac {x}{2}\right )\right )-\log \left (\sinh \left (\frac {x}{2}\right )\right )\right )-\sqrt {a} (a+b) \text {sech}^2\left (\frac {x}{2}\right )}{8 \sqrt {a} (a+b)^2} \]
(8*b^(3/2)*ArcTan[(Sqrt[b] - I*Sqrt[a + b]*Tanh[x/2])/Sqrt[a]] + 8*b^(3/2) *ArcTan[(Sqrt[b] + I*Sqrt[a + b]*Tanh[x/2])/Sqrt[a]] - Sqrt[a]*(a + b)*Csc h[x/2]^2 + 4*Sqrt[a]*(a + 3*b)*(Log[Cosh[x/2]] - Log[Sinh[x/2]]) - Sqrt[a] *(a + b)*Sech[x/2]^2)/(8*Sqrt[a]*(a + b)^2)
Time = 0.29 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.26, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3042, 26, 3669, 316, 397, 218, 219}
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.
| \(\displaystyle \int \frac {\text {csch}^3(x)}{a+b \cosh ^2(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i}{\cos \left (\frac {\pi }{2}+i x\right )^3 \left (a+b \sin \left (\frac {\pi }{2}+i x\right )^2\right )}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {1}{\cos \left (i x+\frac {\pi }{2}\right )^3 \left (b \sin \left (i x+\frac {\pi }{2}\right )^2+a\right )}dx\) |
| \(\Big \downarrow \) 3669 |
| \(\displaystyle \int \frac {1}{\left (1-\cosh ^2(x)\right )^2 \left (a+b \cosh ^2(x)\right )}d\cosh (x)\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\int \frac {b \cosh ^2(x)+a+2 b}{\left (1-\cosh ^2(x)\right ) \left (b \cosh ^2(x)+a\right )}d\cosh (x)}{2 (a+b)}+\frac {\cosh (x)}{2 (a+b) \left (1-\cosh ^2(x)\right )}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {2 b^2 \int \frac {1}{b \cosh ^2(x)+a}d\cosh (x)}{a+b}+\frac {(a+3 b) \int \frac {1}{1-\cosh ^2(x)}d\cosh (x)}{a+b}}{2 (a+b)}+\frac {\cosh (x)}{2 (a+b) \left (1-\cosh ^2(x)\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {(a+3 b) \int \frac {1}{1-\cosh ^2(x)}d\cosh (x)}{a+b}+\frac {2 b^{3/2} \arctan \left (\frac {\sqrt {b} \cosh (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)}}{2 (a+b)}+\frac {\cosh (x)}{2 (a+b) \left (1-\cosh ^2(x)\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {2 b^{3/2} \arctan \left (\frac {\sqrt {b} \cosh (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)}+\frac {(a+3 b) \text {arctanh}(\cosh (x))}{a+b}}{2 (a+b)}+\frac {\cosh (x)}{2 (a+b) \left (1-\cosh ^2(x)\right )}\) |
((2*b^(3/2)*ArcTan[(Sqrt[b]*Cosh[x])/Sqrt[a]])/(Sqrt[a]*(a + b)) + ((a + 3 *b)*ArcTanh[Cosh[x]])/(a + b))/(2*(a + b)) + Cosh[x]/(2*(a + b)*(1 - Cosh[ x]^2))
3.1.11.3.1 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[((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)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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[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 = 6.49 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.43
| method | result | size |
| default | \(\frac {\tanh \left (\frac {x}{2}\right )^{2}}{8 a +8 b}-\frac {1}{8 \left (a +b \right ) \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\left (-2 a -6 b \right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{4 \left (a +b \right )^{2}}+\frac {b^{2} \arctan \left (\frac {2 \left (a +b \right ) \tanh \left (\frac {x}{2}\right )^{2}-2 a +2 b}{4 \sqrt {a b}}\right )}{\left (a +b \right )^{2} \sqrt {a b}}\) | \(87\) |
| risch | \(-\frac {{\mathrm e}^{x} \left (1+{\mathrm e}^{2 x}\right )}{\left ({\mathrm e}^{2 x}-1\right )^{2} \left (a +b \right )}+\frac {a \ln \left ({\mathrm e}^{x}+1\right )}{2 a^{2}+4 a b +2 b^{2}}+\frac {3 \ln \left ({\mathrm e}^{x}+1\right ) b}{2 \left (a^{2}+2 a b +b^{2}\right )}-\frac {\ln \left ({\mathrm e}^{x}-1\right ) a}{2 \left (a^{2}+2 a b +b^{2}\right )}-\frac {3 \ln \left ({\mathrm e}^{x}-1\right ) b}{2 \left (a^{2}+2 a b +b^{2}\right )}+\frac {\sqrt {-a b}\, b \ln \left ({\mathrm e}^{2 x}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{x}}{b}+1\right )}{2 a \left (a +b \right )^{2}}-\frac {\sqrt {-a b}\, b \ln \left ({\mathrm e}^{2 x}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{x}}{b}+1\right )}{2 a \left (a +b \right )^{2}}\) | \(183\) |
1/8*tanh(1/2*x)^2/(a+b)-1/8/(a+b)/tanh(1/2*x)^2+1/4/(a+b)^2*(-2*a-6*b)*ln( tanh(1/2*x))+b^2/(a+b)^2/(a*b)^(1/2)*arctan(1/4*(2*(a+b)*tanh(1/2*x)^2-2*a +2*b)/(a*b)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 638 vs. \(2 (49) = 98\).
Time = 0.30 (sec) , antiderivative size = 1332, normalized size of antiderivative = 21.84 \[ \int \frac {\text {csch}^3(x)}{a+b \cosh ^2(x)} \, dx=\text {Too large to display} \]
[-1/2*(2*(a + b)*cosh(x)^3 + 6*(a + b)*cosh(x)*sinh(x)^2 + 2*(a + b)*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)*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*(a*cosh(x)^3 + 3*a*cosh(x)*sinh(x)^2 + a*sinh(x)^3 + a*cosh(x) + (3*a*cosh(x)^2 + a)*sinh(x))*sqrt(-b/a) + b)/(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) + b)) + 2*(a + b)*cosh(x) - ((a + 3*b)*cosh(x)^4 + 4*(a + 3*b)*c osh(x)*sinh(x)^3 + (a + 3*b)*sinh(x)^4 - 2*(a + 3*b)*cosh(x)^2 + 2*(3*(a + 3*b)*cosh(x)^2 - a - 3*b)*sinh(x)^2 + 4*((a + 3*b)*cosh(x)^3 - (a + 3*b)* cosh(x))*sinh(x) + a + 3*b)*log(cosh(x) + sinh(x) + 1) + ((a + 3*b)*cosh(x )^4 + 4*(a + 3*b)*cosh(x)*sinh(x)^3 + (a + 3*b)*sinh(x)^4 - 2*(a + 3*b)*co sh(x)^2 + 2*(3*(a + 3*b)*cosh(x)^2 - a - 3*b)*sinh(x)^2 + 4*((a + 3*b)*cos h(x)^3 - (a + 3*b)*cosh(x))*sinh(x) + a + 3*b)*log(cosh(x) + sinh(x) - 1) + 2*(3*(a + b)*cosh(x)^2 + a + b)*sinh(x))/((a^2 + 2*a*b + b^2)*cosh(x)^4 + 4*(a^2 + 2*a*b + b^2)*cosh(x)*sinh(x)^3 + (a^2 + 2*a*b + b^2)*sinh(x)^4 - 2*(a^2 + 2*a*b + b^2)*cosh(x)^2 + 2*(3*(a^2 + 2*a*b + b^2)*cosh(x)^2 - a ^2 - 2*a*b - b^2)*sinh(x)^2 + a^2 + 2*a*b + b^2 + 4*((a^2 + 2*a*b + b^2...
\[ \int \frac {\text {csch}^3(x)}{a+b \cosh ^2(x)} \, dx=\int \frac {\operatorname {csch}^{3}{\left (x \right )}}{a + b \cosh ^{2}{\left (x \right )}}\, dx \]
\[ \int \frac {\text {csch}^3(x)}{a+b \cosh ^2(x)} \, dx=\int { \frac {\operatorname {csch}\left (x\right )^{3}}{b \cosh \left (x\right )^{2} + a} \,d x } \]
1/2*(a + 3*b)*log(e^x + 1)/(a^2 + 2*a*b + b^2) - 1/2*(a + 3*b)*log(e^x - 1 )/(a^2 + 2*a*b + b^2) - (e^(3*x) + e^x)/((a + b)*e^(4*x) - 2*(a + b)*e^(2* x) + a + b) + 8*integrate(1/4*(b^2*e^(3*x) - b^2*e^x)/(a^2*b + 2*a*b^2 + b ^3 + (a^2*b + 2*a*b^2 + b^3)*e^(4*x) + 2*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3) *e^(2*x)), x)
\[ \int \frac {\text {csch}^3(x)}{a+b \cosh ^2(x)} \, dx=\int { \frac {\operatorname {csch}\left (x\right )^{3}}{b \cosh \left (x\right )^{2} + a} \,d x } \]
Time = 8.26 (sec) , antiderivative size = 2225, normalized size of antiderivative = 36.48 \[ \int \frac {\text {csch}^3(x)}{a+b \cosh ^2(x)} \, dx=\text {Too large to display} \]
((2*atan((b^2*exp(x)*(a*(a + b)^4)^(1/2))/(2*a*(a + b)^2*(b^3)^(1/2))) - 2 *atan((exp(x)*((32*(b^8*(a*b^4 + 4*a^4*b + a^5 + 4*a^2*b^3 + 6*a^3*b^2)^(1 /2) + 36*a^2*b^6*(a*b^4 + 4*a^4*b + a^5 + 4*a^2*b^3 + 6*a^3*b^2)^(1/2) + 4 7*a^3*b^5*(a*b^4 + 4*a^4*b + a^5 + 4*a^2*b^3 + 6*a^3*b^2)^(1/2) + 30*a^4*b ^4*(a*b^4 + 4*a^4*b + a^5 + 4*a^2*b^3 + 6*a^3*b^2)^(1/2) + 9*a^5*b^3*(a*b^ 4 + 4*a^4*b + a^5 + 4*a^2*b^3 + 6*a^3*b^2)^(1/2) + a^6*b^2*(a*b^4 + 4*a^4* b + a^5 + 4*a^2*b^3 + 6*a^3*b^2)^(1/2) + 12*a*b^7*(a*b^4 + 4*a^4*b + a^5 + 4*a^2*b^3 + 6*a^3*b^2)^(1/2)))/(a^2*b^2*(a + b)^7*(a*b + a^2)*(b^3)^(1/2) *(2*a*b + a^2 + b^2)*(3*a*b^2 + 3*a^2*b + a^3 + b^3)*(9*a*b^2 + 6*a^2*b + a^3 + b^3)*(a*b^4 + 4*a^4*b + a^5 + 4*a^2*b^3 + 6*a^3*b^2)^(1/2)) + (64*(2 0*a^3*(b^3)^(5/2) + 232*a^6*(b^3)^(3/2) + 2*a^9*(b^3)^(1/2) + 10*a^2*b^4*( b^3)^(3/2) + 20*a^4*b^2*(b^3)^(3/2) + 18*a^2*b^7*(b^3)^(1/2) + 102*a^3*b^6 *(b^3)^(1/2) + 242*a^4*b^5*(b^3)^(1/2) + 310*a^5*b^4*(b^3)^(1/2) + 98*a^7* b^2*(b^3)^(1/2) + 2*a*b^5*(b^3)^(3/2) + 10*a^5*b*(b^3)^(3/2) + 22*a^8*b*(b ^3)^(1/2)))/(a*b^4*(a + b)^5*(a*b + a^2)*(a*(a + b)^4)^(1/2)*(2*a*b + a^2 + b^2)*(3*a*b^2 + 3*a^2*b + a^3 + b^3)*(9*a*b^2 + 6*a^2*b + a^3 + b^3)*(a* b^4 + 4*a^4*b + a^5 + 4*a^2*b^3 + 6*a^3*b^2)^(1/2))) + (32*exp(3*x)*(b^8*( a*b^4 + 4*a^4*b + a^5 + 4*a^2*b^3 + 6*a^3*b^2)^(1/2) + 36*a^2*b^6*(a*b^4 + 4*a^4*b + a^5 + 4*a^2*b^3 + 6*a^3*b^2)^(1/2) + 47*a^3*b^5*(a*b^4 + 4*a^4* b + a^5 + 4*a^2*b^3 + 6*a^3*b^2)^(1/2) + 30*a^4*b^4*(a*b^4 + 4*a^4*b + ...