Integrand size = 23, antiderivative size = 142 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=-\frac {(4 a-3 b) b \text {arctanh}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{5/2} (a-b)^{3/2} d}-\frac {\coth (c+d x)}{a d \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac {\left (2 a^2-4 a b+3 b^2\right ) \tanh (c+d x)}{2 a^2 (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )} \]
-1/2*(4*a-3*b)*b*arctanh((a-b)^(1/2)*tanh(d*x+c)/a^(1/2))/a^(5/2)/(a-b)^(3 /2)/d-coth(d*x+c)/a/d/(a-(a-b)*tanh(d*x+c)^2)+1/2*(2*a^2-4*a*b+3*b^2)*tanh (d*x+c)/a^2/(a-b)/d/(a-(a-b)*tanh(d*x+c)^2)
Time = 1.31 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.20 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=-\frac {(2 a-b+b \cosh (2 (c+d x))) \text {csch}^5(c+d x) \left (2 \sqrt {a} \sqrt {a-b} \cosh (c+d x) \left (4 a^2-6 a b+3 b^2+(2 a-3 b) b \cosh (2 (c+d x))\right )+2 (4 a-3 b) b \text {arctanh}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right ) (2 a-b+b \cosh (2 (c+d x))) \sinh (c+d x)\right )}{16 a^{5/2} (a-b)^{3/2} d \left (b+a \text {csch}^2(c+d x)\right )^2} \]
-1/16*((2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^5*(2*Sqrt[a]*Sqrt[a - b]*Cosh[c + d*x]*(4*a^2 - 6*a*b + 3*b^2 + (2*a - 3*b)*b*Cosh[2*(c + d*x)] ) + 2*(4*a - 3*b)*b*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]]*(2*a - b + b*Cosh[2*(c + d*x)])*Sinh[c + d*x]))/(a^(5/2)*(a - b)^(3/2)*d*(b + a*Csc h[c + d*x]^2)^2)
Time = 0.34 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 25, 3666, 365, 298, 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.
| \(\displaystyle \int \frac {\text {csch}^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\sin (i c+i d x)^2 \left (a-b \sin (i c+i d x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\sin (i c+i d x)^2 \left (a-b \sin (i c+i d x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 3666 |
| \(\displaystyle \frac {\int \frac {\coth ^2(c+d x) \left (1-\tanh ^2(c+d x)\right )^2}{\left (a-(a-b) \tanh ^2(c+d x)\right )^2}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 365 |
| \(\displaystyle \frac {\frac {\int \frac {a \tanh ^2(c+d x)+a-3 b}{\left (a-(a-b) \tanh ^2(c+d x)\right )^2}d\tanh (c+d x)}{a}-\frac {\coth (c+d x)}{a \left (a-(a-b) \tanh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 298 |
| \(\displaystyle \frac {\frac {\frac {\left (2 a^2-4 a b+3 b^2\right ) \tanh (c+d x)}{2 a (a-b) \left (a-(a-b) \tanh ^2(c+d x)\right )}-\frac {b (4 a-3 b) \int \frac {1}{a-(a-b) \tanh ^2(c+d x)}d\tanh (c+d x)}{2 a (a-b)}}{a}-\frac {\coth (c+d x)}{a \left (a-(a-b) \tanh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {\left (2 a^2-4 a b+3 b^2\right ) \tanh (c+d x)}{2 a (a-b) \left (a-(a-b) \tanh ^2(c+d x)\right )}-\frac {b (4 a-3 b) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} (a-b)^{3/2}}}{a}-\frac {\coth (c+d x)}{a \left (a-(a-b) \tanh ^2(c+d x)\right )}}{d}\) |
(-(Coth[c + d*x]/(a*(a - (a - b)*Tanh[c + d*x]^2))) + (-1/2*((4*a - 3*b)*b *ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a^(3/2)*(a - b)^(3/2)) + ( (2*a^2 - 4*a*b + 3*b^2)*Tanh[c + d*x])/(2*a*(a - b)*(a - (a - b)*Tanh[c + d*x]^2)))/a)/d
3.1.48.3.1 Defintions of rubi rules used
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_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x _Symbol] :> Simp[c^2*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^2)^p*Simp[2*b*c^2*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*d^2*(m + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff^(m + 1 )/f Subst[Int[x^m*((a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1)) , x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] & & IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(315\) vs. \(2(130)=260\).
Time = 0.37 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.23
| method | result | size |
| derivativedivides | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{2}}-\frac {1}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {4 b \left (\frac {\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4 a -4 b}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a -4 b}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a}+\frac {\left (4 a -3 b \right ) a \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{4 a -4 b}\right )}{a^{2}}}{d}\) | \(316\) |
| default | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{2}}-\frac {1}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {4 b \left (\frac {\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4 a -4 b}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a -4 b}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a}+\frac {\left (4 a -3 b \right ) a \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{4 a -4 b}\right )}{a^{2}}}{d}\) | \(316\) |
| risch | \(-\frac {4 \,{\mathrm e}^{4 d x +4 c} a b -3 b^{2} {\mathrm e}^{4 d x +4 c}+8 \,{\mathrm e}^{2 d x +2 c} a^{2}-14 \,{\mathrm e}^{2 d x +2 c} b a +6 b^{2} {\mathrm e}^{2 d x +2 c}+2 a b -3 b^{2}}{a^{2} \left (a -b \right ) d \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right ) \left ({\mathrm e}^{2 d x +2 c}-1\right )}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}+2 a^{2}-2 a b}{b \sqrt {a^{2}-a b}}\right ) b}{\sqrt {a^{2}-a b}\, \left (a -b \right ) d a}-\frac {3 b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}+2 a^{2}-2 a b}{b \sqrt {a^{2}-a b}}\right )}{4 \sqrt {a^{2}-a b}\, \left (a -b \right ) d \,a^{2}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}-2 a^{2}+2 a b}{b \sqrt {a^{2}-a b}}\right ) b}{\sqrt {a^{2}-a b}\, \left (a -b \right ) d a}+\frac {3 b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}-2 a^{2}+2 a b}{b \sqrt {a^{2}-a b}}\right )}{4 \sqrt {a^{2}-a b}\, \left (a -b \right ) d \,a^{2}}\) | \(498\) |
1/d*(-1/2*tanh(1/2*d*x+1/2*c)/a^2-1/2/a^2/tanh(1/2*d*x+1/2*c)+4*b/a^2*((1/ 4*b/(a-b)*tanh(1/2*d*x+1/2*c)^3+1/4*b/(a-b)*tanh(1/2*d*x+1/2*c))/(tanh(1/2 *d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*b*tanh(1/2*d*x+1/2*c)^2+a)+1/4 *(4*a-3*b)/(a-b)*a*(1/2*((-b*(a-b))^(1/2)+b)/a/(-b*(a-b))^(1/2)/((2*(-b*(a -b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/ 2)-a+2*b)*a)^(1/2))-1/2*((-b*(a-b))^(1/2)-b)/a/(-b*(a-b))^(1/2)/((2*(-b*(a -b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1 /2)+a-2*b)*a)^(1/2)))))
Leaf count of result is larger than twice the leaf count of optimal. 1366 vs. \(2 (131) = 262\).
Time = 0.33 (sec) , antiderivative size = 2988, normalized size of antiderivative = 21.04 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \]
[-1/4*(4*(4*a^3*b - 7*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^4 + 16*(4*a^3*b - 7 *a^2*b^2 + 3*a*b^3)*cosh(d*x + c)*sinh(d*x + c)^3 + 4*(4*a^3*b - 7*a^2*b^2 + 3*a*b^3)*sinh(d*x + c)^4 + 8*a^3*b - 20*a^2*b^2 + 12*a*b^3 + 8*(4*a^4 - 11*a^3*b + 10*a^2*b^2 - 3*a*b^3)*cosh(d*x + c)^2 + 8*(4*a^4 - 11*a^3*b + 10*a^2*b^2 - 3*a*b^3 + 3*(4*a^3*b - 7*a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^2)* sinh(d*x + c)^2 - ((4*a*b^2 - 3*b^3)*cosh(d*x + c)^6 + 6*(4*a*b^2 - 3*b^3) *cosh(d*x + c)*sinh(d*x + c)^5 + (4*a*b^2 - 3*b^3)*sinh(d*x + c)^6 + (16*a ^2*b - 24*a*b^2 + 9*b^3)*cosh(d*x + c)^4 + (16*a^2*b - 24*a*b^2 + 9*b^3 + 15*(4*a*b^2 - 3*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5*(4*a*b^2 - 3* b^3)*cosh(d*x + c)^3 + (16*a^2*b - 24*a*b^2 + 9*b^3)*cosh(d*x + c))*sinh(d *x + c)^3 - 4*a*b^2 + 3*b^3 - (16*a^2*b - 24*a*b^2 + 9*b^3)*cosh(d*x + c)^ 2 + (15*(4*a*b^2 - 3*b^3)*cosh(d*x + c)^4 - 16*a^2*b + 24*a*b^2 - 9*b^3 + 6*(16*a^2*b - 24*a*b^2 + 9*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(3*(4 *a*b^2 - 3*b^3)*cosh(d*x + c)^5 + 2*(16*a^2*b - 24*a*b^2 + 9*b^3)*cosh(d*x + c)^3 - (16*a^2*b - 24*a*b^2 + 9*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt (a^2 - a*b)*log((b^2*cosh(d*x + c)^4 + 4*b^2*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*sinh(d*x + c)^4 + 2*(2*a*b - b^2)*cosh(d*x + c)^2 + 2*(3*b^2*cosh(d *x + c)^2 + 2*a*b - b^2)*sinh(d*x + c)^2 + 8*a^2 - 8*a*b + b^2 + 4*(b^2*co sh(d*x + c)^3 + (2*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c) + 4*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + 2*a - b...
Timed out. \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for more details)Is
\[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{2}}{{\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\int \frac {1}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \]