Integrand size = 23, antiderivative size = 139 \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx=-\frac {(4 a-b) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{3/2} (a-b)^{5/2} d}+\frac {\cosh (c+d x) \sinh (c+d x)}{4 (a-b) d \left (a+b \sinh ^2(c+d x)\right )^2}+\frac {(2 a+b) \cosh (c+d x) \sinh (c+d x)}{8 a (a-b)^2 d \left (a+b \sinh ^2(c+d x)\right )} \]
-1/8*(4*a-b)*arctanh((a-b)^(1/2)*tanh(d*x+c)/a^(1/2))/a^(3/2)/(a-b)^(5/2)/ d+1/4*cosh(d*x+c)*sinh(d*x+c)/(a-b)/d/(a+b*sinh(d*x+c)^2)^2+1/8*(2*a+b)*co sh(d*x+c)*sinh(d*x+c)/a/(a-b)^2/d/(a+b*sinh(d*x+c)^2)
Time = 11.64 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.87 \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx=\frac {-\frac {(4 a-b) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{3/2} (a-b)^{5/2}}+\frac {\left (8 a^2-4 a b-b^2+b (2 a+b) \cosh (2 (c+d x))\right ) \sinh (2 (c+d x))}{a (a-b)^2 (2 a-b+b \cosh (2 (c+d x)))^2}}{8 d} \]
(-(((4*a - b)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a^(3/2)*(a - b)^(5/2))) + ((8*a^2 - 4*a*b - b^2 + b*(2*a + b)*Cosh[2*(c + d*x)])*Sinh[2 *(c + d*x)])/(a*(a - b)^2*(2*a - b + b*Cosh[2*(c + d*x)])^2))/(8*d)
Time = 0.56 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3042, 25, 3652, 3042, 3652, 27, 3042, 3660, 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.
| \(\displaystyle \int \frac {\sinh ^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\sin (i c+i d x)^2}{\left (a-b \sin (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\sin (i c+i d x)^2}{\left (a-b \sin (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 3652 |
| \(\displaystyle \frac {\sinh (c+d x) \cosh (c+d x)}{4 d (a-b) \left (a+b \sinh ^2(c+d x)\right )^2}-\frac {\int \frac {a-2 a \sinh ^2(c+d x)}{\left (b \sinh ^2(c+d x)+a\right )^2}dx}{4 a (a-b)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh (c+d x) \cosh (c+d x)}{4 d (a-b) \left (a+b \sinh ^2(c+d x)\right )^2}-\frac {\int \frac {2 a \sin (i c+i d x)^2+a}{\left (a-b \sin (i c+i d x)^2\right )^2}dx}{4 a (a-b)}\) |
| \(\Big \downarrow \) 3652 |
| \(\displaystyle \frac {\sinh (c+d x) \cosh (c+d x)}{4 d (a-b) \left (a+b \sinh ^2(c+d x)\right )^2}-\frac {\frac {\int \frac {a (4 a-b)}{b \sinh ^2(c+d x)+a}dx}{2 a (a-b)}-\frac {(2 a+b) \sinh (c+d x) \cosh (c+d x)}{2 d (a-b) \left (a+b \sinh ^2(c+d x)\right )}}{4 a (a-b)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sinh (c+d x) \cosh (c+d x)}{4 d (a-b) \left (a+b \sinh ^2(c+d x)\right )^2}-\frac {\frac {(4 a-b) \int \frac {1}{b \sinh ^2(c+d x)+a}dx}{2 (a-b)}-\frac {(2 a+b) \sinh (c+d x) \cosh (c+d x)}{2 d (a-b) \left (a+b \sinh ^2(c+d x)\right )}}{4 a (a-b)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh (c+d x) \cosh (c+d x)}{4 d (a-b) \left (a+b \sinh ^2(c+d x)\right )^2}-\frac {-\frac {(2 a+b) \sinh (c+d x) \cosh (c+d x)}{2 d (a-b) \left (a+b \sinh ^2(c+d x)\right )}+\frac {(4 a-b) \int \frac {1}{a-b \sin (i c+i d x)^2}dx}{2 (a-b)}}{4 a (a-b)}\) |
| \(\Big \downarrow \) 3660 |
| \(\displaystyle \frac {\sinh (c+d x) \cosh (c+d x)}{4 d (a-b) \left (a+b \sinh ^2(c+d x)\right )^2}-\frac {\frac {(4 a-b) \int \frac {1}{a-(a-b) \tanh ^2(c+d x)}d\tanh (c+d x)}{2 d (a-b)}-\frac {(2 a+b) \sinh (c+d x) \cosh (c+d x)}{2 d (a-b) \left (a+b \sinh ^2(c+d x)\right )}}{4 a (a-b)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\sinh (c+d x) \cosh (c+d x)}{4 d (a-b) \left (a+b \sinh ^2(c+d x)\right )^2}-\frac {\frac {(4 a-b) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 \sqrt {a} d (a-b)^{3/2}}-\frac {(2 a+b) \sinh (c+d x) \cosh (c+d x)}{2 d (a-b) \left (a+b \sinh ^2(c+d x)\right )}}{4 a (a-b)}\) |
(Cosh[c + d*x]*Sinh[c + d*x])/(4*(a - b)*d*(a + b*Sinh[c + d*x]^2)^2) - (( (4*a - b)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(2*Sqrt[a]*(a - b) ^(3/2)*d) - ((2*a + b)*Cosh[c + d*x]*Sinh[c + d*x])/(2*(a - b)*d*(a + b*Si nh[c + d*x]^2)))/(4*a*(a - b))
3.1.53.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b - a*B))*Cos[e + f*x]*Sin[e + f*x ]*((a + b*Sin[e + f*x]^2)^(p + 1)/(2*a*f*(a + b)*(p + 1))), x] - Simp[1/(2* a*(a + b)*(p + 1)) Int[(a + b*Sin[e + f*x]^2)^(p + 1)*Simp[a*B - A*(2*a*( p + 1) + b*(2*p + 3)) + 2*(A*b - a*B)*(p + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && LtQ[p, -1] && NeQ[a + b, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[1/(a + (a + b)*ff^2*x^ 2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(400\) vs. \(2(125)=250\).
Time = 0.46 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.88
| method | result | size |
| derivativedivides | \(\frac {-\frac {8 \left (-\frac {\left (4 a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{32 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (4 a^{2}-9 a b -4 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{32 a \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (4 a^{2}-9 a b -4 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{32 a \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (4 a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 \left (a^{2}-2 a b +b^{2}\right )}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a \right )^{2}}-\frac {\left (4 a -b \right ) \left (\frac {\left (-\sqrt {-b \left (a -b \right )}-b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (-\sqrt {-b \left (a -b \right )}+b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{4 \left (a^{2}-2 a b +b^{2}\right )}}{d}\) | \(401\) |
| default | \(\frac {-\frac {8 \left (-\frac {\left (4 a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{32 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (4 a^{2}-9 a b -4 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{32 a \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (4 a^{2}-9 a b -4 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{32 a \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (4 a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 \left (a^{2}-2 a b +b^{2}\right )}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a \right )^{2}}-\frac {\left (4 a -b \right ) \left (\frac {\left (-\sqrt {-b \left (a -b \right )}-b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (-\sqrt {-b \left (a -b \right )}+b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{4 \left (a^{2}-2 a b +b^{2}\right )}}{d}\) | \(401\) |
| risch | \(-\frac {4 a \,b^{2} {\mathrm e}^{6 d x +6 c}-b^{3} {\mathrm e}^{6 d x +6 c}+16 a^{3} {\mathrm e}^{4 d x +4 c}-8 a^{2} b \,{\mathrm e}^{4 d x +4 c}-2 \,{\mathrm e}^{4 d x +4 c} b^{2} a +3 b^{3} {\mathrm e}^{4 d x +4 c}+16 a^{2} b \,{\mathrm e}^{2 d x +2 c}-4 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}-3 b^{3} {\mathrm e}^{2 d x +2 c}+2 a \,b^{2}+b^{3}}{4 b d a \left (a -b \right )^{2} \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}+2 a^{2}-2 a b}{b \sqrt {a^{2}-a b}}\right )}{4 \sqrt {a^{2}-a b}\, \left (a -b \right )^{2} d}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}+2 a^{2}-2 a b}{b \sqrt {a^{2}-a b}}\right ) b}{16 \sqrt {a^{2}-a b}\, \left (a -b \right )^{2} d a}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}-2 a^{2}+2 a b}{b \sqrt {a^{2}-a b}}\right )}{4 \sqrt {a^{2}-a b}\, \left (a -b \right )^{2} d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}-2 a^{2}+2 a b}{b \sqrt {a^{2}-a b}}\right ) b}{16 \sqrt {a^{2}-a b}\, \left (a -b \right )^{2} d a}\) | \(540\) |
1/d*(-8*(-1/32*(4*a-b)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^7+1/32*(4*a^2-9 *a*b-4*b^2)/a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^5+1/32*(4*a^2-9*a*b-4*b^ 2)/a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3-1/32*(4*a-b)/(a^2-2*a*b+b^2)*ta nh(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)^2-1/4*(4*a-b)/(a^2-2*a*b+b^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)*arctan(a*t anh(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. 2632 vs. \(2 (125) = 250\).
Time = 0.36 (sec) , antiderivative size = 5519, normalized size of antiderivative = 39.71 \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx=\int { \frac {\sinh \left (d x + c\right )^{2}}{{\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^2}{{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \]