Integrand size = 23, antiderivative size = 84 \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 \sqrt {a} (a-b)^{3/2} d}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 (a-b) d \left (a+b \sinh ^2(c+d x)\right )} \]
1/2*cosh(d*x+c)*sinh(d*x+c)/(a-b)/d/(a+b*sinh(d*x+c)^2)-1/2*arctanh((a-b)^ (1/2)*tanh(d*x+c)/a^(1/2))/(a-b)^(3/2)/d/a^(1/2)
Time = 11.18 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.96 \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\frac {-\frac {\text {arctanh}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a-b)^{3/2}}+\frac {\sinh (2 (c+d x))}{(a-b) (2 a-b+b \cosh (2 (c+d x)))}}{2 d} \]
(-(ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]]/(Sqrt[a]*(a - b)^(3/2))) + Sinh[2*(c + d*x)]/((a - b)*(2*a - b + b*Cosh[2*(c + d*x)])))/(2*d)
Time = 0.36 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 25, 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.
\(\displaystyle \int \frac {\sinh ^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\sin (i c+i d x)^2}{\left (a-b \sin (i c+i d x)^2\right )^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\sin (i c+i d x)^2}{\left (a-b \sin (i c+i d x)^2\right )^2}dx\) |
\(\Big \downarrow \) 3652 |
\(\displaystyle \frac {\sinh (c+d x) \cosh (c+d x)}{2 d (a-b) \left (a+b \sinh ^2(c+d x)\right )}-\frac {\int \frac {a}{b \sinh ^2(c+d x)+a}dx}{2 a (a-b)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sinh (c+d x) \cosh (c+d x)}{2 d (a-b) \left (a+b \sinh ^2(c+d x)\right )}-\frac {\int \frac {1}{b \sinh ^2(c+d x)+a}dx}{2 (a-b)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sinh (c+d x) \cosh (c+d x)}{2 d (a-b) \left (a+b \sinh ^2(c+d x)\right )}-\frac {\int \frac {1}{a-b \sin (i c+i d x)^2}dx}{2 (a-b)}\) |
\(\Big \downarrow \) 3660 |
\(\displaystyle \frac {\sinh (c+d x) \cosh (c+d x)}{2 d (a-b) \left (a+b \sinh ^2(c+d x)\right )}-\frac {\int \frac {1}{a-(a-b) \tanh ^2(c+d x)}d\tanh (c+d x)}{2 d (a-b)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\sinh (c+d x) \cosh (c+d x)}{2 d (a-b) \left (a+b \sinh ^2(c+d x)\right )}-\frac {\text {arctanh}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 \sqrt {a} d (a-b)^{3/2}}\) |
-1/2*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]]/(Sqrt[a]*(a - b)^(3/2)*d ) + (Cosh[c + d*x]*Sinh[c + d*x])/(2*(a - b)*d*(a + b*Sinh[c + d*x]^2))
3.1.44.3.1 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. \(247\) vs. \(2(72)=144\).
Time = 0.19 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.95
method | result | size |
risch | \(-\frac {2 a \,{\mathrm e}^{2 d x +2 c}-b \,{\mathrm e}^{2 d x +2 c}+b}{b d \left (a -b \right ) \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}+2 a^{2}-2 a b}{b \sqrt {a^{2}-a b}}\right )}{4 \sqrt {a^{2}-a b}\, \left (a -b \right ) d}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}-2 a^{2}+2 a b}{b \sqrt {a^{2}-a b}}\right )}{4 \sqrt {a^{2}-a b}\, \left (a -b \right ) d}\) | \(248\) |
derivativedivides | \(\frac {-\frac {8 \left (-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 \left (a -b \right )}-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a -b \right )}\right )}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a}-\frac {a \left (\frac {\left (-\sqrt {-b \left (a -b \right )}-b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (-\sqrt {-b \left (a -b \right )}+b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{a -b}}{d}\) | \(275\) |
default | \(\frac {-\frac {8 \left (-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 \left (a -b \right )}-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a -b \right )}\right )}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a}-\frac {a \left (\frac {\left (-\sqrt {-b \left (a -b \right )}-b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (-\sqrt {-b \left (a -b \right )}+b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{a -b}}{d}\) | \(275\) |
-(2*a*exp(2*d*x+2*c)-b*exp(2*d*x+2*c)+b)/b/d/(a-b)/(b*exp(4*d*x+4*c)+4*a*e xp(2*d*x+2*c)-2*b*exp(2*d*x+2*c)+b)+1/4/(a^2-a*b)^(1/2)/(a-b)/d*ln(exp(2*d *x+2*c)+(2*a*(a^2-a*b)^(1/2)-b*(a^2-a*b)^(1/2)+2*a^2-2*a*b)/b/(a^2-a*b)^(1 /2))-1/4/(a^2-a*b)^(1/2)/(a-b)/d*ln(exp(2*d*x+2*c)+(2*a*(a^2-a*b)^(1/2)-b* (a^2-a*b)^(1/2)-2*a^2+2*a*b)/b/(a^2-a*b)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 634 vs. \(2 (72) = 144\).
Time = 0.31 (sec) , antiderivative size = 1523, normalized size of antiderivative = 18.13 \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \]
[-1/4*(4*a^2*b - 4*a*b^2 + 4*(2*a^3 - 3*a^2*b + a*b^2)*cosh(d*x + c)^2 + 8 *(2*a^3 - 3*a^2*b + a*b^2)*cosh(d*x + c)*sinh(d*x + c) + 4*(2*a^3 - 3*a^2* b + a*b^2)*sinh(d*x + c)^2 + (b^2*cosh(d*x + c)^4 + 4*b^2*cosh(d*x + c)*si nh(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 + b^2 + 4*(b^2*cosh( d*x + c)^3 + (2*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a^2 - a*b)*l og((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*cosh(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)*sqrt(a^2 - a*b ))/(b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c )^4 + 2*(2*a - b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 + 2*a - b)*sinh (d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + (2*a - b)*cosh(d*x + c))*sinh(d*x + c ) + b)))/((a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^4 + 4*(a^3*b^2 - 2 *a^2*b^3 + a*b^4)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*sinh(d*x + c)^4 + 2*(2*a^4*b - 5*a^3*b^2 + 4*a^2*b^3 - a*b^4)*d* cosh(d*x + c)^2 + 2*(3*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^2 + ( 2*a^4*b - 5*a^3*b^2 + 4*a^2*b^3 - a*b^4)*d)*sinh(d*x + c)^2 + (a^3*b^2 - 2 *a^2*b^3 + a*b^4)*d + 4*((a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^...
Timed out. \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\int { \frac {\sinh \left (d x + c\right )^{2}}{{\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\sinh ^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^2}{{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \]