Integrand size = 21, antiderivative size = 81 \[ \int \frac {\sinh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\frac {\arctan \left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{2 (a-b)^{3/2} \sqrt {b} d}+\frac {\cosh (c+d x)}{2 (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )} \]
1/2*cosh(d*x+c)/(a-b)/d/(a-b+b*cosh(d*x+c)^2)+1/2*arctan(cosh(d*x+c)*b^(1/ 2)/(a-b)^(1/2))/(a-b)^(3/2)/d/b^(1/2)
Result contains complex when optimal does not.
Time = 0.54 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.60 \[ \int \frac {\sinh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\frac {\frac {\arctan \left (\frac {\sqrt {b}-i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )+\arctan \left (\frac {\sqrt {b}+i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )}{(a-b)^{3/2} \sqrt {b}}+\frac {2 \cosh (c+d x)}{(a-b) (2 a-b+b \cosh (2 (c+d x)))}}{2 d} \]
((ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]] + ArcTan[(Sq rt[b] + I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]])/((a - b)^(3/2)*Sqrt[b]) + (2*Cosh[c + d*x])/((a - b)*(2*a - b + b*Cosh[2*(c + d*x)])))/(2*d)
Time = 0.25 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 26, 3665, 215, 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.
\(\displaystyle \int \frac {\sinh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \sin (i c+i d x)}{\left (a-b \sin (i c+i d x)^2\right )^2}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\sin (i c+i d x)}{\left (a-b \sin (i c+i d x)^2\right )^2}dx\) |
\(\Big \downarrow \) 3665 |
\(\displaystyle \frac {\int \frac {1}{\left (b \cosh ^2(c+d x)+a-b\right )^2}d\cosh (c+d x)}{d}\) |
\(\Big \downarrow \) 215 |
\(\displaystyle \frac {\frac {\int \frac {1}{b \cosh ^2(c+d x)+a-b}d\cosh (c+d x)}{2 (a-b)}+\frac {\cosh (c+d x)}{2 (a-b) \left (a+b \cosh ^2(c+d x)-b\right )}}{d}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{2 \sqrt {b} (a-b)^{3/2}}+\frac {\cosh (c+d x)}{2 (a-b) \left (a+b \cosh ^2(c+d x)-b\right )}}{d}\) |
(ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]]/(2*(a - b)^(3/2)*Sqrt[b]) + C osh[c + d*x]/(2*(a - b)*(a - b + b*Cosh[c + d*x]^2)))/d
3.1.45.3.1 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)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
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[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. \(145\) vs. \(2(69)=138\).
Time = 0.20 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.80
method | result | size |
derivativedivides | \(\frac {\frac {-\frac {\left (a -2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{\left (a -b \right ) a}+\frac {2}{2 a -2 b}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a}+\frac {\arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{2 \left (a -b \right ) \sqrt {a b -b^{2}}}}{d}\) | \(146\) |
default | \(\frac {\frac {-\frac {\left (a -2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{\left (a -b \right ) a}+\frac {2}{2 a -2 b}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a}+\frac {\arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{2 \left (a -b \right ) \sqrt {a b -b^{2}}}}{d}\) | \(146\) |
risch | \(\frac {{\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}+1\right )}{d \left (a -b \right ) \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a -b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a b +b^{2}}}+1\right )}{4 \sqrt {-a b +b^{2}}\, \left (a -b \right ) d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a -b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a b +b^{2}}}+1\right )}{4 \sqrt {-a b +b^{2}}\, \left (a -b \right ) d}\) | \(183\) |
1/d*(2*(-1/2*(a-2*b)/(a-b)/a*tanh(1/2*d*x+1/2*c)^2+1/2/(a-b))/(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)+1/2/(a -b)/(a*b-b^2)^(1/2)*arctan(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a-2*a+4*b)/(a*b-b^ 2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 787 vs. \(2 (69) = 138\).
Time = 0.32 (sec) , antiderivative size = 1628, normalized size of antiderivative = 20.10 \[ \int \frac {\sinh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \]
[1/4*(4*(a*b - b^2)*cosh(d*x + c)^3 + 12*(a*b - b^2)*cosh(d*x + c)*sinh(d* x + c)^2 + 4*(a*b - b^2)*sinh(d*x + c)^3 + (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)*sqrt(-a*b + b^2)*log((b*cos h(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 - 2*( 2*a - 3*b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 - 2*a + 3*b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 - (2*a - 3*b)*cosh(d*x + c))*sinh(d*x + c) + 4*(cosh(d*x + c)^3 + 3*cosh(d*x + c)*sinh(d*x + c)^2 + sinh(d*x + c)^3 + (3*cosh(d*x + c)^2 + 1)*sinh(d*x + c) + cosh(d*x + c))*sqrt(-a*b + b^2) + 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)) + 4*(a*b - b^2)*cosh(d*x + c) + 4*(3*(a*b - b^2)*cosh(d*x + c)^2 + a*b - b^2)*sinh(d*x + c))/((a^2*b^2 - 2*a*b^3 + b^4)*d*cosh(d*x + c)^4 + 4*(a^2*b^2 - 2*a*b^3 + b^4)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2*b^2 - 2 *a*b^3 + b^4)*d*sinh(d*x + c)^4 + 2*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)* d*cosh(d*x + c)^2 + 2*(3*(a^2*b^2 - 2*a*b^3 + b^4)*d*cosh(d*x + c)^2 + (2* a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*d)*sinh(d*x + c)^2 + (a^2*b^2 - 2*a*b^3 + b^4)*d + 4*((a^2*b^2 - 2*a*b^3 + b^4)*d*cosh(d*x + c)^3 + (2*a^3*b -...
Timed out. \[ \int \frac {\sinh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {\sinh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\int { \frac {\sinh \left (d x + c\right )}{{\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
(e^(3*d*x + 3*c) + e^(d*x + c))/(a*b*d - b^2*d + (a*b*d*e^(4*c) - b^2*d*e^ (4*c))*e^(4*d*x) + 2*(2*a^2*d*e^(2*c) - 3*a*b*d*e^(2*c) + b^2*d*e^(2*c))*e ^(2*d*x)) + 1/2*integrate(2*(e^(3*d*x + 3*c) - e^(d*x + c))/(a*b - b^2 + ( a*b*e^(4*c) - b^2*e^(4*c))*e^(4*d*x) + 2*(2*a^2*e^(2*c) - 3*a*b*e^(2*c) + b^2*e^(2*c))*e^(2*d*x)), x)
\[ \int \frac {\sinh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\int { \frac {\sinh \left (d x + c\right )}{{\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\sinh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\int \frac {\mathrm {sinh}\left (c+d\,x\right )}{{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \]