Integrand size = 23, antiderivative size = 42 \[ \int \frac {\text {csch}(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{\sqrt {a} f} \]
Time = 0.31 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.17 \[ \int \frac {\text {csch}(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \cosh (e+f x)}{\sqrt {2 a-b+b \cosh (2 (e+f x))}}\right )}{\sqrt {a} f} \]
Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 26, 3665, 291, 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.
\(\displaystyle \int \frac {\text {csch}(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i}{\sin (i e+i f x) \sqrt {a-b \sin (i e+i f x)^2}}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {1}{\sin (i e+i f x) \sqrt {a-b \sin (i e+i f x)^2}}dx\) |
\(\Big \downarrow \) 3665 |
\(\displaystyle -\frac {\int \frac {1}{\left (1-\cosh ^2(e+f x)\right ) \sqrt {b \cosh ^2(e+f x)+a-b}}d\cosh (e+f x)}{f}\) |
\(\Big \downarrow \) 291 |
\(\displaystyle -\frac {\int \frac {1}{1-\frac {a \cosh ^2(e+f x)}{b \cosh ^2(e+f x)+a-b}}d\frac {\cosh (e+f x)}{\sqrt {b \cosh ^2(e+f x)+a-b}}}{f}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\text {arctanh}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a+b \cosh ^2(e+f x)-b}}\right )}{\sqrt {a} f}\) |
3.1.100.3.1 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[(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[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
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. \(112\) vs. \(2(36)=72\).
Time = 0.10 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.69
method | result | size |
default | \(-\frac {\sqrt {\left (a +b \sinh \left (f x +e \right )^{2}\right ) \cosh \left (f x +e \right )^{2}}\, \ln \left (\frac {\left (a +b \right ) \cosh \left (f x +e \right )^{2}+2 \sqrt {a}\, \sqrt {b \cosh \left (f x +e \right )^{4}+\left (a -b \right ) \cosh \left (f x +e \right )^{2}}+a -b}{\sinh \left (f x +e \right )^{2}}\right )}{2 \sqrt {a}\, \cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\) | \(113\) |
-1/2*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)/a^(1/2)*ln(((a+b)*cosh(f*x+ e)^2+2*a^(1/2)*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)+a-b)/sinh(f*x+e )^2)/cosh(f*x+e)/(a+b*sinh(f*x+e)^2)^(1/2)/f
Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (36) = 72\).
Time = 0.27 (sec) , antiderivative size = 572, normalized size of antiderivative = 13.62 \[ \int \frac {\text {csch}(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\left [\frac {\log \left (-\frac {{\left (a + b\right )} \cosh \left (f x + e\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + {\left (a + b\right )} \sinh \left (f x + e\right )^{4} + 2 \, {\left (3 \, a - b\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (f x + e\right )^{2} + 3 \, a - b\right )} \sinh \left (f x + e\right )^{2} - 2 \, \sqrt {2} {\left (\cosh \left (f x + e\right )^{2} + 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2} + 1\right )} \sqrt {a} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}} + 4 \, {\left ({\left (a + b\right )} \cosh \left (f x + e\right )^{3} + {\left (3 \, a - b\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + a + b}{\cosh \left (f x + e\right )^{4} + 4 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + \sinh \left (f x + e\right )^{4} + 2 \, {\left (3 \, \cosh \left (f x + e\right )^{2} - 1\right )} \sinh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right )^{2} + 4 \, {\left (\cosh \left (f x + e\right )^{3} - \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + 1}\right )}{2 \, \sqrt {a} f}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {2} {\left (\cosh \left (f x + e\right )^{2} + 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2} + 1\right )} \sqrt {-a} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}}}{b \cosh \left (f x + e\right )^{4} + 4 \, b \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + b \sinh \left (f x + e\right )^{4} + 2 \, {\left (2 \, a - b\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, b \cosh \left (f x + e\right )^{2} + 2 \, a - b\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left (b \cosh \left (f x + e\right )^{3} + {\left (2 \, a - b\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + b}\right )}{a f}\right ] \]
[1/2*log(-((a + b)*cosh(f*x + e)^4 + 4*(a + b)*cosh(f*x + e)*sinh(f*x + e) ^3 + (a + b)*sinh(f*x + e)^4 + 2*(3*a - b)*cosh(f*x + e)^2 + 2*(3*(a + b)* cosh(f*x + e)^2 + 3*a - b)*sinh(f*x + e)^2 - 2*sqrt(2)*(cosh(f*x + e)^2 + 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 + 1)*sqrt(a)*sqrt((b*cosh( f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)) + 4*((a + b)*cosh(f*x + e)^3 + (3*a - b)*cosh(f*x + e))*sinh(f*x + e) + a + b)/(cosh(f*x + e)^4 + 4*cosh(f*x + e)*sinh(f*x + e)^3 + sinh(f*x + e)^4 + 2*(3*cosh(f*x + e)^2 - 1)*sinh(f*x + e)^2 - 2*cosh(f*x + e)^2 + 4*(cosh(f*x + e)^3 - cosh(f*x + e))*sinh(f*x + e) + 1))/(sqrt(a)*f), sqrt(-a)*arctan(sqrt(2)*(cosh(f*x + e)^2 + 2*cosh( f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 + 1)*sqrt(-a)*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sin h(f*x + e) + sinh(f*x + e)^2))/(b*cosh(f*x + e)^4 + 4*b*cosh(f*x + e)*sinh (f*x + e)^3 + b*sinh(f*x + e)^4 + 2*(2*a - b)*cosh(f*x + e)^2 + 2*(3*b*cos h(f*x + e)^2 + 2*a - b)*sinh(f*x + e)^2 + 4*(b*cosh(f*x + e)^3 + (2*a - b) *cosh(f*x + e))*sinh(f*x + e) + b))/(a*f)]
\[ \int \frac {\text {csch}(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\int \frac {\operatorname {csch}{\left (e + f x \right )}}{\sqrt {a + b \sinh ^{2}{\left (e + f x \right )}}}\, dx \]
\[ \int \frac {\text {csch}(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\int { \frac {\operatorname {csch}\left (f x + e\right )}{\sqrt {b \sinh \left (f x + e\right )^{2} + a}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (36) = 72\).
Time = 0.36 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.86 \[ \int \frac {\text {csch}(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\frac {2 \, \arctan \left (-\frac {\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b} - \sqrt {b}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} f} \]
2*arctan(-1/2*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2 *f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b) - sqrt(b))/sqrt(-a))/(sqrt(-a)*f)
Timed out. \[ \int \frac {\text {csch}(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\int \frac {1}{\mathrm {sinh}\left (e+f\,x\right )\,\sqrt {b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a}} \,d x \]