Integrand size = 23, antiderivative size = 84 \[ \int \frac {\text {csch}(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{a^{3/2} f}-\frac {b \cosh (e+f x)}{a (a-b) f \sqrt {a-b+b \cosh ^2(e+f x)}} \]
-arctanh(cosh(f*x+e)*a^(1/2)/(a-b+b*cosh(f*x+e)^2)^(1/2))/a^(3/2)/f-b*cosh (f*x+e)/a/(a-b)/f/(a-b+b*cosh(f*x+e)^2)^(1/2)
Time = 0.49 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.17 \[ \int \frac {\text {csch}(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\frac {-\text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \cosh (e+f x)}{\sqrt {2 a-b+b \cosh (2 (e+f x))}}\right )-\frac {\sqrt {2} \sqrt {a} b \cosh (e+f x)}{(a-b) \sqrt {2 a-b+b \cosh (2 (e+f x))}}}{a^{3/2} f} \]
(-ArcTanh[(Sqrt[2]*Sqrt[a]*Cosh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x )]]] - (Sqrt[2]*Sqrt[a]*b*Cosh[e + f*x])/((a - b)*Sqrt[2*a - b + b*Cosh[2* (e + f*x)]]))/(a^(3/2)*f)
Time = 0.29 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 26, 3665, 296, 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.
\(\displaystyle \int \frac {\text {csch}(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i}{\sin (i e+i f x) \left (a-b \sin (i e+i f x)^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {1}{\sin (i e+i f x) \left (a-b \sin (i e+i f x)^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3665 |
\(\displaystyle -\frac {\int \frac {1}{\left (1-\cosh ^2(e+f x)\right ) \left (b \cosh ^2(e+f x)+a-b\right )^{3/2}}d\cosh (e+f x)}{f}\) |
\(\Big \downarrow \) 296 |
\(\displaystyle -\frac {\frac {\int \frac {1}{\left (1-\cosh ^2(e+f x)\right ) \sqrt {b \cosh ^2(e+f x)+a-b}}d\cosh (e+f x)}{a}+\frac {b \cosh (e+f x)}{a (a-b) \sqrt {a+b \cosh ^2(e+f x)-b}}}{f}\) |
\(\Big \downarrow \) 291 |
\(\displaystyle -\frac {\frac {\int \frac {1}{1-\frac {a \cosh ^2(e+f x)}{b \cosh ^2(e+f x)+a-b}}d\frac {\cosh (e+f x)}{\sqrt {b \cosh ^2(e+f x)+a-b}}}{a}+\frac {b \cosh (e+f x)}{a (a-b) \sqrt {a+b \cosh ^2(e+f x)-b}}}{f}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\frac {\text {arctanh}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a+b \cosh ^2(e+f x)-b}}\right )}{a^{3/2}}+\frac {b \cosh (e+f x)}{a (a-b) \sqrt {a+b \cosh ^2(e+f x)-b}}}{f}\) |
-((ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]]/a^(3/2 ) + (b*Cosh[e + f*x])/(a*(a - b)*Sqrt[a - b + b*Cosh[e + f*x]^2]))/f)
3.2.9.3.1 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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[(b*c + 2*(p + 1)*(b*c - a*d))/(2*a*(p + 1)*(b*c - a*d)) Int[ (a + b*x^2)^(p + 1)*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, q}, x] && N eQ[b*c - a*d, 0] && EqQ[2*(p + q + 2) + 1, 0] && (LtQ[p, -1] || !LtQ[q, -1 ]) && NeQ[p, -1]
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. \(153\) vs. \(2(76)=152\).
Time = 0.12 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.83
method | result | size |
default | \(\frac {\sqrt {\left (a +b \sinh \left (f x +e \right )^{2}\right ) \cosh \left (f x +e \right )^{2}}\, \left (-\frac {\ln \left (\frac {2 a +\left (a +b \right ) \sinh \left (f x +e \right )^{2}+2 \sqrt {a}\, \sqrt {\left (a +b \sinh \left (f x +e \right )^{2}\right ) \cosh \left (f x +e \right )^{2}}}{\sinh \left (f x +e \right )^{2}}\right )}{2 a^{\frac {3}{2}}}-\frac {b \cosh \left (f x +e \right )^{2}}{a \left (a -b \right ) \sqrt {\left (a +b \sinh \left (f x +e \right )^{2}\right ) \cosh \left (f x +e \right )^{2}}}\right )}{\cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\) | \(154\) |
((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*(-1/2/a^(3/2)*ln((2*a+(a+b)*sinh (f*x+e)^2+2*a^(1/2)*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2))/sinh(f*x+e) ^2)-b/a*cosh(f*x+e)^2/(a-b)/((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2))/cos h(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. 769 vs. \(2 (76) = 152\).
Time = 0.31 (sec) , antiderivative size = 1641, normalized size of antiderivative = 19.54 \[ \int \frac {\text {csch}(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\text {Too large to display} \]
[1/2*(((a*b - b^2)*cosh(f*x + e)^4 + 4*(a*b - b^2)*cosh(f*x + e)*sinh(f*x + e)^3 + (a*b - b^2)*sinh(f*x + e)^4 + 2*(2*a^2 - 3*a*b + b^2)*cosh(f*x + e)^2 + 2*(3*(a*b - b^2)*cosh(f*x + e)^2 + 2*a^2 - 3*a*b + b^2)*sinh(f*x + e)^2 + a*b - b^2 + 4*((a*b - b^2)*cosh(f*x + e)^3 + (2*a^2 - 3*a*b + b^2)* cosh(f*x + e))*sinh(f*x + e))*sqrt(a)*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)) - 2*sqrt(2)*(a*b*cosh(f*x + e) ^2 + 2*a*b*cosh(f*x + e)*sinh(f*x + e) + a*b*sinh(f*x + e)^2 + a*b)*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)))/((a^3*b - a^2*b^2)*f*cosh(f*x + e)^4 + 4*(a^3*b - a^2*b^2)*f*cosh(f*x + e)*sinh(f*x + e)^3 + (a^3*b - a ^2*b^2)*f*sinh(f*x + e)^4 + 2*(2*a^4 - 3*a^3*b + a^2*b^2)*f*cosh(f*x + e)^ 2 + 2*(3*(a^3*b - a^2*b^2)*f*cosh(f*x + e)^2 + (2*a^4 - 3*a^3*b + a^2*b...
\[ \int \frac {\text {csch}(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\operatorname {csch}{\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\text {csch}(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\operatorname {csch}\left (f x + e\right )}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (76) = 152\).
Time = 0.48 (sec) , antiderivative size = 198, normalized size of antiderivative = 2.36 \[ \int \frac {\text {csch}(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=-\frac {{\left (\frac {\frac {a^{2} b e^{\left (2 \, f x + 4 \, e\right )}}{a^{4} e^{\left (6 \, e\right )} - a^{3} b e^{\left (6 \, e\right )}} + \frac {a^{2} b e^{\left (2 \, e\right )}}{a^{4} e^{\left (6 \, e\right )} - a^{3} b e^{\left (6 \, e\right )}}}{\sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b}} - \frac {2 \, \arctan \left (-\frac {\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b} - \sqrt {b}}{2 \, \sqrt {-a}}\right ) e^{\left (-4 \, e\right )}}{\sqrt {-a} a}\right )} e^{\left (4 \, e\right )}}{f} \]
-((a^2*b*e^(2*f*x + 4*e)/(a^4*e^(6*e) - a^3*b*e^(6*e)) + a^2*b*e^(2*e)/(a^ 4*e^(6*e) - a^3*b*e^(6*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) - 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))*e^(-4*e)/(sqrt(-a)*a))*e^(4*e)/f
Timed out. \[ \int \frac {\text {csch}(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {1}{\mathrm {sinh}\left (e+f\,x\right )\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \]