Integrand size = 25, antiderivative size = 89 \[ \int \frac {\text {csch}^3(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\frac {(a+b) \text {arctanh}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{2 a^{3/2} f}-\frac {\sqrt {a-b+b \cosh ^2(e+f x)} \coth (e+f x) \text {csch}(e+f x)}{2 a f} \]
1/2*(a+b)*arctanh(cosh(f*x+e)*a^(1/2)/(a-b+b*cosh(f*x+e)^2)^(1/2))/a^(3/2) /f-1/2*coth(f*x+e)*csch(f*x+e)*(a-b+b*cosh(f*x+e)^2)^(1/2)/a/f
Time = 0.52 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.15 \[ \int \frac {\text {csch}^3(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\frac {2 (a+b) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \cosh (e+f x)}{\sqrt {2 a-b+b \cosh (2 (e+f x))}}\right )-\sqrt {2} \sqrt {a} \sqrt {2 a-b+b \cosh (2 (e+f x))} \coth (e+f x) \text {csch}(e+f x)}{4 a^{3/2} f} \]
(2*(a + b)*ArcTanh[(Sqrt[2]*Sqrt[a]*Cosh[e + f*x])/Sqrt[2*a - b + b*Cosh[2 *(e + f*x)]]] - Sqrt[2]*Sqrt[a]*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]*Coth[e + f*x]*Csch[e + f*x])/(4*a^(3/2)*f)
Time = 0.30 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, 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.
| \(\displaystyle \int \frac {\text {csch}^3(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i}{\sin (i e+i f x)^3 \sqrt {a-b \sin (i e+i f x)^2}}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {1}{\sin (i e+i f x)^3 \sqrt {a-b \sin (i e+i f x)^2}}dx\) |
| \(\Big \downarrow \) 3665 |
| \(\displaystyle \frac {\int \frac {1}{\left (1-\cosh ^2(e+f x)\right )^2 \sqrt {b \cosh ^2(e+f x)+a-b}}d\cosh (e+f x)}{f}\) |
| \(\Big \downarrow \) 296 |
| \(\displaystyle \frac {\frac {(a+b) \int \frac {1}{\left (1-\cosh ^2(e+f x)\right ) \sqrt {b \cosh ^2(e+f x)+a-b}}d\cosh (e+f x)}{2 a}+\frac {\cosh (e+f x) \sqrt {a+b \cosh ^2(e+f x)-b}}{2 a \left (1-\cosh ^2(e+f x)\right )}}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {\frac {(a+b) \int \frac {1}{1-\frac {a \cosh ^2(e+f x)}{b \cosh ^2(e+f x)+a-b}}d\frac {\cosh (e+f x)}{\sqrt {b \cosh ^2(e+f x)+a-b}}}{2 a}+\frac {\cosh (e+f x) \sqrt {a+b \cosh ^2(e+f x)-b}}{2 a \left (1-\cosh ^2(e+f x)\right )}}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {(a+b) \text {arctanh}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a+b \cosh ^2(e+f x)-b}}\right )}{2 a^{3/2}}+\frac {\cosh (e+f x) \sqrt {a+b \cosh ^2(e+f x)-b}}{2 a \left (1-\cosh ^2(e+f x)\right )}}{f}\) |
(((a + b)*ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]] )/(2*a^(3/2)) + (Cosh[e + f*x]*Sqrt[a - b + b*Cosh[e + f*x]^2])/(2*a*(1 - Cosh[e + f*x]^2)))/f
3.2.1.3.1 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]
Time = 1.82 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.67
| method | result | size |
| default | \(\frac {\sqrt {\left (a +b \sinh \left (f x +e \right )^{2}\right ) \cosh \left (f x +e \right )^{2}}\, \left (-\frac {\sqrt {\left (a +b \sinh \left (f x +e \right )^{2}\right ) \cosh \left (f x +e \right )^{2}}}{2 a \sinh \left (f x +e \right )^{2}}+\frac {\left (a +b \right ) \ln \left (\frac {2 a +\left (a +b \right ) \sinh \left (f x +e \right )^{2}+2 \sqrt {a}\, \sqrt {\left (a +b \sinh \left (f x +e \right )^{2}\right ) \cosh \left (f x +e \right )^{2}}}{\sinh \left (f x +e \right )^{2}}\right )}{4 a^{\frac {3}{2}}}\right )}{\cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\) | \(149\) |
| risch | \(\text {Expression too large to display}\) | \(107926\) |
((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*(-1/2/a/sinh(f*x+e)^2*((a+b*sinh (f*x+e)^2)*cosh(f*x+e)^2)^(1/2)+1/4*(a+b)/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))/c osh(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. 591 vs. \(2 (77) = 154\).
Time = 0.31 (sec) , antiderivative size = 1285, normalized size of antiderivative = 14.44 \[ \int \frac {\text {csch}^3(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\text {Too large to display} \]
[1/4*(((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*(a + b)*cosh(f*x + e)^2 + 2*(3*(a + b)*cosh(f *x + e)^2 - a - b)*sinh(f*x + e)^2 + 4*((a + b)*cosh(f*x + e)^3 - (a + b)* cosh(f*x + e))*sinh(f*x + e) + a + b)*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) + s inh(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*(c osh(f*x + e)^3 - cosh(f*x + e))*sinh(f*x + e) + 1)) - 2*sqrt(2)*(a*cosh(f* x + e)^2 + 2*a*cosh(f*x + e)*sinh(f*x + e) + a*sinh(f*x + e)^2 + 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)))/(a^2*f*cosh(f*x + e)^4 + 4*a^ 2*f*cosh(f*x + e)*sinh(f*x + e)^3 + a^2*f*sinh(f*x + e)^4 - 2*a^2*f*cosh(f *x + e)^2 + a^2*f + 2*(3*a^2*f*cosh(f*x + e)^2 - a^2*f)*sinh(f*x + e)^2 + 4*(a^2*f*cosh(f*x + e)^3 - a^2*f*cosh(f*x + e))*sinh(f*x + e)), -1/2*(((a + b)*cosh(f*x + e)^4 + 4*(a + b)*cosh(f*x + e)*sinh(f*x + e)^3 + (a + b...
\[ \int \frac {\text {csch}^3(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\int \frac {\operatorname {csch}^{3}{\left (e + f x \right )}}{\sqrt {a + b \sinh ^{2}{\left (e + f x \right )}}}\, dx \]
\[ \int \frac {\text {csch}^3(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\int { \frac {\operatorname {csch}\left (f x + e\right )^{3}}{\sqrt {b \sinh \left (f x + e\right )^{2} + a}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 669 vs. \(2 (77) = 154\).
Time = 0.45 (sec) , antiderivative size = 669, normalized size of antiderivative = 7.52 \[ \int \frac {\text {csch}^3(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=-\frac {{\left (\frac {{\left (a + b\right )} \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} - \frac {2 \, {\left ({\left (\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}\right )}^{3} a + {\left (\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}\right )}^{3} b + 5 \, {\left (\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}\right )}^{2} a \sqrt {b} - 3 \, {\left (\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}\right )}^{2} b^{\frac {3}{2}} + 4 \, {\left (\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}\right )} a^{2} - 9 \, {\left (\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}\right )} a b + 3 \, {\left (\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}\right )} b^{2} - 4 \, a^{2} \sqrt {b} + 3 \, a b^{\frac {3}{2}} - b^{\frac {5}{2}}\right )} e^{\left (-4 \, e\right )}}{{\left ({\left (\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}\right )}^{2} - 2 \, {\left (\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}\right )} \sqrt {b} - 4 \, a + b\right )}^{2} a}\right )} e^{\left (4 \, e\right )}}{f} \]
-((a + b)*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) - 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))^3*a + (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))^3*b + 5*(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))^2*a*sqrt(b) - 3*(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))^2*b^(3/2) + 4*(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))*a^2 - 9*(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))*a*b + 3*(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))*b^2 - 4*a^2*sqrt(b) + 3*a*b^(3/2) - b^(5/2))*e^(-4*e)/(((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))^2 - 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) - 4*a + b)^2*a))*e^(4*e)/f
Timed out. \[ \int \frac {\text {csch}^3(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\int \frac {1}{{\mathrm {sinh}\left (e+f\,x\right )}^3\,\sqrt {b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a}} \,d x \]