Integrand size = 25, antiderivative size = 139 \[ \int \frac {\text {csch}^3(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\frac {(a+3 b) \text {arctanh}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{2 a^{5/2} f}-\frac {(a-3 b) b \cosh (e+f x)}{2 a^2 (a-b) f \sqrt {a-b+b \cosh ^2(e+f x)}}-\frac {\coth (e+f x) \text {csch}(e+f x)}{2 a f \sqrt {a-b+b \cosh ^2(e+f x)}} \]
1/2*(a+3*b)*arctanh(cosh(f*x+e)*a^(1/2)/(a-b+b*cosh(f*x+e)^2)^(1/2))/a^(5/ 2)/f-1/2*(a-3*b)*b*cosh(f*x+e)/a^2/(a-b)/f/(a-b+b*cosh(f*x+e)^2)^(1/2)-1/2 *coth(f*x+e)*csch(f*x+e)/a/f/(a-b+b*cosh(f*x+e)^2)^(1/2)
Time = 0.79 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.96 \[ \int \frac {\text {csch}^3(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\frac {\frac {(a+3 b) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \cosh (e+f x)}{\sqrt {2 a-b+b \cosh (2 (e+f x))}}\right )}{a^{5/2}}-\frac {\left (2 a^2-3 a b+3 b^2+(a-3 b) b \cosh (2 (e+f x))\right ) \coth (e+f x) \text {csch}(e+f x)}{a^2 (a-b) \sqrt {4 a-2 b+2 b \cosh (2 (e+f x))}}}{2 f} \]
(((a + 3*b)*ArcTanh[(Sqrt[2]*Sqrt[a]*Cosh[e + f*x])/Sqrt[2*a - b + b*Cosh[ 2*(e + f*x)]]])/a^(5/2) - ((2*a^2 - 3*a*b + 3*b^2 + (a - 3*b)*b*Cosh[2*(e + f*x)])*Coth[e + f*x]*Csch[e + f*x])/(a^2*(a - b)*Sqrt[4*a - 2*b + 2*b*Co sh[2*(e + f*x)]]))/(2*f)
Time = 0.35 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 26, 3665, 316, 402, 25, 27, 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.
| \(\displaystyle \int \frac {\text {csch}^3(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i}{\sin (i e+i f x)^3 \left (a-b \sin (i e+i f x)^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {1}{\sin (i e+i f x)^3 \left (a-b \sin (i e+i f x)^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3665 |
| \(\displaystyle \frac {\int \frac {1}{\left (1-\cosh ^2(e+f x)\right )^2 \left (b \cosh ^2(e+f x)+a-b\right )^{3/2}}d\cosh (e+f x)}{f}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\frac {\int \frac {2 b \cosh ^2(e+f x)+a+b}{\left (1-\cosh ^2(e+f x)\right ) \left (b \cosh ^2(e+f x)+a-b\right )^{3/2}}d\cosh (e+f x)}{2 a}+\frac {\cosh (e+f x)}{2 a \left (1-\cosh ^2(e+f x)\right ) \sqrt {a+b \cosh ^2(e+f x)-b}}}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {(a-b) (a+3 b)}{\left (1-\cosh ^2(e+f x)\right ) \sqrt {b \cosh ^2(e+f x)+a-b}}d\cosh (e+f x)}{a (a-b)}-\frac {b (a-3 b) \cosh (e+f x)}{a (a-b) \sqrt {a+b \cosh ^2(e+f x)-b}}}{2 a}+\frac {\cosh (e+f x)}{2 a \left (1-\cosh ^2(e+f x)\right ) \sqrt {a+b \cosh ^2(e+f x)-b}}}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {(a-b) (a+3 b)}{\left (1-\cosh ^2(e+f x)\right ) \sqrt {b \cosh ^2(e+f x)+a-b}}d\cosh (e+f x)}{a (a-b)}-\frac {b (a-3 b) \cosh (e+f x)}{a (a-b) \sqrt {a+b \cosh ^2(e+f x)-b}}}{2 a}+\frac {\cosh (e+f x)}{2 a \left (1-\cosh ^2(e+f x)\right ) \sqrt {a+b \cosh ^2(e+f x)-b}}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {(a+3 b) \int \frac {1}{\left (1-\cosh ^2(e+f x)\right ) \sqrt {b \cosh ^2(e+f x)+a-b}}d\cosh (e+f x)}{a}-\frac {b (a-3 b) \cosh (e+f x)}{a (a-b) \sqrt {a+b \cosh ^2(e+f x)-b}}}{2 a}+\frac {\cosh (e+f x)}{2 a \left (1-\cosh ^2(e+f x)\right ) \sqrt {a+b \cosh ^2(e+f x)-b}}}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {\frac {\frac {(a+3 b) \int \frac {1}{1-\frac {a \cosh ^2(e+f x)}{b \cosh ^2(e+f x)+a-b}}d\frac {\cosh (e+f x)}{\sqrt {b \cosh ^2(e+f x)+a-b}}}{a}-\frac {b (a-3 b) \cosh (e+f x)}{a (a-b) \sqrt {a+b \cosh ^2(e+f x)-b}}}{2 a}+\frac {\cosh (e+f x)}{2 a \left (1-\cosh ^2(e+f x)\right ) \sqrt {a+b \cosh ^2(e+f x)-b}}}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {(a+3 b) \text {arctanh}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a+b \cosh ^2(e+f x)-b}}\right )}{a^{3/2}}-\frac {b (a-3 b) \cosh (e+f x)}{a (a-b) \sqrt {a+b \cosh ^2(e+f x)-b}}}{2 a}+\frac {\cosh (e+f x)}{2 a \left (1-\cosh ^2(e+f x)\right ) \sqrt {a+b \cosh ^2(e+f x)-b}}}{f}\) |
(Cosh[e + f*x]/(2*a*(1 - Cosh[e + f*x]^2)*Sqrt[a - b + b*Cosh[e + f*x]^2]) + (((a + 3*b)*ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x ]^2]])/a^(3/2) - ((a - 3*b)*b*Cosh[e + f*x])/(a*(a - b)*Sqrt[a - b + b*Cos h[e + f*x]^2]))/(2*a))/f
3.2.10.3.1 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_)*(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[(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[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[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. \(250\) vs. \(2(123)=246\).
Time = 0.10 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.81
\[\frac {\sqrt {\left (a +b \sinh \left (f x +e \right )^{2}\right ) \cosh \left (f x +e \right )^{2}}\, \left (-\frac {\sqrt {\left (a +b \sinh \left (f x +e \right )^{2}\right ) \cosh \left (f x +e \right )^{2}}}{2 a^{2} \sinh \left (f x +e \right )^{2}}+\frac {\ln \left (\frac {2 a +\left (a +b \right ) \sinh \left (f x +e \right )^{2}+2 \sqrt {a}\, \sqrt {\left (a +b \sinh \left (f x +e \right )^{2}\right ) \cosh \left (f x +e \right )^{2}}}{\sinh \left (f x +e \right )^{2}}\right )}{4 a^{\frac {3}{2}}}+\frac {3 b \ln \left (\frac {2 a +\left (a +b \right ) \sinh \left (f x +e \right )^{2}+2 \sqrt {a}\, \sqrt {\left (a +b \sinh \left (f x +e \right )^{2}\right ) \cosh \left (f x +e \right )^{2}}}{\sinh \left (f x +e \right )^{2}}\right )}{4 a^{\frac {5}{2}}}+\frac {b^{2} \cosh \left (f x +e \right )^{2}}{a^{2} \left (a -b \right ) \sqrt {\left (a +b \sinh \left (f x +e \right )^{2}\right ) \cosh \left (f x +e \right )^{2}}}\right )}{\cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\]
((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*(-1/2/a^2/sinh(f*x+e)^2*((a+b*si nh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)+1/4/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)+3/4*b/ a^(5/2)*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^2/a^2*cosh(f*x+e)^2/(a-b)/((a+b*sinh(f*x+e )^2)*cosh(f*x+e)^2)^(1/2))/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. 2169 vs. \(2 (123) = 246\).
Time = 0.48 (sec) , antiderivative size = 4441, normalized size of antiderivative = 31.95 \[ \int \frac {\text {csch}^3(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\text {Too large to display} \]
[1/4*(((a^2*b + 2*a*b^2 - 3*b^3)*cosh(f*x + e)^8 + 8*(a^2*b + 2*a*b^2 - 3* b^3)*cosh(f*x + e)*sinh(f*x + e)^7 + (a^2*b + 2*a*b^2 - 3*b^3)*sinh(f*x + e)^8 + 4*(a^3 + a^2*b - 5*a*b^2 + 3*b^3)*cosh(f*x + e)^6 + 4*(a^3 + a^2*b - 5*a*b^2 + 3*b^3 + 7*(a^2*b + 2*a*b^2 - 3*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^6 + 8*(7*(a^2*b + 2*a*b^2 - 3*b^3)*cosh(f*x + e)^3 + 3*(a^3 + a^2*b - 5*a*b^2 + 3*b^3)*cosh(f*x + e))*sinh(f*x + e)^5 - 2*(4*a^3 + 5*a^2*b - 18 *a*b^2 + 9*b^3)*cosh(f*x + e)^4 + 2*(35*(a^2*b + 2*a*b^2 - 3*b^3)*cosh(f*x + e)^4 - 4*a^3 - 5*a^2*b + 18*a*b^2 - 9*b^3 + 30*(a^3 + a^2*b - 5*a*b^2 + 3*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 8*(7*(a^2*b + 2*a*b^2 - 3*b^3)* cosh(f*x + e)^5 + 10*(a^3 + a^2*b - 5*a*b^2 + 3*b^3)*cosh(f*x + e)^3 - (4* a^3 + 5*a^2*b - 18*a*b^2 + 9*b^3)*cosh(f*x + e))*sinh(f*x + e)^3 + a^2*b + 2*a*b^2 - 3*b^3 + 4*(a^3 + a^2*b - 5*a*b^2 + 3*b^3)*cosh(f*x + e)^2 + 4*( 7*(a^2*b + 2*a*b^2 - 3*b^3)*cosh(f*x + e)^6 + 15*(a^3 + a^2*b - 5*a*b^2 + 3*b^3)*cosh(f*x + e)^4 + a^3 + a^2*b - 5*a*b^2 + 3*b^3 - 3*(4*a^3 + 5*a^2* b - 18*a*b^2 + 9*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 8*((a^2*b + 2*a*b ^2 - 3*b^3)*cosh(f*x + e)^7 + 3*(a^3 + a^2*b - 5*a*b^2 + 3*b^3)*cosh(f*x + e)^5 - (4*a^3 + 5*a^2*b - 18*a*b^2 + 9*b^3)*cosh(f*x + e)^3 + (a^3 + a^2* b - 5*a*b^2 + 3*b^3)*cosh(f*x + e))*sinh(f*x + e))*sqrt(a)*log(-((a + b)*c osh(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 +...
\[ \int \frac {\text {csch}^3(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\operatorname {csch}^{3}{\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\text {csch}^3(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\operatorname {csch}\left (f x + e\right )^{3}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
Exception generated. \[ \int \frac {\text {csch}^3(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {\text {csch}^3(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {1}{{\mathrm {sinh}\left (e+f\,x\right )}^3\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \]