Integrand size = 16, antiderivative size = 82 \[ \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a-b+b \coth ^2(c+d x)}}\right )}{a^{3/2} d}+\frac {b \coth (c+d x)}{a (a-b) d \sqrt {a-b+b \coth ^2(c+d x)}} \]
arctanh(coth(d*x+c)*a^(1/2)/(a-b+b*coth(d*x+c)^2)^(1/2))/a^(3/2)/d+b*coth( d*x+c)/a/(a-b)/d/(a-b+b*coth(d*x+c)^2)^(1/2)
Time = 0.35 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.80 \[ \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^{3/2}} \, dx=\frac {\text {csch}^2(c+d x) \left (\frac {2 \sqrt {a} b (-a+2 b+a \cosh (2 (c+d x))) \coth (c+d x)}{a-b}+\sqrt {2} (-a+2 b+a \cosh (2 (c+d x)))^{3/2} \text {csch}(c+d x) \log \left (\sqrt {2} \sqrt {a} \cosh (c+d x)+\sqrt {-a+2 b+a \cosh (2 (c+d x))}\right )\right )}{4 a^{3/2} d \left (a+b \text {csch}^2(c+d x)\right )^{3/2}} \]
(Csch[c + d*x]^2*((2*Sqrt[a]*b*(-a + 2*b + a*Cosh[2*(c + d*x)])*Coth[c + d *x])/(a - b) + Sqrt[2]*(-a + 2*b + a*Cosh[2*(c + d*x)])^(3/2)*Csch[c + d*x ]*Log[Sqrt[2]*Sqrt[a]*Cosh[c + d*x] + Sqrt[-a + 2*b + a*Cosh[2*(c + d*x)]] ]))/(4*a^(3/2)*d*(a + b*Csch[c + d*x]^2)^(3/2))
Time = 0.25 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3042, 4616, 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.
\(\displaystyle \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (a-b \sec \left (i c+i d x+\frac {\pi }{2}\right )^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4616 |
\(\displaystyle \frac {\int \frac {1}{\left (1-\coth ^2(c+d x)\right ) \left (b \coth ^2(c+d x)+a-b\right )^{3/2}}d\coth (c+d x)}{d}\) |
\(\Big \downarrow \) 296 |
\(\displaystyle \frac {\frac {\int \frac {1}{\left (1-\coth ^2(c+d x)\right ) \sqrt {b \coth ^2(c+d x)+a-b}}d\coth (c+d x)}{a}+\frac {b \coth (c+d x)}{a (a-b) \sqrt {a+b \coth ^2(c+d x)-b}}}{d}\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \frac {\frac {\int \frac {1}{1-\frac {a \coth ^2(c+d x)}{b \coth ^2(c+d x)+a-b}}d\frac {\coth (c+d x)}{\sqrt {b \coth ^2(c+d x)+a-b}}}{a}+\frac {b \coth (c+d x)}{a (a-b) \sqrt {a+b \coth ^2(c+d x)-b}}}{d}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {\text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+b \coth ^2(c+d x)-b}}\right )}{a^{3/2}}+\frac {b \coth (c+d x)}{a (a-b) \sqrt {a+b \coth ^2(c+d x)-b}}}{d}\) |
(ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a - b + b*Coth[c + d*x]^2]]/a^(3/2) + (b*Coth[c + d*x])/(a*(a - b)*Sqrt[a - b + b*Coth[c + d*x]^2]))/d
3.1.13.3.1 Defintions of rubi rules used
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[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[(a + b + b*ff^2*x^2)^p /(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && NeQ[a + b, 0] && NeQ[p, -1]
\[\int \frac {1}{\left (a +b \operatorname {csch}\left (d x +c \right )^{2}\right )^{\frac {3}{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 1166 vs. \(2 (74) = 148\).
Time = 0.35 (sec) , antiderivative size = 3009, normalized size of antiderivative = 36.70 \[ \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^{3/2}} \, dx=\text {Too large to display} \]
[1/4*(((a^2 - a*b)*cosh(d*x + c)^4 + 4*(a^2 - a*b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 - a*b)*sinh(d*x + c)^4 - 2*(a^2 - 3*a*b + 2*b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 - a*b)*cosh(d*x + c)^2 - a^2 + 3*a*b - 2*b^2)*sinh(d*x + c)^2 + a^2 - a*b + 4*((a^2 - a*b)*cosh(d*x + c)^3 - (a^2 - 3*a*b + 2*b^2)* cosh(d*x + c))*sinh(d*x + c))*sqrt(a)*log((a*b^2*cosh(d*x + c)^8 + 8*a*b^2 *cosh(d*x + c)*sinh(d*x + c)^7 + a*b^2*sinh(d*x + c)^8 + 2*(a*b^2 + b^3)*c osh(d*x + c)^6 + 2*(14*a*b^2*cosh(d*x + c)^2 + a*b^2 + b^3)*sinh(d*x + c)^ 6 + 4*(14*a*b^2*cosh(d*x + c)^3 + 3*(a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + (a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x + c)^4 + (70*a*b^2*cosh(d*x + c)^4 + a^3 - 4*a^2*b + 9*a*b^2 + 30*(a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d* x + c)^4 + 4*(14*a*b^2*cosh(d*x + c)^5 + 10*(a*b^2 + b^3)*cosh(d*x + c)^3 + (a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + a^3 - 2*(a^3 - 3*a^2*b)*cosh(d*x + c)^2 + 2*(14*a*b^2*cosh(d*x + c)^6 + 15*(a*b^2 + b^3 )*cosh(d*x + c)^4 - a^3 + 3*a^2*b + 3*(a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + sqrt(2)*(b^2*cosh(d*x + c)^6 + 6*b^2*cosh(d*x + c )*sinh(d*x + c)^5 + b^2*sinh(d*x + c)^6 + 3*b^2*cosh(d*x + c)^4 + 3*(5*b^2 *cosh(d*x + c)^2 + b^2)*sinh(d*x + c)^4 + 4*(5*b^2*cosh(d*x + c)^3 + 3*b^2 *cosh(d*x + c))*sinh(d*x + c)^3 - (a^2 - 4*a*b)*cosh(d*x + c)^2 + (15*b^2* cosh(d*x + c)^4 + 18*b^2*cosh(d*x + c)^2 - a^2 + 4*a*b)*sinh(d*x + c)^2 + a^2 + 2*(3*b^2*cosh(d*x + c)^5 + 6*b^2*cosh(d*x + c)^3 - (a^2 - 4*a*b)*...
\[ \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^{3/2}} \, dx=\int \frac {1}{\left (a + b \operatorname {csch}^{2}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {csch}\left (d x + c\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {csch}\left (d x + c\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (a+\frac {b}{{\mathrm {sinh}\left (c+d\,x\right )}^2}\right )}^{3/2}} \,d x \]