Integrand size = 14, antiderivative size = 65 \[ \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{3/2}} \, dx=\frac {\coth (c+d x) \log (\cosh (c+d x))}{b d \sqrt {b \coth ^2(c+d x)}}-\frac {\tanh (c+d x)}{2 b d \sqrt {b \coth ^2(c+d x)}} \]
coth(d*x+c)*ln(cosh(d*x+c))/b/d/(b*coth(d*x+c)^2)^(1/2)-1/2*tanh(d*x+c)/b/ d/(b*coth(d*x+c)^2)^(1/2)
Time = 0.15 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{3/2}} \, dx=\frac {2 \coth (c+d x) \log (\cosh (c+d x))-\tanh (c+d x)}{2 b d \sqrt {b \coth ^2(c+d x)}} \]
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.91, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {3042, 4141, 3042, 26, 3954, 26, 3042, 26, 3956}
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.
\(\displaystyle \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (-b \tan \left (i c+i d x+\frac {\pi }{2}\right )^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4141 |
\(\displaystyle \frac {\coth (c+d x) \int \tanh ^3(c+d x)dx}{b \sqrt {b \coth ^2(c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\coth (c+d x) \int i \tan (i c+i d x)^3dx}{b \sqrt {b \coth ^2(c+d x)}}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {i \coth (c+d x) \int \tan (i c+i d x)^3dx}{b \sqrt {b \coth ^2(c+d x)}}\) |
\(\Big \downarrow \) 3954 |
\(\displaystyle \frac {i \coth (c+d x) \left (\frac {i \tanh ^2(c+d x)}{2 d}-\int i \tanh (c+d x)dx\right )}{b \sqrt {b \coth ^2(c+d x)}}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {i \coth (c+d x) \left (\frac {i \tanh ^2(c+d x)}{2 d}-i \int \tanh (c+d x)dx\right )}{b \sqrt {b \coth ^2(c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {i \coth (c+d x) \left (\frac {i \tanh ^2(c+d x)}{2 d}-i \int -i \tan (i c+i d x)dx\right )}{b \sqrt {b \coth ^2(c+d x)}}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {i \coth (c+d x) \left (\frac {i \tanh ^2(c+d x)}{2 d}-\int \tan (i c+i d x)dx\right )}{b \sqrt {b \coth ^2(c+d x)}}\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle \frac {i \coth (c+d x) \left (\frac {i \tanh ^2(c+d x)}{2 d}-\frac {i \log (\cosh (c+d x))}{d}\right )}{b \sqrt {b \coth ^2(c+d x)}}\) |
(I*Coth[c + d*x]*(((-I)*Log[Cosh[c + d*x]])/d + ((I/2)*Tanh[c + d*x]^2)/d) )/(b*Sqrt[b*Coth[c + d*x]^2])
3.1.21.3.1 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[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^ n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Ta n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / ; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
Time = 0.10 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.22
method | result | size |
derivativedivides | \(-\frac {\coth \left (d x +c \right ) \left (\ln \left (\coth \left (d x +c \right )+1\right ) \coth \left (d x +c \right )^{2}+\ln \left (\coth \left (d x +c \right )-1\right ) \coth \left (d x +c \right )^{2}-2 \ln \left (\coth \left (d x +c \right )\right ) \coth \left (d x +c \right )^{2}+1\right )}{2 d \left (\coth \left (d x +c \right )^{2} b \right )^{\frac {3}{2}}}\) | \(79\) |
default | \(-\frac {\coth \left (d x +c \right ) \left (\ln \left (\coth \left (d x +c \right )+1\right ) \coth \left (d x +c \right )^{2}+\ln \left (\coth \left (d x +c \right )-1\right ) \coth \left (d x +c \right )^{2}-2 \ln \left (\coth \left (d x +c \right )\right ) \coth \left (d x +c \right )^{2}+1\right )}{2 d \left (\coth \left (d x +c \right )^{2} b \right )^{\frac {3}{2}}}\) | \(79\) |
risch | \(\frac {-{\mathrm e}^{4 d x +4 c} d x +{\mathrm e}^{4 d x +4 c} \ln \left ({\mathrm e}^{2 d x +2 c}+1\right )-2 \,{\mathrm e}^{4 d x +4 c} c -2 \,{\mathrm e}^{2 d x +2 c} d x +2 \,{\mathrm e}^{2 d x +2 c} \ln \left ({\mathrm e}^{2 d x +2 c}+1\right )-4 \,{\mathrm e}^{2 d x +2 c} c -d x +2 \,{\mathrm e}^{2 d x +2 c}+\ln \left ({\mathrm e}^{2 d x +2 c}+1\right )-2 c}{b \left ({\mathrm e}^{2 d x +2 c}+1\right ) \left ({\mathrm e}^{2 d x +2 c}-1\right ) \sqrt {\frac {\left ({\mathrm e}^{2 d x +2 c}+1\right )^{2} b}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}}\, d}\) | \(190\) |
-1/2/d*coth(d*x+c)*(ln(coth(d*x+c)+1)*coth(d*x+c)^2+ln(coth(d*x+c)-1)*coth (d*x+c)^2-2*ln(coth(d*x+c))*coth(d*x+c)^2+1)/(coth(d*x+c)^2*b)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 817 vs. \(2 (59) = 118\).
Time = 0.27 (sec) , antiderivative size = 817, normalized size of antiderivative = 12.57 \[ \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{3/2}} \, dx=\text {Too large to display} \]
(d*x*cosh(d*x + c)^4 - (d*x*e^(2*d*x + 2*c) - d*x)*sinh(d*x + c)^4 - 4*(d* x*cosh(d*x + c)*e^(2*d*x + 2*c) - d*x*cosh(d*x + c))*sinh(d*x + c)^3 + 2*( d*x - 1)*cosh(d*x + c)^2 + 2*(3*d*x*cosh(d*x + c)^2 + d*x - (3*d*x*cosh(d* x + c)^2 + d*x - 1)*e^(2*d*x + 2*c) - 1)*sinh(d*x + c)^2 + d*x - (d*x*cosh (d*x + c)^4 + 2*(d*x - 1)*cosh(d*x + c)^2 + d*x)*e^(2*d*x + 2*c) + ((e^(2* d*x + 2*c) - 1)*sinh(d*x + c)^4 - cosh(d*x + c)^4 + 4*(cosh(d*x + c)*e^(2* d*x + 2*c) - cosh(d*x + c))*sinh(d*x + c)^3 - 2*(3*cosh(d*x + c)^2 - (3*co sh(d*x + c)^2 + 1)*e^(2*d*x + 2*c) + 1)*sinh(d*x + c)^2 - 2*cosh(d*x + c)^ 2 + (cosh(d*x + c)^4 + 2*cosh(d*x + c)^2 + 1)*e^(2*d*x + 2*c) - 4*(cosh(d* x + c)^3 - (cosh(d*x + c)^3 + cosh(d*x + c))*e^(2*d*x + 2*c) + cosh(d*x + c))*sinh(d*x + c) - 1)*log(2*cosh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c)) ) + 4*(d*x*cosh(d*x + c)^3 + (d*x - 1)*cosh(d*x + c) - (d*x*cosh(d*x + c)^ 3 + (d*x - 1)*cosh(d*x + c))*e^(2*d*x + 2*c))*sinh(d*x + c))*sqrt((b*e^(4* d*x + 4*c) + 2*b*e^(2*d*x + 2*c) + b)/(e^(4*d*x + 4*c) - 2*e^(2*d*x + 2*c) + 1))/(b^2*d*cosh(d*x + c)^4 + 2*b^2*d*cosh(d*x + c)^2 + (b^2*d*e^(2*d*x + 2*c) + b^2*d)*sinh(d*x + c)^4 + 4*(b^2*d*cosh(d*x + c)*e^(2*d*x + 2*c) + b^2*d*cosh(d*x + c))*sinh(d*x + c)^3 + b^2*d + 2*(3*b^2*d*cosh(d*x + c)^2 + b^2*d + (3*b^2*d*cosh(d*x + c)^2 + b^2*d)*e^(2*d*x + 2*c))*sinh(d*x + c )^2 + (b^2*d*cosh(d*x + c)^4 + 2*b^2*d*cosh(d*x + c)^2 + b^2*d)*e^(2*d*x + 2*c) + 4*(b^2*d*cosh(d*x + c)^3 + b^2*d*cosh(d*x + c) + (b^2*d*cosh(d*...
\[ \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{3/2}} \, dx=\int \frac {1}{\left (b \coth ^{2}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Time = 0.27 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.29 \[ \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {b} e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (2 \, b^{2} e^{\left (-2 \, d x - 2 \, c\right )} + b^{2} e^{\left (-4 \, d x - 4 \, c\right )} + b^{2}\right )} d} - \frac {d x + c}{b^{\frac {3}{2}} d} - \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{b^{\frac {3}{2}} d} \]
-2*sqrt(b)*e^(-2*d*x - 2*c)/((2*b^2*e^(-2*d*x - 2*c) + b^2*e^(-4*d*x - 4*c ) + b^2)*d) - (d*x + c)/(b^(3/2)*d) - log(e^(-2*d*x - 2*c) + 1)/(b^(3/2)*d )
Time = 0.30 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.60 \[ \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{3/2}} \, dx=-\frac {\frac {d x + c}{\sqrt {b} \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )} - \frac {\log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{\sqrt {b} \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )} - \frac {2 \, e^{\left (2 \, d x + 2 \, c\right )}}{\sqrt {b} {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{2} \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )}}{b d} \]
-((d*x + c)/(sqrt(b)*sgn(e^(4*d*x + 4*c) - 1)) - log(e^(2*d*x + 2*c) + 1)/ (sqrt(b)*sgn(e^(4*d*x + 4*c) - 1)) - 2*e^(2*d*x + 2*c)/(sqrt(b)*(e^(2*d*x + 2*c) + 1)^2*sgn(e^(4*d*x + 4*c) - 1)))/(b*d)
Timed out. \[ \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^2\right )}^{3/2}} \,d x \]