Integrand size = 14, antiderivative size = 141 \[ \int \frac {1}{\left (b \coth ^3(c+d x)\right )^{3/2}} \, dx=-\frac {2}{3 b d \sqrt {b \coth ^3(c+d x)}}+\frac {\arctan \left (\sqrt {\coth (c+d x)}\right ) \coth ^{\frac {3}{2}}(c+d x)}{b d \sqrt {b \coth ^3(c+d x)}}+\frac {\text {arctanh}\left (\sqrt {\coth (c+d x)}\right ) \coth ^{\frac {3}{2}}(c+d x)}{b d \sqrt {b \coth ^3(c+d x)}}-\frac {2 \tanh ^2(c+d x)}{7 b d \sqrt {b \coth ^3(c+d x)}} \]
-2/3/b/d/(b*coth(d*x+c)^3)^(1/2)+arctan(coth(d*x+c)^(1/2))*coth(d*x+c)^(3/ 2)/b/d/(b*coth(d*x+c)^3)^(1/2)+arctanh(coth(d*x+c)^(1/2))*coth(d*x+c)^(3/2 )/b/d/(b*coth(d*x+c)^3)^(1/2)-2/7*tanh(d*x+c)^2/b/d/(b*coth(d*x+c)^3)^(1/2 )
Time = 0.22 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\left (b \coth ^3(c+d x)\right )^{3/2}} \, dx=\frac {-14+21 \arctan \left (\sqrt [4]{\coth ^2(c+d x)}\right ) \coth ^2(c+d x)^{3/4}+21 \text {arctanh}\left (\sqrt [4]{\coth ^2(c+d x)}\right ) \coth ^2(c+d x)^{3/4}-6 \tanh ^2(c+d x)}{21 b d \sqrt {b \coth ^3(c+d x)}} \]
(-14 + 21*ArcTan[(Coth[c + d*x]^2)^(1/4)]*(Coth[c + d*x]^2)^(3/4) + 21*Arc Tanh[(Coth[c + d*x]^2)^(1/4)]*(Coth[c + d*x]^2)^(3/4) - 6*Tanh[c + d*x]^2) /(21*b*d*Sqrt[b*Coth[c + d*x]^3])
Time = 0.44 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.70, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {3042, 4141, 3042, 3955, 3042, 3955, 3042, 3957, 25, 266, 756, 216, 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.
| \(\displaystyle \int \frac {1}{\left (b \coth ^3(c+d x)\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (i b \tan \left (i c+i d x+\frac {\pi }{2}\right )^3\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4141 |
| \(\displaystyle \frac {\coth ^{\frac {3}{2}}(c+d x) \int \frac {1}{\coth ^{\frac {9}{2}}(c+d x)}dx}{b \sqrt {b \coth ^3(c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\coth ^{\frac {3}{2}}(c+d x) \int \frac {1}{\left (-i \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^{9/2}}dx}{b \sqrt {b \coth ^3(c+d x)}}\) |
| \(\Big \downarrow \) 3955 |
| \(\displaystyle \frac {\coth ^{\frac {3}{2}}(c+d x) \left (\int \frac {1}{\coth ^{\frac {5}{2}}(c+d x)}dx-\frac {2}{7 d \coth ^{\frac {7}{2}}(c+d x)}\right )}{b \sqrt {b \coth ^3(c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\coth ^{\frac {3}{2}}(c+d x) \left (-\frac {2}{7 d \coth ^{\frac {7}{2}}(c+d x)}+\int \frac {1}{\left (-i \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\right )}{b \sqrt {b \coth ^3(c+d x)}}\) |
| \(\Big \downarrow \) 3955 |
| \(\displaystyle \frac {\coth ^{\frac {3}{2}}(c+d x) \left (\int \frac {1}{\sqrt {\coth (c+d x)}}dx-\frac {2}{3 d \coth ^{\frac {3}{2}}(c+d x)}-\frac {2}{7 d \coth ^{\frac {7}{2}}(c+d x)}\right )}{b \sqrt {b \coth ^3(c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\coth ^{\frac {3}{2}}(c+d x) \left (\int \frac {1}{\sqrt {-i \tan \left (i c+i d x+\frac {\pi }{2}\right )}}dx-\frac {2}{3 d \coth ^{\frac {3}{2}}(c+d x)}-\frac {2}{7 d \coth ^{\frac {7}{2}}(c+d x)}\right )}{b \sqrt {b \coth ^3(c+d x)}}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {\coth ^{\frac {3}{2}}(c+d x) \left (-\frac {\int -\frac {1}{\sqrt {\coth (c+d x)} \left (1-\coth ^2(c+d x)\right )}d\coth (c+d x)}{d}-\frac {2}{3 d \coth ^{\frac {3}{2}}(c+d x)}-\frac {2}{7 d \coth ^{\frac {7}{2}}(c+d x)}\right )}{b \sqrt {b \coth ^3(c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\coth ^{\frac {3}{2}}(c+d x) \left (\frac {\int \frac {1}{\sqrt {\coth (c+d x)} \left (1-\coth ^2(c+d x)\right )}d\coth (c+d x)}{d}-\frac {2}{3 d \coth ^{\frac {3}{2}}(c+d x)}-\frac {2}{7 d \coth ^{\frac {7}{2}}(c+d x)}\right )}{b \sqrt {b \coth ^3(c+d x)}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\coth ^{\frac {3}{2}}(c+d x) \left (\frac {2 \int \frac {1}{1-\coth ^2(c+d x)}d\sqrt {\coth (c+d x)}}{d}-\frac {2}{3 d \coth ^{\frac {3}{2}}(c+d x)}-\frac {2}{7 d \coth ^{\frac {7}{2}}(c+d x)}\right )}{b \sqrt {b \coth ^3(c+d x)}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {\coth ^{\frac {3}{2}}(c+d x) \left (\frac {2 \left (\frac {1}{2} \int \frac {1}{1-\coth (c+d x)}d\sqrt {\coth (c+d x)}+\frac {1}{2} \int \frac {1}{\coth (c+d x)+1}d\sqrt {\coth (c+d x)}\right )}{d}-\frac {2}{3 d \coth ^{\frac {3}{2}}(c+d x)}-\frac {2}{7 d \coth ^{\frac {7}{2}}(c+d x)}\right )}{b \sqrt {b \coth ^3(c+d x)}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\coth ^{\frac {3}{2}}(c+d x) \left (\frac {2 \left (\frac {1}{2} \int \frac {1}{1-\coth (c+d x)}d\sqrt {\coth (c+d x)}+\frac {1}{2} \arctan \left (\sqrt {\coth (c+d x)}\right )\right )}{d}-\frac {2}{3 d \coth ^{\frac {3}{2}}(c+d x)}-\frac {2}{7 d \coth ^{\frac {7}{2}}(c+d x)}\right )}{b \sqrt {b \coth ^3(c+d x)}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\coth ^{\frac {3}{2}}(c+d x) \left (\frac {2 \left (\frac {1}{2} \arctan \left (\sqrt {\coth (c+d x)}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt {\coth (c+d x)}\right )\right )}{d}-\frac {2}{3 d \coth ^{\frac {3}{2}}(c+d x)}-\frac {2}{7 d \coth ^{\frac {7}{2}}(c+d x)}\right )}{b \sqrt {b \coth ^3(c+d x)}}\) |
(((2*(ArcTan[Sqrt[Coth[c + d*x]]]/2 + ArcTanh[Sqrt[Coth[c + d*x]]]/2))/d - 2/(7*d*Coth[c + d*x]^(7/2)) - 2/(3*d*Coth[c + d*x]^(3/2)))*Coth[c + d*x]^ (3/2))/(b*Sqrt[b*Coth[c + d*x]^3])
3.1.32.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Tan[c + d*x] )^(n + 1)/(b*d*(n + 1)), x] - Simp[1/b^2 Int[(b*Tan[c + d*x])^(n + 2), x] , x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Subst[Int [x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
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 = 0.15 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.75
| method | result | size |
| derivativedivides | \(\frac {\coth \left (d x +c \right ) \left (-14 b^{\frac {15}{2}} \coth \left (d x +c \right )^{2}-6 b^{\frac {15}{2}}+21 \,\operatorname {arctanh}\left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right ) b^{4} \left (b \coth \left (d x +c \right )\right )^{\frac {7}{2}}+21 \arctan \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right ) b^{4} \left (b \coth \left (d x +c \right )\right )^{\frac {7}{2}}\right )}{21 d \,b^{\frac {15}{2}} \left (b \coth \left (d x +c \right )^{3}\right )^{\frac {3}{2}}}\) | \(106\) |
| default | \(\frac {\coth \left (d x +c \right ) \left (-14 b^{\frac {15}{2}} \coth \left (d x +c \right )^{2}-6 b^{\frac {15}{2}}+21 \,\operatorname {arctanh}\left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right ) b^{4} \left (b \coth \left (d x +c \right )\right )^{\frac {7}{2}}+21 \arctan \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right ) b^{4} \left (b \coth \left (d x +c \right )\right )^{\frac {7}{2}}\right )}{21 d \,b^{\frac {15}{2}} \left (b \coth \left (d x +c \right )^{3}\right )^{\frac {3}{2}}}\) | \(106\) |
1/21/d*coth(d*x+c)/b^(15/2)*(-14*b^(15/2)*coth(d*x+c)^2-6*b^(15/2)+21*arct anh((b*coth(d*x+c))^(1/2)/b^(1/2))*b^4*(b*coth(d*x+c))^(7/2)+21*arctan((b* coth(d*x+c))^(1/2)/b^(1/2))*b^4*(b*coth(d*x+c))^(7/2))/(b*coth(d*x+c)^3)^( 3/2)
Leaf count of result is larger than twice the leaf count of optimal. 1486 vs. \(2 (121) = 242\).
Time = 0.34 (sec) , antiderivative size = 3022, normalized size of antiderivative = 21.43 \[ \int \frac {1}{\left (b \coth ^3(c+d x)\right )^{3/2}} \, dx=\text {Too large to display} \]
[-1/84*(42*(cosh(d*x + c)^8 + 8*cosh(d*x + c)*sinh(d*x + c)^7 + sinh(d*x + c)^8 + 4*(7*cosh(d*x + c)^2 + 1)*sinh(d*x + c)^6 + 4*cosh(d*x + c)^6 + 8* (7*cosh(d*x + c)^3 + 3*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*cosh(d*x + c )^4 + 30*cosh(d*x + c)^2 + 3)*sinh(d*x + c)^4 + 6*cosh(d*x + c)^4 + 8*(7*c osh(d*x + c)^5 + 10*cosh(d*x + c)^3 + 3*cosh(d*x + c))*sinh(d*x + c)^3 + 4 *(7*cosh(d*x + c)^6 + 15*cosh(d*x + c)^4 + 9*cosh(d*x + c)^2 + 1)*sinh(d*x + c)^2 + 4*cosh(d*x + c)^2 + 8*(cosh(d*x + c)^7 + 3*cosh(d*x + c)^5 + 3*c osh(d*x + c)^3 + cosh(d*x + c))*sinh(d*x + c) + 1)*sqrt(-b)*arctan((cosh(d *x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)*sqrt(-b)*sqrt (b*cosh(d*x + c)/sinh(d*x + c))/(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sin h(d*x + c) + b*sinh(d*x + c)^2 + b)) + 21*(cosh(d*x + c)^8 + 8*cosh(d*x + c)*sinh(d*x + c)^7 + sinh(d*x + c)^8 + 4*(7*cosh(d*x + c)^2 + 1)*sinh(d*x + c)^6 + 4*cosh(d*x + c)^6 + 8*(7*cosh(d*x + c)^3 + 3*cosh(d*x + c))*sinh( d*x + c)^5 + 2*(35*cosh(d*x + c)^4 + 30*cosh(d*x + c)^2 + 3)*sinh(d*x + c) ^4 + 6*cosh(d*x + c)^4 + 8*(7*cosh(d*x + c)^5 + 10*cosh(d*x + c)^3 + 3*cos h(d*x + c))*sinh(d*x + c)^3 + 4*(7*cosh(d*x + c)^6 + 15*cosh(d*x + c)^4 + 9*cosh(d*x + c)^2 + 1)*sinh(d*x + c)^2 + 4*cosh(d*x + c)^2 + 8*(cosh(d*x + c)^7 + 3*cosh(d*x + c)^5 + 3*cosh(d*x + c)^3 + cosh(d*x + c))*sinh(d*x + c) + 1)*sqrt(-b)*log(-(b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)^3*sinh(d*x + c) + 6*b*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*b*cosh(d*x + c)*sinh(d*x +...
\[ \int \frac {1}{\left (b \coth ^3(c+d x)\right )^{3/2}} \, dx=\int \frac {1}{\left (b \coth ^{3}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{\left (b \coth ^3(c+d x)\right )^{3/2}} \, dx=\int { \frac {1}{\left (b \coth \left (d x + c\right )^{3}\right )^{\frac {3}{2}}} \,d x } \]
Exception generated. \[ \int \frac {1}{\left (b \coth ^3(c+d x)\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {1}{\left (b \coth ^3(c+d x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^3\right )}^{3/2}} \,d x \]