Integrand size = 14, antiderivative size = 105 \[ \int \frac {1}{\sqrt {b \coth ^3(c+d x)}} \, dx=-\frac {2 \coth (c+d x)}{d \sqrt {b \coth ^3(c+d x)}}-\frac {\arctan \left (\sqrt {\coth (c+d x)}\right ) \coth ^{\frac {3}{2}}(c+d x)}{d \sqrt {b \coth ^3(c+d x)}}+\frac {\text {arctanh}\left (\sqrt {\coth (c+d x)}\right ) \coth ^{\frac {3}{2}}(c+d x)}{d \sqrt {b \coth ^3(c+d x)}} \]
-2*coth(d*x+c)/d/(b*coth(d*x+c)^3)^(1/2)-arctan(coth(d*x+c)^(1/2))*coth(d* x+c)^(3/2)/d/(b*coth(d*x+c)^3)^(1/2)+arctanh(coth(d*x+c)^(1/2))*coth(d*x+c )^(3/2)/d/(b*coth(d*x+c)^3)^(1/2)
Time = 0.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\sqrt {b \coth ^3(c+d x)}} \, dx=-\frac {\coth (c+d x) \left (2+\arctan \left (\sqrt [4]{\coth ^2(c+d x)}\right ) \sqrt [4]{\coth ^2(c+d x)}-\text {arctanh}\left (\sqrt [4]{\coth ^2(c+d x)}\right ) \sqrt [4]{\coth ^2(c+d x)}\right )}{d \sqrt {b \coth ^3(c+d x)}} \]
-((Coth[c + d*x]*(2 + ArcTan[(Coth[c + d*x]^2)^(1/4)]*(Coth[c + d*x]^2)^(1 /4) - ArcTanh[(Coth[c + d*x]^2)^(1/4)]*(Coth[c + d*x]^2)^(1/4)))/(d*Sqrt[b *Coth[c + d*x]^3]))
Time = 0.38 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.73, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {3042, 4141, 3042, 3955, 3042, 3957, 25, 266, 827, 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.
\(\displaystyle \int \frac {1}{\sqrt {b \coth ^3(c+d x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt {i b \tan \left (i c+i d x+\frac {\pi }{2}\right )^3}}dx\) |
\(\Big \downarrow \) 4141 |
\(\displaystyle \frac {\coth ^{\frac {3}{2}}(c+d x) \int \frac {1}{\coth ^{\frac {3}{2}}(c+d x)}dx}{\sqrt {b \coth ^3(c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\coth ^{\frac {3}{2}}(c+d x) \int \frac {1}{\left (-i \tan \left (i c+i d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{\sqrt {b \coth ^3(c+d x)}}\) |
\(\Big \downarrow \) 3955 |
\(\displaystyle \frac {\coth ^{\frac {3}{2}}(c+d x) \left (\int \sqrt {\coth (c+d x)}dx-\frac {2}{d \sqrt {\coth (c+d x)}}\right )}{\sqrt {b \coth ^3(c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\coth ^{\frac {3}{2}}(c+d x) \left (-\frac {2}{d \sqrt {\coth (c+d x)}}+\int \sqrt {-i \tan \left (i c+i d x+\frac {\pi }{2}\right )}dx\right )}{\sqrt {b \coth ^3(c+d x)}}\) |
\(\Big \downarrow \) 3957 |
\(\displaystyle \frac {\coth ^{\frac {3}{2}}(c+d x) \left (-\frac {\int -\frac {\sqrt {\coth (c+d x)}}{1-\coth ^2(c+d x)}d\coth (c+d x)}{d}-\frac {2}{d \sqrt {\coth (c+d x)}}\right )}{\sqrt {b \coth ^3(c+d x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\coth ^{\frac {3}{2}}(c+d x) \left (\frac {\int \frac {\sqrt {\coth (c+d x)}}{1-\coth ^2(c+d x)}d\coth (c+d x)}{d}-\frac {2}{d \sqrt {\coth (c+d x)}}\right )}{\sqrt {b \coth ^3(c+d x)}}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {\coth ^{\frac {3}{2}}(c+d x) \left (\frac {2 \int \frac {\coth (c+d x)}{1-\coth ^2(c+d x)}d\sqrt {\coth (c+d x)}}{d}-\frac {2}{d \sqrt {\coth (c+d x)}}\right )}{\sqrt {b \coth ^3(c+d x)}}\) |
\(\Big \downarrow \) 827 |
\(\displaystyle \frac {\coth ^{\frac {3}{2}}(c+d x) \left (\frac {2 \left (\frac {1}{2} \int \frac {1}{1-\coth (c+d x)}d\sqrt {\coth (c+d x)}-\frac {1}{2} \int \frac {1}{\coth (c+d x)+1}d\sqrt {\coth (c+d x)}\right )}{d}-\frac {2}{d \sqrt {\coth (c+d x)}}\right )}{\sqrt {b \coth ^3(c+d x)}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\coth ^{\frac {3}{2}}(c+d x) \left (\frac {2 \left (\frac {1}{2} \int \frac {1}{1-\coth (c+d x)}d\sqrt {\coth (c+d x)}-\frac {1}{2} \arctan \left (\sqrt {\coth (c+d x)}\right )\right )}{d}-\frac {2}{d \sqrt {\coth (c+d x)}}\right )}{\sqrt {b \coth ^3(c+d x)}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\coth ^{\frac {3}{2}}(c+d x) \left (\frac {2 \left (\frac {1}{2} \text {arctanh}\left (\sqrt {\coth (c+d x)}\right )-\frac {1}{2} \arctan \left (\sqrt {\coth (c+d x)}\right )\right )}{d}-\frac {2}{d \sqrt {\coth (c+d x)}}\right )}{\sqrt {b \coth ^3(c+d x)}}\) |
(((2*(-1/2*ArcTan[Sqrt[Coth[c + d*x]]] + ArcTanh[Sqrt[Coth[c + d*x]]]/2))/ d - 2/(d*Sqrt[Coth[c + d*x]]))*Coth[c + d*x]^(3/2))/Sqrt[b*Coth[c + d*x]^3 ]
3.1.31.3.1 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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) 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 = 0.17 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(\frac {\coth \left (d x +c \right ) \left (-2 b^{\frac {5}{2}}+\operatorname {arctanh}\left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right ) b^{2} \sqrt {b \coth \left (d x +c \right )}-\arctan \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right ) b^{2} \sqrt {b \coth \left (d x +c \right )}\right )}{d \sqrt {b \coth \left (d x +c \right )^{3}}\, b^{\frac {5}{2}}}\) | \(91\) |
default | \(\frac {\coth \left (d x +c \right ) \left (-2 b^{\frac {5}{2}}+\operatorname {arctanh}\left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right ) b^{2} \sqrt {b \coth \left (d x +c \right )}-\arctan \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right ) b^{2} \sqrt {b \coth \left (d x +c \right )}\right )}{d \sqrt {b \coth \left (d x +c \right )^{3}}\, b^{\frac {5}{2}}}\) | \(91\) |
1/d*coth(d*x+c)*(-2*b^(5/2)+arctanh((b*coth(d*x+c))^(1/2)/b^(1/2))*b^2*(b* coth(d*x+c))^(1/2)-arctan((b*coth(d*x+c))^(1/2)/b^(1/2))*b^2*(b*coth(d*x+c ))^(1/2))/(b*coth(d*x+c)^3)^(1/2)/b^(5/2)
Leaf count of result is larger than twice the leaf count of optimal. 429 vs. \(2 (91) = 182\).
Time = 0.30 (sec) , antiderivative size = 907, normalized size of antiderivative = 8.64 \[ \int \frac {1}{\sqrt {b \coth ^3(c+d x)}} \, dx=\text {Too large to display} \]
[-1/4*(2*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^ 2 + 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)*sinh(d*x + c) + b*sinh(d*x + c)^2 + b)) + (cosh (d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 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 + c)^3 + b*sinh(d*x + c)^4 - 2*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*sqrt(-b)*sqrt(b*cosh(d*x + c)/sinh(d*x + c)) - 2*b)/(cosh(d*x + c)^4 + 4*cosh(d*x + c)^3*sinh(d*x + c) + 6*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4)) + 8*(cosh(d*x + c)^ 2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*sqrt(b*cosh(d*x + c)/sinh(d*x + c)))/(b*d*cosh(d*x + c)^2 + 2*b*d*cosh(d*x + c)*sinh(d*x + c) + b*d*sinh(d*x + c)^2 + b*d), -1/4*(2*(cosh(d*x + c)^2 + 2*cosh(d*x + c )*sinh(d*x + c) + sinh(d*x + c)^2 + 1)*sqrt(b)*arctan(sqrt(b)*sqrt(b*cosh( d*x + c)/sinh(d*x + c))/(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + b)) - (cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)*sqrt(b)*log(2*b*cosh(d*x + c)^4 + 8*b*cosh(d* x + c)^3*sinh(d*x + c) + 12*b*cosh(d*x + c)^2*sinh(d*x + c)^2 + 8*b*cosh(d *x + c)*sinh(d*x + c)^3 + 2*b*sinh(d*x + c)^4 + 2*(cosh(d*x + c)^4 + 4*...
\[ \int \frac {1}{\sqrt {b \coth ^3(c+d x)}} \, dx=\int \frac {1}{\sqrt {b \coth ^{3}{\left (c + d x \right )}}}\, dx \]
\[ \int \frac {1}{\sqrt {b \coth ^3(c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \coth \left (d x + c\right )^{3}}} \,d x } \]
Exception generated. \[ \int \frac {1}{\sqrt {b \coth ^3(c+d x)}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {1}{\sqrt {b \coth ^3(c+d x)}} \, dx=\int \frac {1}{\sqrt {b\,{\mathrm {coth}\left (c+d\,x\right )}^3}} \,d x \]