Integrand size = 23, antiderivative size = 85 \[ \int \frac {e^{\text {csch}^{-1}(c x)} (d x)^m}{1+c^2 x^2} \, dx=-\frac {d (d x)^{-1+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-m}{2},\frac {3-m}{2},-\frac {1}{c^2 x^2}\right )}{c^2 (1-m)}+\frac {(d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {2+m}{2},-c^2 x^2\right )}{c m} \] Output:
-d*(d*x)^(-1+m)*hypergeom([1/2, 1/2-1/2*m],[3/2-1/2*m],-1/c^2/x^2)/c^2/(1- m)+(d*x)^m*hypergeom([1, 1/2*m],[1+1/2*m],-c^2*x^2)/c/m
Time = 0.73 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.04 \[ \int \frac {e^{\text {csch}^{-1}(c x)} (d x)^m}{1+c^2 x^2} \, dx=\frac {(d x)^m \left (\frac {\sqrt {1+\frac {1}{c^2 x^2}} x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m}{2},1+\frac {m}{2},-c^2 x^2\right )}{\sqrt {1+c^2 x^2}}+\frac {\operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},1+\frac {m}{2},-c^2 x^2\right )}{c}\right )}{m} \] Input:
Integrate[(E^ArcCsch[c*x]*(d*x)^m)/(1 + c^2*x^2),x]
Output:
((d*x)^m*((Sqrt[1 + 1/(c^2*x^2)]*x*Hypergeometric2F1[1/2, m/2, 1 + m/2, -( c^2*x^2)])/Sqrt[1 + c^2*x^2] + Hypergeometric2F1[1, m/2, 1 + m/2, -(c^2*x^ 2)]/c))/m
Time = 0.57 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6896, 278, 862, 278}
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.
\(\displaystyle \int \frac {e^{\text {csch}^{-1}(c x)} (d x)^m}{c^2 x^2+1} \, dx\) |
\(\Big \downarrow \) 6896 |
\(\displaystyle \frac {d^2 \int \frac {(d x)^{m-2}}{\sqrt {1+\frac {1}{c^2 x^2}}}dx}{c^2}+\frac {d \int \frac {(d x)^{m-1}}{c^2 x^2+1}dx}{c}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {d^2 \int \frac {(d x)^{m-2}}{\sqrt {1+\frac {1}{c^2 x^2}}}dx}{c^2}+\frac {(d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {m+2}{2},-c^2 x^2\right )}{c m}\) |
\(\Big \downarrow \) 862 |
\(\displaystyle \frac {(d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {m+2}{2},-c^2 x^2\right )}{c m}-\frac {d \left (\frac {1}{x}\right )^{m-1} (d x)^{m-1} \int \frac {\left (\frac {1}{x}\right )^{-m}}{\sqrt {1+\frac {1}{c^2 x^2}}}d\frac {1}{x}}{c^2}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {(d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {m+2}{2},-c^2 x^2\right )}{c m}-\frac {d (d x)^{m-1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-m}{2},\frac {3-m}{2},-\frac {1}{c^2 x^2}\right )}{c^2 (1-m)}\) |
Input:
Int[(E^ArcCsch[c*x]*(d*x)^m)/(1 + c^2*x^2),x]
Output:
-((d*(d*x)^(-1 + m)*Hypergeometric2F1[1/2, (1 - m)/2, (3 - m)/2, -(1/(c^2* x^2))])/(c^2*(1 - m))) + ((d*x)^m*Hypergeometric2F1[1, m/2, (2 + m)/2, -(c ^2*x^2)])/(c*m)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-c^ (-1))*(c*x)^(m + 1)*(1/x)^(m + 1) Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, m, p}, x] && ILtQ[n, 0] && !RationalQ[m]
Int[(E^ArcCsch[(c_.)*(x_)]*((d_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^2), x_Sym bol] :> Simp[d^2/(a*c^2) Int[(d*x)^(m - 2)/Sqrt[1 + 1/(c^2*x^2)], x], x] + Simp[d/c Int[(d*x)^(m - 1)/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d, m} , x] && EqQ[b - a*c^2, 0]
\[\int \frac {\left (\frac {1}{c x}+\sqrt {1+\frac {1}{c^{2} x^{2}}}\right ) \left (d x \right )^{m}}{c^{2} x^{2}+1}d x\]
Input:
int((1/c/x+(1+1/c^2/x^2)^(1/2))*(d*x)^m/(c^2*x^2+1),x)
Output:
int((1/c/x+(1+1/c^2/x^2)^(1/2))*(d*x)^m/(c^2*x^2+1),x)
\[ \int \frac {e^{\text {csch}^{-1}(c x)} (d x)^m}{1+c^2 x^2} \, dx=\int { \frac {\left (d x\right )^{m} {\left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + \frac {1}{c x}\right )}}{c^{2} x^{2} + 1} \,d x } \] Input:
integrate((1/c/x+(1+1/c^2/x^2)^(1/2))*(d*x)^m/(c^2*x^2+1),x, algorithm="fr icas")
Output:
integral(((d*x)^m*c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + (d*x)^m)/(c^3*x^3 + c*x), x)
\[ \int \frac {e^{\text {csch}^{-1}(c x)} (d x)^m}{1+c^2 x^2} \, dx=\frac {\int \frac {\left (d x\right )^{m}}{c^{2} x^{3} + x}\, dx + \int \frac {c x \left (d x\right )^{m} \sqrt {1 + \frac {1}{c^{2} x^{2}}}}{c^{2} x^{3} + x}\, dx}{c} \] Input:
integrate((1/c/x+(1+1/c**2/x**2)**(1/2))*(d*x)**m/(c**2*x**2+1),x)
Output:
(Integral((d*x)**m/(c**2*x**3 + x), x) + Integral(c*x*(d*x)**m*sqrt(1 + 1/ (c**2*x**2))/(c**2*x**3 + x), x))/c
\[ \int \frac {e^{\text {csch}^{-1}(c x)} (d x)^m}{1+c^2 x^2} \, dx=\int { \frac {\left (d x\right )^{m} {\left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + \frac {1}{c x}\right )}}{c^{2} x^{2} + 1} \,d x } \] Input:
integrate((1/c/x+(1+1/c^2/x^2)^(1/2))*(d*x)^m/(c^2*x^2+1),x, algorithm="ma xima")
Output:
integrate((d*x)^m*(sqrt(1/(c^2*x^2) + 1) + 1/(c*x))/(c^2*x^2 + 1), x)
Exception generated. \[ \int \frac {e^{\text {csch}^{-1}(c x)} (d x)^m}{1+c^2 x^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((1/c/x+(1+1/c^2/x^2)^(1/2))*(d*x)^m/(c^2*x^2+1),x, algorithm="gi ac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {e^{\text {csch}^{-1}(c x)} (d x)^m}{1+c^2 x^2} \, dx=\int \frac {\left (\sqrt {\frac {1}{c^2\,x^2}+1}+\frac {1}{c\,x}\right )\,{\left (d\,x\right )}^m}{c^2\,x^2+1} \,d x \] Input:
int((((1/(c^2*x^2) + 1)^(1/2) + 1/(c*x))*(d*x)^m)/(c^2*x^2 + 1),x)
Output:
int((((1/(c^2*x^2) + 1)^(1/2) + 1/(c*x))*(d*x)^m)/(c^2*x^2 + 1), x)
\[ \int \frac {e^{\text {csch}^{-1}(c x)} (d x)^m}{1+c^2 x^2} \, dx=\frac {d^{m} \left (\int \frac {x^{m}}{c^{2} x^{3}+x}d x +\int \frac {x^{m} \sqrt {c^{2} x^{2}+1}}{c^{2} x^{3}+x}d x \right )}{c} \] Input:
int((1/c/x+(1+1/c^2/x^2)^(1/2))*(d*x)^m/(c^2*x^2+1),x)
Output:
(d**m*(int(x**m/(c**2*x**3 + x),x) + int((x**m*sqrt(c**2*x**2 + 1))/(c**2* x**3 + x),x)))/c