Integrand size = 24, antiderivative size = 101 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )} \, dx=-\frac {a+b \text {arcsinh}(c x)}{d x}-\frac {2 c (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d}-\frac {b c \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{d}+\frac {i b c \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{d}-\frac {i b c \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{d} \]
(-a-b*arcsinh(c*x))/d/x-2*c*(a+b*arcsinh(c*x))*arctan(c*x+(c^2*x^2+1)^(1/2 ))/d-b*c*arctanh((c^2*x^2+1)^(1/2))/d+I*b*c*polylog(2,-I*(c*x+(c^2*x^2+1)^ (1/2)))/d-I*b*c*polylog(2,I*(c*x+(c^2*x^2+1)^(1/2)))/d
Time = 0.11 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.80 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )} \, dx=-\frac {a+b \text {arcsinh}(c x)+a c x \arctan (c x)+b c x \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )+b \sqrt {-c^2} x \text {arcsinh}(c x) \log \left (1+\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )-b \sqrt {-c^2} x \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )-b \sqrt {-c^2} x \operatorname {PolyLog}\left (2,\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+b \sqrt {-c^2} x \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )}{d x} \]
-((a + b*ArcSinh[c*x] + a*c*x*ArcTan[c*x] + b*c*x*ArcTanh[Sqrt[1 + c^2*x^2 ]] + b*Sqrt[-c^2]*x*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - b*Sqrt[-c^2]*x*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] - b*Sqr t[-c^2]*x*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + b*Sqrt[-c^2]*x*PolyL og[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c])/(d*x))
Time = 0.57 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.95, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6224, 27, 243, 73, 221, 6204, 3042, 4668, 2715, 2838}
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.
\(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (c^2 d x^2+d\right )} \, dx\) |
\(\Big \downarrow \) 6224 |
\(\displaystyle c^2 \left (-\int \frac {a+b \text {arcsinh}(c x)}{d \left (c^2 x^2+1\right )}dx\right )+\frac {b c \int \frac {1}{x \sqrt {c^2 x^2+1}}dx}{d}-\frac {a+b \text {arcsinh}(c x)}{d x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {c^2 \int \frac {a+b \text {arcsinh}(c x)}{c^2 x^2+1}dx}{d}+\frac {b c \int \frac {1}{x \sqrt {c^2 x^2+1}}dx}{d}-\frac {a+b \text {arcsinh}(c x)}{d x}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -\frac {c^2 \int \frac {a+b \text {arcsinh}(c x)}{c^2 x^2+1}dx}{d}+\frac {b c \int \frac {1}{x^2 \sqrt {c^2 x^2+1}}dx^2}{2 d}-\frac {a+b \text {arcsinh}(c x)}{d x}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {c^2 \int \frac {a+b \text {arcsinh}(c x)}{c^2 x^2+1}dx}{d}+\frac {b \int \frac {1}{\frac {x^4}{c^2}-\frac {1}{c^2}}d\sqrt {c^2 x^2+1}}{c d}-\frac {a+b \text {arcsinh}(c x)}{d x}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {c^2 \int \frac {a+b \text {arcsinh}(c x)}{c^2 x^2+1}dx}{d}-\frac {a+b \text {arcsinh}(c x)}{d x}-\frac {b c \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )}{d}\) |
\(\Big \downarrow \) 6204 |
\(\displaystyle -\frac {c \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{d}-\frac {a+b \text {arcsinh}(c x)}{d x}-\frac {b c \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {c \int (a+b \text {arcsinh}(c x)) \csc \left (i \text {arcsinh}(c x)+\frac {\pi }{2}\right )d\text {arcsinh}(c x)}{d}-\frac {a+b \text {arcsinh}(c x)}{d x}-\frac {b c \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )}{d}\) |
\(\Big \downarrow \) 4668 |
\(\displaystyle -\frac {c \left (-i b \int \log \left (1-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+i b \int \log \left (1+i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )}{d}-\frac {a+b \text {arcsinh}(c x)}{d x}-\frac {b c \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )}{d}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {c \left (-i b \int e^{-\text {arcsinh}(c x)} \log \left (1-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+i b \int e^{-\text {arcsinh}(c x)} \log \left (1+i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )}{d}-\frac {a+b \text {arcsinh}(c x)}{d x}-\frac {b c \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )}{d}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {c \left (2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )}{d}-\frac {a+b \text {arcsinh}(c x)}{d x}-\frac {b c \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )}{d}\) |
-((a + b*ArcSinh[c*x])/(d*x)) - (b*c*ArcTanh[Sqrt[1 + c^2*x^2]])/d - (c*(2 *(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]] - I*b*PolyLog[2, (-I)*E^ArcSi nh[c*x]] + I*b*PolyLog[2, I*E^ArcSinh[c*x]]))/d
3.1.34.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ ))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[ 1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c , d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[1/(c*d) Subst[Int[(a + b*x)^n*Sech[x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] + (-Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Sim p[b*c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{ a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[m, -1]
Time = 0.20 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.74
method | result | size |
parts | \(\frac {a \left (-c \arctan \left (c x \right )-\frac {1}{x}\right )}{d}+\frac {b c \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{c x}-\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )-\arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )+\arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )\right )}{d}\) | \(176\) |
derivativedivides | \(c \left (\frac {a \left (-\frac {1}{c x}-\arctan \left (c x \right )\right )}{d}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{c x}-\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )-\arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )+\arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )\right )}{d}\right )\) | \(179\) |
default | \(c \left (\frac {a \left (-\frac {1}{c x}-\arctan \left (c x \right )\right )}{d}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{c x}-\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )-\arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )+\arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )\right )}{d}\right )\) | \(179\) |
a/d*(-c*arctan(c*x)-1/x)+b/d*c*(-arcsinh(c*x)/c/x-arcsinh(c*x)*arctan(c*x) -arctanh(1/(c^2*x^2+1)^(1/2))-arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/ 2))+arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+I*dilog(1+I*(1+I*c*x)/ (c^2*x^2+1)^(1/2))-I*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2)))
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )} x^{2}} \,d x } \]
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )} \, dx=\frac {\int \frac {a}{c^{2} x^{4} + x^{2}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{4} + x^{2}}\, dx}{d} \]
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )} x^{2}} \,d x } \]
-a*(c*arctan(c*x)/d + 1/(d*x)) + b*integrate(log(c*x + sqrt(c^2*x^2 + 1))/ (c^2*d*x^4 + d*x^2), x)
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^2\,\left (d\,c^2\,x^2+d\right )} \,d x \]