Integrand size = 16, antiderivative size = 187 \[ \int \frac {a+b \text {arcsinh}(c x)}{d+e x} \, dx=-\frac {(a+b \text {arcsinh}(c x))^2}{2 b e}+\frac {(a+b \text {arcsinh}(c x)) \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {(a+b \text {arcsinh}(c x)) \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e} \]
-1/2*(a+b*arcsinh(c*x))^2/b/e+(a+b*arcsinh(c*x))*ln(1+e*(c*x+(c^2*x^2+1)^( 1/2))/(c*d-(c^2*d^2+e^2)^(1/2)))/e+(a+b*arcsinh(c*x))*ln(1+e*(c*x+(c^2*x^2 +1)^(1/2))/(c*d+(c^2*d^2+e^2)^(1/2)))/e+b*polylog(2,-e*(c*x+(c^2*x^2+1)^(1 /2))/(c*d-(c^2*d^2+e^2)^(1/2)))/e+b*polylog(2,-e*(c*x+(c^2*x^2+1)^(1/2))/( c*d+(c^2*d^2+e^2)^(1/2)))/e
Time = 0.04 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.94 \[ \int \frac {a+b \text {arcsinh}(c x)}{d+e x} \, dx=\frac {-\left ((a+b \text {arcsinh}(c x)) \left (a+b \text {arcsinh}(c x)-2 b \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )-2 b \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )\right )\right )+2 b^2 \operatorname {PolyLog}\left (2,\frac {e e^{\text {arcsinh}(c x)}}{-c d+\sqrt {c^2 d^2+e^2}}\right )+2 b^2 \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{2 b e} \]
(-((a + b*ArcSinh[c*x])*(a + b*ArcSinh[c*x] - 2*b*Log[1 + (e*E^ArcSinh[c*x ])/(c*d - Sqrt[c^2*d^2 + e^2])] - 2*b*Log[1 + (e*E^ArcSinh[c*x])/(c*d + Sq rt[c^2*d^2 + e^2])])) + 2*b^2*PolyLog[2, (e*E^ArcSinh[c*x])/(-(c*d) + Sqrt [c^2*d^2 + e^2])] + 2*b^2*PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2* d^2 + e^2]))])/(2*b*e)
Time = 0.64 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6242, 6095, 2620, 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.
\(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{d+e x} \, dx\) |
\(\Big \downarrow \) 6242 |
\(\displaystyle \int \frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c d+c e x}d\text {arcsinh}(c x)\) |
\(\Big \downarrow \) 6095 |
\(\displaystyle \int \frac {e^{\text {arcsinh}(c x)} (a+b \text {arcsinh}(c x))}{c d+e e^{\text {arcsinh}(c x)}-\sqrt {c^2 d^2+e^2}}d\text {arcsinh}(c x)+\int \frac {e^{\text {arcsinh}(c x)} (a+b \text {arcsinh}(c x))}{c d+e e^{\text {arcsinh}(c x)}+\sqrt {c^2 d^2+e^2}}d\text {arcsinh}(c x)-\frac {(a+b \text {arcsinh}(c x))^2}{2 b e}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -\frac {b \int \log \left (\frac {e^{\text {arcsinh}(c x)} e}{c d-\sqrt {c^2 d^2+e^2}}+1\right )d\text {arcsinh}(c x)}{e}-\frac {b \int \log \left (\frac {e^{\text {arcsinh}(c x)} e}{c d+\sqrt {c^2 d^2+e^2}}+1\right )d\text {arcsinh}(c x)}{e}+\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}+1\right )}{e}+\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {e e^{\text {arcsinh}(c x)}}{\sqrt {c^2 d^2+e^2}+c d}+1\right )}{e}-\frac {(a+b \text {arcsinh}(c x))^2}{2 b e}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {b \int e^{-\text {arcsinh}(c x)} \log \left (\frac {e^{\text {arcsinh}(c x)} e}{c d-\sqrt {c^2 d^2+e^2}}+1\right )de^{\text {arcsinh}(c x)}}{e}-\frac {b \int e^{-\text {arcsinh}(c x)} \log \left (\frac {e^{\text {arcsinh}(c x)} e}{c d+\sqrt {c^2 d^2+e^2}}+1\right )de^{\text {arcsinh}(c x)}}{e}+\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}+1\right )}{e}+\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {e e^{\text {arcsinh}(c x)}}{\sqrt {c^2 d^2+e^2}+c d}+1\right )}{e}-\frac {(a+b \text {arcsinh}(c x))^2}{2 b e}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}+1\right )}{e}+\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {e e^{\text {arcsinh}(c x)}}{\sqrt {c^2 d^2+e^2}+c d}+1\right )}{e}-\frac {(a+b \text {arcsinh}(c x))^2}{2 b e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}\) |
-1/2*(a + b*ArcSinh[c*x])^2/(b*e) + ((a + b*ArcSinh[c*x])*Log[1 + (e*E^Arc Sinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2])])/e + ((a + b*ArcSinh[c*x])*Log[1 + (e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2])])/e + (b*PolyLog[2, -((e*E ^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2]))])/e + (b*PolyLog[2, -((e*E^Arc Sinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))])/e
3.1.8.3.1 Defintions of rubi rules used
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
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[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin h[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d*x))) , x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))) , x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbo l] :> Subst[Int[(a + b*x)^n*(Cosh[x]/(c*d + e*Sinh[x])), x], x, ArcSinh[c*x ]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]
Time = 0.54 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.49
method | result | size |
parts | \(\frac {a \ln \left (e x +d \right )}{e}-\frac {b \operatorname {arcsinh}\left (c x \right )^{2}}{2 e}+\frac {b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {b \operatorname {dilog}\left (\frac {-c d -e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {b \operatorname {dilog}\left (\frac {c d +e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}\) | \(279\) |
derivativedivides | \(\frac {\frac {a c \ln \left (e c x +c d \right )}{e}+b c \left (-\frac {\operatorname {arcsinh}\left (c x \right )^{2}}{2 e}+\frac {\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {\operatorname {dilog}\left (\frac {-c d -e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {\operatorname {dilog}\left (\frac {c d +e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}\right )}{c}\) | \(286\) |
default | \(\frac {\frac {a c \ln \left (e c x +c d \right )}{e}+b c \left (-\frac {\operatorname {arcsinh}\left (c x \right )^{2}}{2 e}+\frac {\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {\operatorname {dilog}\left (\frac {-c d -e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {\operatorname {dilog}\left (\frac {c d +e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}\right )}{c}\) | \(286\) |
a*ln(e*x+d)/e-1/2*b*arcsinh(c*x)^2/e+b/e*arcsinh(c*x)*ln((-c*d-e*(c*x+(c^2 *x^2+1)^(1/2))+(c^2*d^2+e^2)^(1/2))/(-c*d+(c^2*d^2+e^2)^(1/2)))+b/e*arcsin h(c*x)*ln((c*d+e*(c*x+(c^2*x^2+1)^(1/2))+(c^2*d^2+e^2)^(1/2))/(c*d+(c^2*d^ 2+e^2)^(1/2)))+b/e*dilog((-c*d-e*(c*x+(c^2*x^2+1)^(1/2))+(c^2*d^2+e^2)^(1/ 2))/(-c*d+(c^2*d^2+e^2)^(1/2)))+b/e*dilog((c*d+e*(c*x+(c^2*x^2+1)^(1/2))+( c^2*d^2+e^2)^(1/2))/(c*d+(c^2*d^2+e^2)^(1/2)))
\[ \int \frac {a+b \text {arcsinh}(c x)}{d+e x} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{e x + d} \,d x } \]
\[ \int \frac {a+b \text {arcsinh}(c x)}{d+e x} \, dx=\int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{d + e x}\, dx \]
\[ \int \frac {a+b \text {arcsinh}(c x)}{d+e x} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{e x + d} \,d x } \]
\[ \int \frac {a+b \text {arcsinh}(c x)}{d+e x} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{e x + d} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{d+e x} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{d+e\,x} \,d x \]