Integrand size = 27, antiderivative size = 82 \[ \int \frac {x^2}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{2 b c^3}-\frac {\log (a+b \text {arcsinh}(c x))}{2 b c^3}-\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{2 b c^3} \]
1/2*Chi(2*(a+b*arcsinh(c*x))/b)*cosh(2*a/b)/b/c^3-1/2*ln(a+b*arcsinh(c*x)) /b/c^3-1/2*Shi(2*(a+b*arcsinh(c*x))/b)*sinh(2*a/b)/b/c^3
Time = 0.29 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.79 \[ \int \frac {x^2}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-\log (a+b \text {arcsinh}(c x))-\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{2 b c^3} \]
(Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcSinh[c*x])] - Log[a + b*ArcSinh[c* x]] - Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])])/(2*b*c^3)
Time = 0.45 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.87, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6234, 3042, 25, 3793, 2009}
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.
\(\displaystyle \int \frac {x^2}{\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))} \, dx\) |
\(\Big \downarrow \) 6234 |
\(\displaystyle \frac {\int \frac {\sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b c^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int -\frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )^2}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b c^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )^2}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b c^3}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle -\frac {\int \left (\frac {1}{2 (a+b \text {arcsinh}(c x))}-\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{2 (a+b \text {arcsinh}(c x))}\right )d(a+b \text {arcsinh}(c x))}{b c^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1}{2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{2} \log (a+b \text {arcsinh}(c x))}{b c^3}\) |
((Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b])/2 - Log[a + b*Ar cSinh[c*x]]/2 - (Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/2 )/(b*c^3)
3.4.91.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2* x^2)^p] Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
Time = 0.23 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.82
method | result | size |
default | \(-\frac {{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (c x \right )+\frac {2 a}{b}\right )+{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (c x \right )-\frac {2 a}{b}\right )+2 \ln \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}{4 c^{3} b}\) | \(67\) |
-1/4*(exp(2*a/b)*Ei(1,2*arcsinh(c*x)+2*a/b)+exp(-2*a/b)*Ei(1,-2*arcsinh(c* x)-2*a/b)+2*ln(a+b*arcsinh(c*x)))/c^3/b
\[ \int \frac {x^2}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {x^{2}}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}} \,d x } \]
\[ \int \frac {x^2}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\int \frac {x^{2}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \sqrt {c^{2} x^{2} + 1}}\, dx \]
\[ \int \frac {x^2}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {x^{2}}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}} \,d x } \]
\[ \int \frac {x^2}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {x^{2}}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}} \,d x } \]
Timed out. \[ \int \frac {x^2}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\int \frac {x^2}{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {c^2\,x^2+1}} \,d x \]