Integrand size = 24, antiderivative size = 127 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=-\frac {b}{2 c^3 d^2 \sqrt {1+c^2 x^2}}-\frac {x (a+b \text {arcsinh}(c x))}{2 c^2 d^2 \left (1+c^2 x^2\right )}+\frac {(a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^3 d^2}-\frac {i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{2 c^3 d^2}+\frac {i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c^3 d^2} \]
-1/2*x*(a+b*arcsinh(c*x))/c^2/d^2/(c^2*x^2+1)+(a+b*arcsinh(c*x))*arctan(c* x+(c^2*x^2+1)^(1/2))/c^3/d^2-1/2*I*b*polylog(2,-I*(c*x+(c^2*x^2+1)^(1/2))) /c^3/d^2+1/2*I*b*polylog(2,I*(c*x+(c^2*x^2+1)^(1/2)))/c^3/d^2-1/2*b/c^3/d^ 2/(c^2*x^2+1)^(1/2)
Time = 0.17 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.74 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=-\frac {a c x+b \sqrt {1+c^2 x^2}+b c x \text {arcsinh}(c x)-a \arctan (c x)-a c^2 x^2 \arctan (c x)-i b \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )-i b c^2 x^2 \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )+i b \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )+i b c^2 x^2 \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )+i b \left (1+c^2 x^2\right ) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )-i b \left (1+c^2 x^2\right ) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c^3 d^2 \left (1+c^2 x^2\right )} \]
-1/2*(a*c*x + b*Sqrt[1 + c^2*x^2] + b*c*x*ArcSinh[c*x] - a*ArcTan[c*x] - a *c^2*x^2*ArcTan[c*x] - I*b*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] - I*b*c^ 2*x^2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + I*b*ArcSinh[c*x]*Log[1 + I* E^ArcSinh[c*x]] + I*b*c^2*x^2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] + I*b *(1 + c^2*x^2)*PolyLog[2, (-I)*E^ArcSinh[c*x]] - I*b*(1 + c^2*x^2)*PolyLog [2, I*E^ArcSinh[c*x]])/(c^3*d^2*(1 + c^2*x^2))
Time = 0.56 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.92, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6225, 27, 241, 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.
\(\displaystyle \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (c^2 d x^2+d\right )^2} \, dx\) |
\(\Big \downarrow \) 6225 |
\(\displaystyle \frac {\int \frac {a+b \text {arcsinh}(c x)}{d \left (c^2 x^2+1\right )}dx}{2 c^2 d}+\frac {b \int \frac {x}{\left (c^2 x^2+1\right )^{3/2}}dx}{2 c d^2}-\frac {x (a+b \text {arcsinh}(c x))}{2 c^2 d^2 \left (c^2 x^2+1\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {a+b \text {arcsinh}(c x)}{c^2 x^2+1}dx}{2 c^2 d^2}+\frac {b \int \frac {x}{\left (c^2 x^2+1\right )^{3/2}}dx}{2 c d^2}-\frac {x (a+b \text {arcsinh}(c x))}{2 c^2 d^2 \left (c^2 x^2+1\right )}\) |
\(\Big \downarrow \) 241 |
\(\displaystyle \frac {\int \frac {a+b \text {arcsinh}(c x)}{c^2 x^2+1}dx}{2 c^2 d^2}-\frac {x (a+b \text {arcsinh}(c x))}{2 c^2 d^2 \left (c^2 x^2+1\right )}-\frac {b}{2 c^3 d^2 \sqrt {c^2 x^2+1}}\) |
\(\Big \downarrow \) 6204 |
\(\displaystyle \frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{2 c^3 d^2}-\frac {x (a+b \text {arcsinh}(c x))}{2 c^2 d^2 \left (c^2 x^2+1\right )}-\frac {b}{2 c^3 d^2 \sqrt {c^2 x^2+1}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int (a+b \text {arcsinh}(c x)) \csc \left (i \text {arcsinh}(c x)+\frac {\pi }{2}\right )d\text {arcsinh}(c x)}{2 c^3 d^2}-\frac {x (a+b \text {arcsinh}(c x))}{2 c^2 d^2 \left (c^2 x^2+1\right )}-\frac {b}{2 c^3 d^2 \sqrt {c^2 x^2+1}}\) |
\(\Big \downarrow \) 4668 |
\(\displaystyle \frac {-i b \int \log \left (1-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+i b \int \log \left (1+i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{2 c^3 d^2}-\frac {x (a+b \text {arcsinh}(c x))}{2 c^2 d^2 \left (c^2 x^2+1\right )}-\frac {b}{2 c^3 d^2 \sqrt {c^2 x^2+1}}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle \frac {-i b \int e^{-\text {arcsinh}(c x)} \log \left (1-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+i b \int e^{-\text {arcsinh}(c x)} \log \left (1+i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{2 c^3 d^2}-\frac {x (a+b \text {arcsinh}(c x))}{2 c^2 d^2 \left (c^2 x^2+1\right )}-\frac {b}{2 c^3 d^2 \sqrt {c^2 x^2+1}}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c^3 d^2}-\frac {x (a+b \text {arcsinh}(c x))}{2 c^2 d^2 \left (c^2 x^2+1\right )}-\frac {b}{2 c^3 d^2 \sqrt {c^2 x^2+1}}\) |
-1/2*b/(c^3*d^2*Sqrt[1 + c^2*x^2]) - (x*(a + b*ArcSinh[c*x]))/(2*c^2*d^2*( 1 + c^2*x^2)) + (2*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]] - I*b*PolyL og[2, (-I)*E^ArcSinh[c*x]] + I*b*PolyLog[2, I*E^ArcSinh[c*x]])/(2*c^3*d^2)
3.1.39.3.1 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[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
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*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] - S imp[b*f*(n/(2*c*(p + 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]) /; Fre eQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && IG tQ[m, 1]
Time = 0.16 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.54
method | result | size |
derivativedivides | \(\frac {\frac {a \left (-\frac {c x}{2 \left (c^{2} x^{2}+1\right )}+\frac {\arctan \left (c x \right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {c x \,\operatorname {arcsinh}\left (c x \right )}{2 \left (c^{2} x^{2}+1\right )}+\frac {\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{2}+\frac {\arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {\arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}+\frac {i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {1}{2 \sqrt {c^{2} x^{2}+1}}\right )}{d^{2}}}{c^{3}}\) | \(195\) |
default | \(\frac {\frac {a \left (-\frac {c x}{2 \left (c^{2} x^{2}+1\right )}+\frac {\arctan \left (c x \right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {c x \,\operatorname {arcsinh}\left (c x \right )}{2 \left (c^{2} x^{2}+1\right )}+\frac {\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{2}+\frac {\arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {\arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}+\frac {i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {1}{2 \sqrt {c^{2} x^{2}+1}}\right )}{d^{2}}}{c^{3}}\) | \(195\) |
parts | \(\frac {a \left (-\frac {x}{2 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\arctan \left (c x \right )}{2 c^{3}}\right )}{d^{2}}+\frac {b \left (-\frac {c x \,\operatorname {arcsinh}\left (c x \right )}{2 \left (c^{2} x^{2}+1\right )}+\frac {\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{2}+\frac {\arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {\arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}+\frac {i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {1}{2 \sqrt {c^{2} x^{2}+1}}\right )}{d^{2} c^{3}}\) | \(199\) |
1/c^3*(a/d^2*(-1/2*c*x/(c^2*x^2+1)+1/2*arctan(c*x))+b/d^2*(-1/2*c*x/(c^2*x ^2+1)*arcsinh(c*x)+1/2*arcsinh(c*x)*arctan(c*x)+1/2*arctan(c*x)*ln(1+I*(1+ I*c*x)/(c^2*x^2+1)^(1/2))-1/2*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/ 2))-1/2*I*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+1/2*I*dilog(1-I*(1+I*c*x) /(c^2*x^2+1)^(1/2))-1/2/(c^2*x^2+1)^(1/2)))
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}{{\left (c^{2} d x^{2} + d\right )}^{2}} \,d x } \]
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a x^{2}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx + \int \frac {b x^{2} \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]
(Integral(a*x**2/(c**4*x**4 + 2*c**2*x**2 + 1), x) + Integral(b*x**2*asinh (c*x)/(c**4*x**4 + 2*c**2*x**2 + 1), x))/d**2
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}{{\left (c^{2} d x^{2} + d\right )}^{2}} \,d x } \]
-1/2*a*(x/(c^4*d^2*x^2 + c^2*d^2) - arctan(c*x)/(c^3*d^2)) + b*integrate(x ^2*log(c*x + sqrt(c^2*x^2 + 1))/(c^4*d^2*x^4 + 2*c^2*d^2*x^2 + d^2), x)
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}{{\left (c^{2} d x^{2} + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^2} \,d x \]