Integrand size = 26, antiderivative size = 198 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\frac {2 b^2 x}{c^2 d}-\frac {2 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c^3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^3 d}+\frac {2 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c^3 d}-\frac {2 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^3 d}-\frac {2 i b^2 \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{c^3 d}+\frac {2 i b^2 \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{c^3 d} \]
2*b^2*x/c^2/d+x*(a+b*arcsinh(c*x))^2/c^2/d-2*(a+b*arcsinh(c*x))^2*arctan(c *x+(c^2*x^2+1)^(1/2))/c^3/d+2*I*b*(a+b*arcsinh(c*x))*polylog(2,-I*(c*x+(c^ 2*x^2+1)^(1/2)))/c^3/d-2*I*b*(a+b*arcsinh(c*x))*polylog(2,I*(c*x+(c^2*x^2+ 1)^(1/2)))/c^3/d-2*I*b^2*polylog(3,-I*(c*x+(c^2*x^2+1)^(1/2)))/c^3/d+2*I*b ^2*polylog(3,I*(c*x+(c^2*x^2+1)^(1/2)))/c^3/d-2*b*(a+b*arcsinh(c*x))*(c^2* x^2+1)^(1/2)/c^3/d
Time = 0.45 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.60 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\frac {a^2 x}{c^2 d}-\frac {a^2 \arctan (c x)}{c^3 d}+\frac {2 a b \left (-\sqrt {1+c^2 x^2}+c x \text {arcsinh}(c x)+\frac {1}{2} i \left (-\frac {1}{2} \text {arcsinh}(c x)^2+2 \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )+2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )\right )-\frac {1}{2} i \left (-\frac {1}{2} \text {arcsinh}(c x)^2+2 \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )+2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )\right )}{c^3 d}+\frac {b^2 \left (-2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+c x \left (2+\text {arcsinh}(c x)^2\right )-i \left (-\text {arcsinh}(c x)^2 \left (\log \left (1-i e^{-\text {arcsinh}(c x)}\right )-\log \left (1+i e^{-\text {arcsinh}(c x)}\right )\right )-2 \text {arcsinh}(c x) \left (\operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )\right )-2 \left (\operatorname {PolyLog}\left (3,-i e^{-\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (3,i e^{-\text {arcsinh}(c x)}\right )\right )\right )\right )}{c^3 d} \]
(a^2*x)/(c^2*d) - (a^2*ArcTan[c*x])/(c^3*d) + (2*a*b*(-Sqrt[1 + c^2*x^2] + c*x*ArcSinh[c*x] + (I/2)*(-1/2*ArcSinh[c*x]^2 + 2*ArcSinh[c*x]*Log[1 + I* E^ArcSinh[c*x]] + 2*PolyLog[2, (-I)*E^ArcSinh[c*x]]) - (I/2)*(-1/2*ArcSinh [c*x]^2 + 2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + 2*PolyLog[2, I*E^ArcS inh[c*x]])))/(c^3*d) + (b^2*(-2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + c*x*(2 + ArcSinh[c*x]^2) - I*(-(ArcSinh[c*x]^2*(Log[1 - I/E^ArcSinh[c*x]] - Log[1 + I/E^ArcSinh[c*x]])) - 2*ArcSinh[c*x]*(PolyLog[2, (-I)/E^ArcSinh[c*x]] - P olyLog[2, I/E^ArcSinh[c*x]]) - 2*(PolyLog[3, (-I)/E^ArcSinh[c*x]] - PolyLo g[3, I/E^ArcSinh[c*x]]))))/(c^3*d)
Time = 1.14 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.87, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6227, 27, 6204, 3042, 4668, 3011, 2720, 6213, 24, 7143}
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.
\(\displaystyle \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{c^2 d x^2+d} \, dx\) |
\(\Big \downarrow \) 6227 |
\(\displaystyle -\frac {2 b \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{c d}-\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{d \left (c^2 x^2+1\right )}dx}{c^2}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2 b \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{c d}-\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{c^2 x^2+1}dx}{c^2 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2 d}\) |
\(\Big \downarrow \) 6204 |
\(\displaystyle -\frac {2 b \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{c d}-\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{c^3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int (a+b \text {arcsinh}(c x))^2 \csc \left (i \text {arcsinh}(c x)+\frac {\pi }{2}\right )d\text {arcsinh}(c x)}{c^3 d}-\frac {2 b \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{c d}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2 d}\) |
\(\Big \downarrow \) 4668 |
\(\displaystyle -\frac {-2 i b \int (a+b \text {arcsinh}(c x)) \log \left (1-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i b \int (a+b \text {arcsinh}(c x)) \log \left (1+i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{c^3 d}-\frac {2 b \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{c d}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2 d}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -\frac {2 i b \left (b \int \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{c^3 d}-\frac {2 b \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{c d}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2 d}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{c^3 d}-\frac {2 b \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{c d}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2 d}\) |
\(\Big \downarrow \) 6213 |
\(\displaystyle -\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{c^3 d}-\frac {2 b \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \int 1dx}{c}\right )}{c d}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2 d}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{c^3 d}-\frac {2 b \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )}{c d}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2 d}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2+2 i b \left (b \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )}{c^3 d}-\frac {2 b \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )}{c d}+\frac {x (a+b \text {arcsinh}(c x))^2}{c^2 d}\) |
(x*(a + b*ArcSinh[c*x])^2)/(c^2*d) - (2*b*(-((b*x)/c) + (Sqrt[1 + c^2*x^2] *(a + b*ArcSinh[c*x]))/c^2))/(c*d) - (2*(a + b*ArcSinh[c*x])^2*ArcTan[E^Ar cSinh[c*x]] + (2*I)*b*(-((a + b*ArcSinh[c*x])*PolyLog[2, (-I)*E^ArcSinh[c* x]]) + b*PolyLog[3, (-I)*E^ArcSinh[c*x]]) - (2*I)*b*(-((a + b*ArcSinh[c*x] )*PolyLog[2, I*E^ArcSinh[c*x]]) + b*PolyLog[3, I*E^ArcSinh[c*x]]))/(c^3*d)
3.3.27.3.1 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[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
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_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ {a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
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/(e*(m + 2*p + 1))), x] + (-Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*f*(n/(c*(m + 2*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] ) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ m, 1] && NeQ[m + 2*p + 1, 0]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
\[\int \frac {x^{2} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}{c^{2} d \,x^{2}+d}d x\]
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2}}{c^{2} d x^{2} + d} \,d x } \]
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\frac {\int \frac {a^{2} x^{2}}{c^{2} x^{2} + 1}\, dx + \int \frac {b^{2} x^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{2} x^{2} + 1}\, dx + \int \frac {2 a b x^{2} \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} + 1}\, dx}{d} \]
(Integral(a**2*x**2/(c**2*x**2 + 1), x) + Integral(b**2*x**2*asinh(c*x)**2 /(c**2*x**2 + 1), x) + Integral(2*a*b*x**2*asinh(c*x)/(c**2*x**2 + 1), x)) /d
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2}}{c^{2} d x^{2} + d} \,d x } \]
a^2*(x/(c^2*d) - arctan(c*x)/(c^3*d)) + integrate(b^2*x^2*log(c*x + sqrt(c ^2*x^2 + 1))^2/(c^2*d*x^2 + d) + 2*a*b*x^2*log(c*x + sqrt(c^2*x^2 + 1))/(c ^2*d*x^2 + d), x)
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2}}{c^{2} d x^{2} + d} \,d x } \]
Timed out. \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\int \frac {x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{d\,c^2\,x^2+d} \,d x \]