\(\int \frac {(a+b \text {arcsinh}(c x))^2}{d+e x} \, dx\) [16]

Optimal result

Integrand size = 18, antiderivative size = 291 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{d+e x} \, dx=-\frac {(a+b \text {arcsinh}(c x))^3}{3 b e}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {2 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {2 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {2 b^2 \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {2 b^2 \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e} \]

[Out]

-1/3*(a+b*arcsinh(c*x))^3/b/e+(a+b*arcsinh(c*x))^2*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))^2*ln(1+e*(c*x+(c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2+e^2)^(1/2)))/e+2*b*(a+b*arcsinh(c*x))*polyl
og(2,-e*(c*x+(c^2*x^2+1)^(1/2))/(c*d-(c^2*d^2+e^2)^(1/2)))/e+2*b*(a+b*arcsinh(c*x))*polylog(2,-e*(c*x+(c^2*x^2
+1)^(1/2))/(c*d+(c^2*d^2+e^2)^(1/2)))/e-2*b^2*polylog(3,-e*(c*x+(c^2*x^2+1)^(1/2))/(c*d-(c^2*d^2+e^2)^(1/2)))/
e-2*b^2*polylog(3,-e*(c*x+(c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2+e^2)^(1/2)))/e

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5827, 5680, 2221, 2611, 2320, 6724} \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{d+e x} \, dx=\frac {2 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {2 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}+1\right )}{e}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (\frac {e e^{\text {arcsinh}(c x)}}{\sqrt {c^2 d^2+e^2}+c d}+1\right )}{e}-\frac {(a+b \text {arcsinh}(c x))^3}{3 b e}-\frac {2 b^2 \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {2 b^2 \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e} \]

[In]

Int[(a + b*ArcSinh[c*x])^2/(d + e*x),x]

[Out]

-1/3*(a + b*ArcSinh[c*x])^3/(b*e) + ((a + b*ArcSinh[c*x])^2*Log[1 + (e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e
^2])])/e + ((a + b*ArcSinh[c*x])^2*Log[1 + (e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2])])/e + (2*b*(a + b*Ar
cSinh[c*x])*PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2]))])/e + (2*b*(a + b*ArcSinh[c*x])*PolyL
og[2, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))])/e - (2*b^2*PolyLog[3, -((e*E^ArcSinh[c*x])/(c*d - Sq
rt[c^2*d^2 + e^2]))])/e - (2*b^2*PolyLog[3, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))])/e

Rule 2221

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]
- Dist[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]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

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] + Dist[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]

Rule 5680

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(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]

Rule 5827

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> 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]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {(a+b x)^2 \cosh (x)}{c d+e \sinh (x)} \, dx,x,\text {arcsinh}(c x)\right ) \\ & = -\frac {(a+b \text {arcsinh}(c x))^3}{3 b e}+\text {Subst}\left (\int \frac {e^x (a+b x)^2}{c d-\sqrt {c^2 d^2+e^2}+e e^x} \, dx,x,\text {arcsinh}(c x)\right )+\text {Subst}\left (\int \frac {e^x (a+b x)^2}{c d+\sqrt {c^2 d^2+e^2}+e e^x} \, dx,x,\text {arcsinh}(c x)\right ) \\ & = -\frac {(a+b \text {arcsinh}(c x))^3}{3 b e}+\frac {(a+b \text {arcsinh}(c x))^2 \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))^2 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {(2 b) \text {Subst}\left (\int (a+b x) \log \left (1+\frac {e e^x}{c d-\sqrt {c^2 d^2+e^2}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{e}-\frac {(2 b) \text {Subst}\left (\int (a+b x) \log \left (1+\frac {e e^x}{c d+\sqrt {c^2 d^2+e^2}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{e} \\ & = -\frac {(a+b \text {arcsinh}(c x))^3}{3 b e}+\frac {(a+b \text {arcsinh}(c x))^2 \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))^2 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {2 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {2 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\frac {e e^x}{c d-\sqrt {c^2 d^2+e^2}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{e}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\frac {e e^x}{c d+\sqrt {c^2 d^2+e^2}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{e} \\ & = -\frac {(a+b \text {arcsinh}(c x))^3}{3 b e}+\frac {(a+b \text {arcsinh}(c x))^2 \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))^2 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {2 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {2 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {e x}{-c d+\sqrt {c^2 d^2+e^2}}\right )}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{e}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {e x}{c d+\sqrt {c^2 d^2+e^2}}\right )}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{e} \\ & = -\frac {(a+b \text {arcsinh}(c x))^3}{3 b e}+\frac {(a+b \text {arcsinh}(c x))^2 \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))^2 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {2 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {2 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {2 b^2 \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {2 b^2 \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{d+e x} \, dx=\frac {-\frac {(a+b \text {arcsinh}(c x))^3}{b}+3 (a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )+3 (a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )+6 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,\frac {e e^{\text {arcsinh}(c x)}}{-c d+\sqrt {c^2 d^2+e^2}}\right )+6 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )-6 b^2 \operatorname {PolyLog}\left (3,\frac {e e^{\text {arcsinh}(c x)}}{-c d+\sqrt {c^2 d^2+e^2}}\right )-6 b^2 \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{3 e} \]

[In]

Integrate[(a + b*ArcSinh[c*x])^2/(d + e*x),x]

[Out]

(-((a + b*ArcSinh[c*x])^3/b) + 3*(a + b*ArcSinh[c*x])^2*Log[1 + (e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2])
] + 3*(a + b*ArcSinh[c*x])^2*Log[1 + (e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2])] + 6*b*(a + b*ArcSinh[c*x]
)*PolyLog[2, (e*E^ArcSinh[c*x])/(-(c*d) + Sqrt[c^2*d^2 + e^2])] + 6*b*(a + b*ArcSinh[c*x])*PolyLog[2, -((e*E^A
rcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))] - 6*b^2*PolyLog[3, (e*E^ArcSinh[c*x])/(-(c*d) + Sqrt[c^2*d^2 + e^2]
)] - 6*b^2*PolyLog[3, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))])/(3*e)

Maple [F]

\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}{e x +d}d x\]

[In]

int((a+b*arcsinh(c*x))^2/(e*x+d),x)

[Out]

int((a+b*arcsinh(c*x))^2/(e*x+d),x)

Fricas [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{e x + d} \,d x } \]

[In]

integrate((a+b*arcsinh(c*x))^2/(e*x+d),x, algorithm="fricas")

[Out]

integral((b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)/(e*x + d), x)

Sympy [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{d+e x} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{d + e x}\, dx \]

[In]

integrate((a+b*asinh(c*x))**2/(e*x+d),x)

[Out]

Integral((a + b*asinh(c*x))**2/(d + e*x), x)

Maxima [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{e x + d} \,d x } \]

[In]

integrate((a+b*arcsinh(c*x))^2/(e*x+d),x, algorithm="maxima")

[Out]

a^2*log(e*x + d)/e + integrate(b^2*log(c*x + sqrt(c^2*x^2 + 1))^2/(e*x + d) + 2*a*b*log(c*x + sqrt(c^2*x^2 + 1
))/(e*x + d), x)

Giac [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{e x + d} \,d x } \]

[In]

integrate((a+b*arcsinh(c*x))^2/(e*x+d),x, algorithm="giac")

[Out]

integrate((b*arcsinh(c*x) + a)^2/(e*x + d), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{d+e x} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{d+e\,x} \,d x \]

[In]

int((a + b*asinh(c*x))^2/(d + e*x),x)

[Out]

int((a + b*asinh(c*x))^2/(d + e*x), x)