\(\int \frac {(d+c^2 d x^2)^2 (a+b \text {arcsinh}(c x))}{x} \, dx\) [15]

Optimal result

Integrand size = 24, antiderivative size = 172 \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x} \, dx=-\frac {11}{32} b c d^2 x \sqrt {1+c^2 x^2}-\frac {1}{16} b c d^2 x \left (1+c^2 x^2\right )^{3/2}-\frac {11}{32} b d^2 \text {arcsinh}(c x)+\frac {1}{2} d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))+\frac {d^2 (a+b \text {arcsinh}(c x))^2}{2 b}+d^2 (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-\frac {1}{2} b d^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right ) \]

[Out]

-1/16*b*c*d^2*x*(c^2*x^2+1)^(3/2)-11/32*b*d^2*arcsinh(c*x)+1/2*d^2*(c^2*x^2+1)*(a+b*arcsinh(c*x))+1/4*d^2*(c^2
*x^2+1)^2*(a+b*arcsinh(c*x))+1/2*d^2*(a+b*arcsinh(c*x))^2/b+d^2*(a+b*arcsinh(c*x))*ln(1-1/(c*x+(c^2*x^2+1)^(1/
2))^2)-1/2*b*d^2*polylog(2,1/(c*x+(c^2*x^2+1)^(1/2))^2)-11/32*b*c*d^2*x*(c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5801, 5775, 3797, 2221, 2317, 2438, 201, 221} \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x} \, dx=\frac {1}{4} d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))+\frac {1}{2} d^2 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))+\frac {d^2 (a+b \text {arcsinh}(c x))^2}{2 b}+d^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b d^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )-\frac {11}{32} b d^2 \text {arcsinh}(c x)-\frac {1}{16} b c d^2 x \left (c^2 x^2+1\right )^{3/2}-\frac {11}{32} b c d^2 x \sqrt {c^2 x^2+1} \]

[In]

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

[Out]

(-11*b*c*d^2*x*Sqrt[1 + c^2*x^2])/32 - (b*c*d^2*x*(1 + c^2*x^2)^(3/2))/16 - (11*b*d^2*ArcSinh[c*x])/32 + (d^2*
(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/2 + (d^2*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))/4 + (d^2*(a + b*ArcSinh[c*x
])^2)/(2*b) + d^2*(a + b*ArcSinh[c*x])*Log[1 - E^(-2*ArcSinh[c*x])] - (b*d^2*PolyLog[2, E^(-2*ArcSinh[c*x])])/
2

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

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 2317

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

Rule 2438

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]

Rule 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 5775

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[1/b, Subst[Int[x^n*Coth[-a/b + x/b], x],
x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 5801

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.))/(x_), x_Symbol] :> Simp[(d + e*x^2)^p*((
a + b*ArcSinh[c*x])/(2*p)), x] + (Dist[d, Int[(d + e*x^2)^(p - 1)*((a + b*ArcSinh[c*x])/x), x], x] - Dist[b*c*
(d^p/(2*p)), Int[(1 + c^2*x^2)^(p - 1/2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))+d \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x} \, dx-\frac {1}{4} \left (b c d^2\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx \\ & = -\frac {1}{16} b c d^2 x \left (1+c^2 x^2\right )^{3/2}+\frac {1}{2} d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))+d^2 \int \frac {a+b \text {arcsinh}(c x)}{x} \, dx-\frac {1}{16} \left (3 b c d^2\right ) \int \sqrt {1+c^2 x^2} \, dx-\frac {1}{2} \left (b c d^2\right ) \int \sqrt {1+c^2 x^2} \, dx \\ & = -\frac {11}{32} b c d^2 x \sqrt {1+c^2 x^2}-\frac {1}{16} b c d^2 x \left (1+c^2 x^2\right )^{3/2}+\frac {1}{2} d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))-\frac {d^2 \text {Subst}\left (\int x \coth \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b}-\frac {1}{32} \left (3 b c d^2\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{4} \left (b c d^2\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {11}{32} b c d^2 x \sqrt {1+c^2 x^2}-\frac {1}{16} b c d^2 x \left (1+c^2 x^2\right )^{3/2}-\frac {11}{32} b d^2 \text {arcsinh}(c x)+\frac {1}{2} d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))+\frac {d^2 (a+b \text {arcsinh}(c x))^2}{2 b}+\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )} x}{1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b} \\ & = -\frac {11}{32} b c d^2 x \sqrt {1+c^2 x^2}-\frac {1}{16} b c d^2 x \left (1+c^2 x^2\right )^{3/2}-\frac {11}{32} b d^2 \text {arcsinh}(c x)+\frac {1}{2} d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))+\frac {d^2 (a+b \text {arcsinh}(c x))^2}{2 b}+d^2 (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-d^2 \text {Subst}\left (\int \log \left (1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right ) \\ & = -\frac {11}{32} b c d^2 x \sqrt {1+c^2 x^2}-\frac {1}{16} b c d^2 x \left (1+c^2 x^2\right )^{3/2}-\frac {11}{32} b d^2 \text {arcsinh}(c x)+\frac {1}{2} d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))+\frac {d^2 (a+b \text {arcsinh}(c x))^2}{2 b}+d^2 (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )+\frac {1}{2} \left (b d^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}\right ) \\ & = -\frac {11}{32} b c d^2 x \sqrt {1+c^2 x^2}-\frac {1}{16} b c d^2 x \left (1+c^2 x^2\right )^{3/2}-\frac {11}{32} b d^2 \text {arcsinh}(c x)+\frac {1}{2} d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))+\frac {d^2 (a+b \text {arcsinh}(c x))^2}{2 b}+d^2 (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-\frac {1}{2} b d^2 \operatorname {PolyLog}\left (2,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.01 \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x} \, dx=\frac {d^2 \left (-16 a^2+24 a b+32 a b c^2 x^2+8 a b c^4 x^4-13 b^2 c x \sqrt {1+c^2 x^2}-2 b^2 c^3 x^3 \sqrt {1+c^2 x^2}-16 b^2 \text {arcsinh}(c x)^2+32 a b \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+b \text {arcsinh}(c x) \left (-32 a+b \left (13+32 c^2 x^2+8 c^4 x^4\right )+32 b \log \left (1-e^{2 \text {arcsinh}(c x)}\right )\right )+16 b^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )\right )}{32 b} \]

[In]

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

[Out]

(d^2*(-16*a^2 + 24*a*b + 32*a*b*c^2*x^2 + 8*a*b*c^4*x^4 - 13*b^2*c*x*Sqrt[1 + c^2*x^2] - 2*b^2*c^3*x^3*Sqrt[1
+ c^2*x^2] - 16*b^2*ArcSinh[c*x]^2 + 32*a*b*Log[1 - E^(2*ArcSinh[c*x])] + b*ArcSinh[c*x]*(-32*a + b*(13 + 32*c
^2*x^2 + 8*c^4*x^4) + 32*b*Log[1 - E^(2*ArcSinh[c*x])]) + 16*b^2*PolyLog[2, E^(2*ArcSinh[c*x])]))/(32*b)

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.29

method	result	size
parts	\(d^{2} a \left (\frac {c^{4} x^{4}}{4}+c^{2} x^{2}+\ln \left (x \right )\right )-\frac {d^{2} b \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{16}-\frac {13 b c \,d^{2} x \sqrt {c^{2} x^{2}+1}}{32}+\frac {d^{2} b \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}+d^{2} b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+d^{2} b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+d^{2} b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+\frac {13 b \,d^{2} \operatorname {arcsinh}\left (c x \right )}{32}+d^{2} b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+d^{2} b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{2} b \operatorname {arcsinh}\left (c x \right )^{2}}{2}\)	\(222\)
derivativedivides	\(d^{2} a \left (\frac {c^{4} x^{4}}{4}+c^{2} x^{2}+\ln \left (c x \right )\right )+\frac {13 b \,d^{2} \operatorname {arcsinh}\left (c x \right )}{32}+d^{2} b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+d^{2} b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{2} b \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{16}-\frac {13 b c \,d^{2} x \sqrt {c^{2} x^{2}+1}}{32}-\frac {d^{2} b \operatorname {arcsinh}\left (c x \right )^{2}}{2}+d^{2} b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+d^{2} b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+\frac {d^{2} b \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}+d^{2} b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}\)	\(224\)
default	\(d^{2} a \left (\frac {c^{4} x^{4}}{4}+c^{2} x^{2}+\ln \left (c x \right )\right )+\frac {13 b \,d^{2} \operatorname {arcsinh}\left (c x \right )}{32}+d^{2} b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+d^{2} b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{2} b \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{16}-\frac {13 b c \,d^{2} x \sqrt {c^{2} x^{2}+1}}{32}-\frac {d^{2} b \operatorname {arcsinh}\left (c x \right )^{2}}{2}+d^{2} b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+d^{2} b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+\frac {d^{2} b \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}+d^{2} b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}\)	\(224\)

[In]

int((c^2*d*x^2+d)^2*(a+b*arcsinh(c*x))/x,x,method=_RETURNVERBOSE)

[Out]

d^2*a*(1/4*c^4*x^4+c^2*x^2+ln(x))-1/16*d^2*b*c^3*x^3*(c^2*x^2+1)^(1/2)-13/32*b*c*d^2*x*(c^2*x^2+1)^(1/2)+1/4*d
^2*b*arcsinh(c*x)*c^4*x^4+d^2*b*arcsinh(c*x)*c^2*x^2+d^2*b*polylog(2,-c*x-(c^2*x^2+1)^(1/2))+d^2*b*polylog(2,c
*x+(c^2*x^2+1)^(1/2))+13/32*b*d^2*arcsinh(c*x)+d^2*b*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))+d^2*b*arcsinh(c*
x)*ln(1-c*x-(c^2*x^2+1)^(1/2))-1/2*d^2*b*arcsinh(c*x)^2

Fricas [F]

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

[In]

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

[Out]

integral((a*c^4*d^2*x^4 + 2*a*c^2*d^2*x^2 + a*d^2 + (b*c^4*d^2*x^4 + 2*b*c^2*d^2*x^2 + b*d^2)*arcsinh(c*x))/x,
x)

Sympy [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x} \, dx=d^{2} \left (\int \frac {a}{x}\, dx + \int 2 a c^{2} x\, dx + \int a c^{4} x^{3}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int 2 b c^{2} x \operatorname {asinh}{\left (c x \right )}\, dx + \int b c^{4} x^{3} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]

[In]

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

[Out]

d**2*(Integral(a/x, x) + Integral(2*a*c**2*x, x) + Integral(a*c**4*x**3, x) + Integral(b*asinh(c*x)/x, x) + In
tegral(2*b*c**2*x*asinh(c*x), x) + Integral(b*c**4*x**3*asinh(c*x), x))

Maxima [F]

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

[In]

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

[Out]

1/4*a*c^4*d^2*x^4 + a*c^2*d^2*x^2 + a*d^2*log(x) + integrate(b*c^4*d^2*x^3*log(c*x + sqrt(c^2*x^2 + 1)) + 2*b*
c^2*d^2*x*log(c*x + sqrt(c^2*x^2 + 1)) + b*d^2*log(c*x + sqrt(c^2*x^2 + 1))/x, x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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