3.1.7 \(\int \frac {(d+c^2 d x^2) (a+b \sinh ^{-1}(c x))}{x^2} \, dx\) [7]

Optimal. Leaf size=66 \[ -b c d \sqrt {1+c^2 x^2}-\frac {d \left (a+b \sinh ^{-1}(c x)\right )}{x}+c^2 d x \left (a+b \sinh ^{-1}(c x)\right )-b c d \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right ) \]

[Out]

-d*(a+b*arcsinh(c*x))/x+c^2*d*x*(a+b*arcsinh(c*x))-b*c*d*arctanh((c^2*x^2+1)^(1/2))-b*c*d*(c^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.06, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {14, 5803, 12, 457, 81, 65, 214} \begin {gather*} c^2 d x \left (a+b \sinh ^{-1}(c x)\right )-\frac {d \left (a+b \sinh ^{-1}(c x)\right )}{x}-b c d \sqrt {c^2 x^2+1}-b c d \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*c*d*Sqrt[1 + c^2*x^2]) - (d*(a + b*ArcSinh[c*x]))/x + c^2*d*x*(a + b*ArcSinh[c*x]) - b*c*d*ArcTanh[Sqrt[1
+ c^2*x^2]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 81

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

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 5803

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1
+ c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {\left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{x^2} \, dx &=-\frac {d \left (a+b \sinh ^{-1}(c x)\right )}{x}+c^2 d x \left (a+b \sinh ^{-1}(c x)\right )-(b c) \int \frac {d \left (-1+c^2 x^2\right )}{x \sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {d \left (a+b \sinh ^{-1}(c x)\right )}{x}+c^2 d x \left (a+b \sinh ^{-1}(c x)\right )-(b c d) \int \frac {-1+c^2 x^2}{x \sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {d \left (a+b \sinh ^{-1}(c x)\right )}{x}+c^2 d x \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{2} (b c d) \text {Subst}\left (\int \frac {-1+c^2 x}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )\\ &=-b c d \sqrt {1+c^2 x^2}-\frac {d \left (a+b \sinh ^{-1}(c x)\right )}{x}+c^2 d x \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{2} (b c d) \text {Subst}\left (\int \frac {1}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )\\ &=-b c d \sqrt {1+c^2 x^2}-\frac {d \left (a+b \sinh ^{-1}(c x)\right )}{x}+c^2 d x \left (a+b \sinh ^{-1}(c x)\right )+\frac {(b d) \text {Subst}\left (\int \frac {1}{-\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {1+c^2 x^2}\right )}{c}\\ &=-b c d \sqrt {1+c^2 x^2}-\frac {d \left (a+b \sinh ^{-1}(c x)\right )}{x}+c^2 d x \left (a+b \sinh ^{-1}(c x)\right )-b c d \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 74, normalized size = 1.12 \begin {gather*} -\frac {a d}{x}+a c^2 d x-b c d \sqrt {1+c^2 x^2}-\frac {b d \sinh ^{-1}(c x)}{x}+b c^2 d x \sinh ^{-1}(c x)-b c d \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*d)/x) + a*c^2*d*x - b*c*d*Sqrt[1 + c^2*x^2] - (b*d*ArcSinh[c*x])/x + b*c^2*d*x*ArcSinh[c*x] - b*c*d*ArcTa
nh[Sqrt[1 + c^2*x^2]]

________________________________________________________________________________________

Maple [A]
time = 0.48, size = 69, normalized size = 1.05

method	result	size
derivativedivides	\(c \left (a d \left (c x -\frac {1}{c x}\right )+b d \left (\arcsinh \left (c x \right ) c x -\frac {\arcsinh \left (c x \right )}{c x}-\sqrt {c^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\right )\)	\(69\)
default	\(c \left (a d \left (c x -\frac {1}{c x}\right )+b d \left (\arcsinh \left (c x \right ) c x -\frac {\arcsinh \left (c x \right )}{c x}-\sqrt {c^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\right )\)	\(69\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*(a*d*(c*x-1/c/x)+b*d*(arcsinh(c*x)*c*x-arcsinh(c*x)/c/x-(c^2*x^2+1)^(1/2)-arctanh(1/(c^2*x^2+1)^(1/2))))

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 64, normalized size = 0.97 \begin {gather*} a c^{2} d x + {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b c d - {\left (c \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arsinh}\left (c x\right )}{x}\right )} b d - \frac {a d}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c^2*d*x + (c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*b*c*d - (c*arcsinh(1/(c*abs(x))) + arcsinh(c*x)/x)*b*d - a*
d/x

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (62) = 124\).
time = 0.37, size = 156, normalized size = 2.36 \begin {gather*} \frac {a c^{2} d x^{2} - b c d x \log \left (-c x + \sqrt {c^{2} x^{2} + 1} + 1\right ) + b c d x \log \left (-c x + \sqrt {c^{2} x^{2} + 1} - 1\right ) - \sqrt {c^{2} x^{2} + 1} b c d x - {\left (b c^{2} - b\right )} d x \log \left (-c x + \sqrt {c^{2} x^{2} + 1}\right ) - a d + {\left (b c^{2} d x^{2} - {\left (b c^{2} - b\right )} d x - b d\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*c^2*d*x^2 - b*c*d*x*log(-c*x + sqrt(c^2*x^2 + 1) + 1) + b*c*d*x*log(-c*x + sqrt(c^2*x^2 + 1) - 1) - sqrt(c^
2*x^2 + 1)*b*c*d*x - (b*c^2 - b)*d*x*log(-c*x + sqrt(c^2*x^2 + 1)) - a*d + (b*c^2*d*x^2 - (b*c^2 - b)*d*x - b*
d)*log(c*x + sqrt(c^2*x^2 + 1)))/x

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} d \left (\int a c^{2}\, dx + \int \frac {a}{x^{2}}\, dx + \int b c^{2} \operatorname {asinh}{\left (c x \right )}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{x^{2}}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*d*x^2+d)*(a+b*arcsinh(c*x))/x^2,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]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\left (d\,c^2\,x^2+d\right )}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________