\(\int \frac {a+b \text {arcsinh}(c x)}{(d+c^2 d x^2)^3} \, dx\) [50]

Optimal result

Integrand size = 21, antiderivative size = 178 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {b}{12 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {3 b}{8 c d^3 \sqrt {1+c^2 x^2}}+\frac {x (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {3 x (a+b \text {arcsinh}(c x))}{8 d^3 \left (1+c^2 x^2\right )}+\frac {3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 c d^3}-\frac {3 i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{8 c d^3}+\frac {3 i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{8 c d^3} \]

[Out]

1/12*b/c/d^3/(c^2*x^2+1)^(3/2)+1/4*x*(a+b*arcsinh(c*x))/d^3/(c^2*x^2+1)^2+3/8*x*(a+b*arcsinh(c*x))/d^3/(c^2*x^
2+1)+3/4*(a+b*arcsinh(c*x))*arctan(c*x+(c^2*x^2+1)^(1/2))/c/d^3-3/8*I*b*polylog(2,-I*(c*x+(c^2*x^2+1)^(1/2)))/
c/d^3+3/8*I*b*polylog(2,I*(c*x+(c^2*x^2+1)^(1/2)))/c/d^3+3/8*b/c/d^3/(c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5788, 5789, 4265, 2317, 2438, 267} \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {3 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{4 c d^3}+\frac {3 x (a+b \text {arcsinh}(c x))}{8 d^3 \left (c^2 x^2+1\right )}+\frac {x (a+b \text {arcsinh}(c x))}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac {3 i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{8 c d^3}+\frac {3 i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{8 c d^3}+\frac {3 b}{8 c d^3 \sqrt {c^2 x^2+1}}+\frac {b}{12 c d^3 \left (c^2 x^2+1\right )^{3/2}} \]

[In]

Int[(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^3,x]

[Out]

b/(12*c*d^3*(1 + c^2*x^2)^(3/2)) + (3*b)/(8*c*d^3*Sqrt[1 + c^2*x^2]) + (x*(a + b*ArcSinh[c*x]))/(4*d^3*(1 + c^
2*x^2)^2) + (3*x*(a + b*ArcSinh[c*x]))/(8*d^3*(1 + c^2*x^2)) + (3*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])
/(4*c*d^3) - (((3*I)/8)*b*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c*d^3) + (((3*I)/8)*b*PolyLog[2, I*E^ArcSinh[c*x]]
)/(c*d^3)

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

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 4265

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] + (-Dist[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] + Dist[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]

Rule 5788

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*(d + e*x^2)^(
p + 1)*((a + b*ArcSinh[c*x])^n/(2*d*(p + 1))), x] + (Dist[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p + 1)*(a
+ b*ArcSinh[c*x])^n, x], x] + Dist[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Int[x*(1 + c^2*x^2
)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] &
& LtQ[p, -1] && NeQ[p, -3/2]

Rule 5789

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {x (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {(b c) \int \frac {x}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{4 d^3}+\frac {3 \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^2} \, dx}{4 d} \\ & = \frac {b}{12 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {x (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {3 x (a+b \text {arcsinh}(c x))}{8 d^3 \left (1+c^2 x^2\right )}-\frac {(3 b c) \int \frac {x}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{8 d^3}+\frac {3 \int \frac {a+b \text {arcsinh}(c x)}{d+c^2 d x^2} \, dx}{8 d^2} \\ & = \frac {b}{12 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {3 b}{8 c d^3 \sqrt {1+c^2 x^2}}+\frac {x (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {3 x (a+b \text {arcsinh}(c x))}{8 d^3 \left (1+c^2 x^2\right )}+\frac {3 \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{8 c d^3} \\ & = \frac {b}{12 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {3 b}{8 c d^3 \sqrt {1+c^2 x^2}}+\frac {x (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {3 x (a+b \text {arcsinh}(c x))}{8 d^3 \left (1+c^2 x^2\right )}+\frac {3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 c d^3}-\frac {(3 i b) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{8 c d^3}+\frac {(3 i b) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{8 c d^3} \\ & = \frac {b}{12 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {3 b}{8 c d^3 \sqrt {1+c^2 x^2}}+\frac {x (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {3 x (a+b \text {arcsinh}(c x))}{8 d^3 \left (1+c^2 x^2\right )}+\frac {3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 c d^3}-\frac {(3 i b) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{8 c d^3}+\frac {(3 i b) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{8 c d^3} \\ & = \frac {b}{12 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {3 b}{8 c d^3 \sqrt {1+c^2 x^2}}+\frac {x (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {3 x (a+b \text {arcsinh}(c x))}{8 d^3 \left (1+c^2 x^2\right )}+\frac {3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 c d^3}-\frac {3 i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{8 c d^3}+\frac {3 i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{8 c d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.92 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {15 a c x+9 a c^3 x^3+11 b \sqrt {1+c^2 x^2}+9 b c^2 x^2 \sqrt {1+c^2 x^2}+15 b c x \text {arcsinh}(c x)+9 b c^3 x^3 \text {arcsinh}(c x)+9 a \arctan (c x)+18 a c^2 x^2 \arctan (c x)+9 a c^4 x^4 \arctan (c x)+9 i b \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )+18 i b c^2 x^2 \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )+9 i b c^4 x^4 \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )-9 i b \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )-18 i b c^2 x^2 \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )-9 i b c^4 x^4 \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )-9 i b \left (1+c^2 x^2\right )^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+9 i b \left (1+c^2 x^2\right )^2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{24 c d^3 \left (1+c^2 x^2\right )^2} \]

[In]

Integrate[(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^3,x]

[Out]

(15*a*c*x + 9*a*c^3*x^3 + 11*b*Sqrt[1 + c^2*x^2] + 9*b*c^2*x^2*Sqrt[1 + c^2*x^2] + 15*b*c*x*ArcSinh[c*x] + 9*b
*c^3*x^3*ArcSinh[c*x] + 9*a*ArcTan[c*x] + 18*a*c^2*x^2*ArcTan[c*x] + 9*a*c^4*x^4*ArcTan[c*x] + (9*I)*b*ArcSinh
[c*x]*Log[1 - I*E^ArcSinh[c*x]] + (18*I)*b*c^2*x^2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + (9*I)*b*c^4*x^4*Ar
cSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] - (9*I)*b*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] - (18*I)*b*c^2*x^2*ArcSi
nh[c*x]*Log[1 + I*E^ArcSinh[c*x]] - (9*I)*b*c^4*x^4*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] - (9*I)*b*(1 + c^2*
x^2)^2*PolyLog[2, (-I)*E^ArcSinh[c*x]] + (9*I)*b*(1 + c^2*x^2)^2*PolyLog[2, I*E^ArcSinh[c*x]])/(24*c*d^3*(1 +
c^2*x^2)^2)

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.39

method	result	size
derivativedivides	\(\frac {\frac {a \left (\frac {c x}{4 \left (c^{2} x^{2}+1\right )^{2}}+\frac {3 c x}{8 \left (c^{2} x^{2}+1\right )}+\frac {3 \arctan \left (c x \right )}{8}\right )}{d^{3}}+\frac {b \left (\frac {c x \,\operatorname {arcsinh}\left (c x \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}+\frac {3 c x \,\operatorname {arcsinh}\left (c x \right )}{8 \left (c^{2} x^{2}+1\right )}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{8}+\frac {11}{24 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {3 c^{2} x^{2}}{8 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {3 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {3 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {3 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}+\frac {3 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}\right )}{d^{3}}}{c}\)	\(248\)
default	\(\frac {\frac {a \left (\frac {c x}{4 \left (c^{2} x^{2}+1\right )^{2}}+\frac {3 c x}{8 \left (c^{2} x^{2}+1\right )}+\frac {3 \arctan \left (c x \right )}{8}\right )}{d^{3}}+\frac {b \left (\frac {c x \,\operatorname {arcsinh}\left (c x \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}+\frac {3 c x \,\operatorname {arcsinh}\left (c x \right )}{8 \left (c^{2} x^{2}+1\right )}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{8}+\frac {11}{24 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {3 c^{2} x^{2}}{8 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {3 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {3 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {3 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}+\frac {3 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}\right )}{d^{3}}}{c}\)	\(248\)
parts	\(\frac {a \left (\frac {x}{4 \left (c^{2} x^{2}+1\right )^{2}}+\frac {3 x}{8 \left (c^{2} x^{2}+1\right )}+\frac {3 \arctan \left (c x \right )}{8 c}\right )}{d^{3}}+\frac {b \left (\frac {c x \,\operatorname {arcsinh}\left (c x \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}+\frac {3 c x \,\operatorname {arcsinh}\left (c x \right )}{8 \left (c^{2} x^{2}+1\right )}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{8}+\frac {11}{24 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {3 c^{2} x^{2}}{8 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {3 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {3 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {3 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}+\frac {3 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}\right )}{d^{3} c}\)	\(248\)

[In]

int((a+b*arcsinh(c*x))/(c^2*d*x^2+d)^3,x,method=_RETURNVERBOSE)

[Out]

1/c*(a/d^3*(1/4*c*x/(c^2*x^2+1)^2+3/8*c*x/(c^2*x^2+1)+3/8*arctan(c*x))+b/d^3*(1/4*c*x/(c^2*x^2+1)^2*arcsinh(c*
x)+3/8*c*x/(c^2*x^2+1)*arcsinh(c*x)+3/8*arcsinh(c*x)*arctan(c*x)+11/24/(c^2*x^2+1)^(3/2)+3/8*c^2*x^2/(c^2*x^2+
1)^(3/2)+3/8*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-3/8*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2
))-3/8*I*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+3/8*I*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))))

Fricas [F]

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

[In]

integrate((a+b*arcsinh(c*x))/(c^2*d*x^2+d)^3,x, algorithm="fricas")

[Out]

integral((b*arcsinh(c*x) + a)/(c^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x)

Sympy [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {\int \frac {a}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx}{d^{3}} \]

[In]

integrate((a+b*asinh(c*x))/(c**2*d*x**2+d)**3,x)

[Out]

(Integral(a/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1), x) + Integral(b*asinh(c*x)/(c**6*x**6 + 3*c**4*x**4 +
3*c**2*x**2 + 1), x))/d**3

Maxima [F]

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

[In]

integrate((a+b*arcsinh(c*x))/(c^2*d*x^2+d)^3,x, algorithm="maxima")

[Out]

1/8*a*((3*c^2*x^3 + 5*x)/(c^4*d^3*x^4 + 2*c^2*d^3*x^2 + d^3) + 3*arctan(c*x)/(c*d^3)) + b*integrate(log(c*x +
sqrt(c^2*x^2 + 1))/(c^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x)

Giac [F]

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

[In]

integrate((a+b*arcsinh(c*x))/(c^2*d*x^2+d)^3,x, algorithm="giac")

[Out]

integrate((b*arcsinh(c*x) + a)/(c^2*d*x^2 + d)^3, x)

Mupad [F(-1)]

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

[In]

int((a + b*asinh(c*x))/(d + c^2*d*x^2)^3,x)

[Out]

int((a + b*asinh(c*x))/(d + c^2*d*x^2)^3, x)