3.130 \(\int \frac{a+b \cosh ^{-1}(c x)}{x (d-c^2 d x^2)^{5/2}} \, dx\)

Optimal. Leaf size=317 \[ -\frac{i b \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{i b \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{a+b \cosh ^{-1}(c x)}{d^2 \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{a+b \cosh ^{-1}(c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{b c x \sqrt{c x-1} \sqrt{c x+1}}{6 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{7 b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}(c x)}{6 d^2 \sqrt{d-c^2 d x^2}} \]

[Out]

(b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (a + b*ArcCosh[c*x])/(3*d*(d
- c^2*d*x^2)^(3/2)) + (a + b*ArcCosh[c*x])/(d^2*Sqrt[d - c^2*d*x^2]) + (2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*
ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) + (7*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTanh[c*
x])/(6*d^2*Sqrt[d - c^2*d*x^2]) - (I*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(d^2*Sqrt
[d - c^2*d*x^2]) + (I*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, I*E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2])

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Rubi [A]  time = 0.831696, antiderivative size = 332, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {5798, 5756, 5761, 4180, 2279, 2391, 207, 199} \[ -\frac{i b \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{i b \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{a+b \cosh ^{-1}(c x)}{3 d^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2}}+\frac{a+b \cosh ^{-1}(c x)}{d^2 \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{b c x \sqrt{c x-1} \sqrt{c x+1}}{6 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{7 b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}(c x)}{6 d^2 \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c*x])/(x*(d - c^2*d*x^2)^(5/2)),x]

[Out]

(b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (a + b*ArcCosh[c*x])/(d^2*Sqr
t[d - c^2*d*x^2]) + (a + b*ArcCosh[c*x])/(3*d^2*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]) + (2*Sqrt[-1 + c*x]*S
qrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) + (7*b*Sqrt[-1 + c*x]*Sqrt
[1 + c*x]*ArcTanh[c*x])/(6*d^2*Sqrt[d - c^2*d*x^2]) - (I*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, (-I)*E^ArcC
osh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) + (I*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, I*E^ArcCosh[c*x]])/(d^2*Sq
rt[d - c^2*d*x^2])

Rule 5798

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

Rule 5756

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> -Simp[((f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*
d1*d2*f*(p + 1)), x] + (Dist[(m + 2*p + 3)/(2*d1*d2*(p + 1)), Int[(f*x)^m*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p +
1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[(b*c*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^Fra
cPart[p])/(2*f*(p + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p + 1/
2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] &&
EqQ[e2 + c*d2, 0] && GtQ[n, 0] && LtQ[p, -1] &&  !GtQ[m, 1] && (IntegerQ[m] || EqQ[n, 1]) && IntegerQ[p + 1/2]

Rule 5761

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]
), x_Symbol] :> Dist[1/(c^(m + 1)*Sqrt[-(d1*d2)]), Subst[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]], x] /
; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[n, 0] && GtQ[d1, 0] &&
LtQ[d2, 0] && IntegerQ[m]

Rule 4180

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 2279

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 2391

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 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 199

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

Rubi steps

\begin{align*} \int \frac{a+b \cosh ^{-1}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{a+b \cosh ^{-1}(c x)}{3 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{\left (-1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b c x \sqrt{-1+c x} \sqrt{1+c x}}{6 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{a+b \cosh ^{-1}(c x)}{d^2 \sqrt{d-c^2 d x^2}}+\frac{a+b \cosh ^{-1}(c x)}{3 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{-1+c^2 x^2} \, dx}{6 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{-1+c^2 x^2} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b c x \sqrt{-1+c x} \sqrt{1+c x}}{6 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{a+b \cosh ^{-1}(c x)}{d^2 \sqrt{d-c^2 d x^2}}+\frac{a+b \cosh ^{-1}(c x)}{3 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{7 b \sqrt{-1+c x} \sqrt{1+c x} \tanh ^{-1}(c x)}{6 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \text{sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b c x \sqrt{-1+c x} \sqrt{1+c x}}{6 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{a+b \cosh ^{-1}(c x)}{d^2 \sqrt{d-c^2 d x^2}}+\frac{a+b \cosh ^{-1}(c x)}{3 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{7 b \sqrt{-1+c x} \sqrt{1+c x} \tanh ^{-1}(c x)}{6 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (i b \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (i b \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b c x \sqrt{-1+c x} \sqrt{1+c x}}{6 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{a+b \cosh ^{-1}(c x)}{d^2 \sqrt{d-c^2 d x^2}}+\frac{a+b \cosh ^{-1}(c x)}{3 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{7 b \sqrt{-1+c x} \sqrt{1+c x} \tanh ^{-1}(c x)}{6 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (i b \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (i b \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b c x \sqrt{-1+c x} \sqrt{1+c x}}{6 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{a+b \cosh ^{-1}(c x)}{d^2 \sqrt{d-c^2 d x^2}}+\frac{a+b \cosh ^{-1}(c x)}{3 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{7 b \sqrt{-1+c x} \sqrt{1+c x} \tanh ^{-1}(c x)}{6 d^2 \sqrt{d-c^2 d x^2}}-\frac{i b \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{i b \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 7.0114, size = 377, normalized size = 1.19 \[ \frac{b \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (-24 i \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )+24 i \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right )-24 i \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )+24 i \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )-\frac{8 \cosh ^{-1}(c x) \sinh ^4\left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\left (\frac{c x-1}{c x+1}\right )^{3/2} (c x+1)^3}-14 \cosh ^{-1}(c x) \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )+14 \cosh ^{-1}(c x) \coth \left (\frac{1}{2} \cosh ^{-1}(c x)\right )-\frac{1}{2} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \text{csch}^4\left (\frac{1}{2} \cosh ^{-1}(c x)\right )-\text{csch}^2\left (\frac{1}{2} \cosh ^{-1}(c x)\right )-\text{sech}^2\left (\frac{1}{2} \cosh ^{-1}(c x)\right )-28 \log \left (\tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )\right )}{24 d^2 \sqrt{-d (c x-1) (c x+1)}}+\sqrt{-d \left (c^2 x^2-1\right )} \left (\frac{a}{3 d^3 \left (c^2 x^2-1\right )^2}-\frac{a}{d^3 \left (c^2 x^2-1\right )}\right )-\frac{a \log \left (\sqrt{d} \sqrt{-d \left (c^2 x^2-1\right )}+d\right )}{d^{5/2}}+\frac{a \log (x)}{d^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCosh[c*x])/(x*(d - c^2*d*x^2)^(5/2)),x]

[Out]

Sqrt[-(d*(-1 + c^2*x^2))]*(a/(3*d^3*(-1 + c^2*x^2)^2) - a/(d^3*(-1 + c^2*x^2))) + (a*Log[x])/d^(5/2) - (a*Log[
d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/d^(5/2) + (b*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(14*ArcCosh[c*x]*Cot
h[ArcCosh[c*x]/2] - Csch[ArcCosh[c*x]/2]^2 - (Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Csch[ArcCosh[c
*x]/2]^4)/2 - (24*I)*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] + (24*I)*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] -
28*Log[Tanh[ArcCosh[c*x]/2]] - (24*I)*PolyLog[2, (-I)/E^ArcCosh[c*x]] + (24*I)*PolyLog[2, I/E^ArcCosh[c*x]] -
Sech[ArcCosh[c*x]/2]^2 - (8*ArcCosh[c*x]*Sinh[ArcCosh[c*x]/2]^4)/(((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3) -
14*ArcCosh[c*x]*Tanh[ArcCosh[c*x]/2]))/(24*d^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))])

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Maple [A]  time = 0.243, size = 619, normalized size = 2. \begin{align*}{\frac{a}{3\,d} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}+{\frac{a}{{d}^{2}}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}}-{a\ln \left ({\frac{1}{x} \left ( 2\,d+2\,\sqrt{d}\sqrt{-{c}^{2}d{x}^{2}+d} \right ) } \right ){d}^{-{\frac{5}{2}}}}-{\frac{b{x}^{2}{\rm arccosh} \left (cx\right ){c}^{2}}{{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) ^{2}}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{xbc}{6\,{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) ^{2}}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}}+{\frac{4\,b{\rm arccosh} \left (cx\right )}{3\,{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) ^{2}}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{7\,b}{6\,{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}\ln \left ( cx+\sqrt{cx-1}\sqrt{cx+1}-1 \right ) }-{\frac{7\,b}{6\,{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}\ln \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{ib{\rm arccosh} \left (cx\right )}{{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}\ln \left ( 1+i \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) \right ) }+{\frac{ib}{{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}{\it dilog} \left ( 1+i \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) \right ) }-{\frac{ib}{{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}{\it dilog} \left ( 1-i \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) \right ) }-{\frac{ib{\rm arccosh} \left (cx\right )}{{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}\ln \left ( 1-i \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(c*x))/x/(-c^2*d*x^2+d)^(5/2),x)

[Out]

1/3*a/d/(-c^2*d*x^2+d)^(3/2)+a/d^2/(-c^2*d*x^2+d)^(1/2)-a/d^(5/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)-b
*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)^2*arccosh(c*x)*x^2*c^2+1/6*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)^2*
(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+4/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)^2*arccosh(c*x)+7/6*b*(-d*(c^2*x^2
-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*ln(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)-1)-7/6*b*(-d*(c^2*x^
2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+I*b*(-d*(c^2*x^2
-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*arccosh(c*x)*ln(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))+
I*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1
/2)))-I*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*dilog(1-I*(c*x+(c*x-1)^(1/2)*(c*x
+1)^(1/2)))-I*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*arccosh(c*x)*ln(1-I*(c*x+(c
*x-1)^(1/2)*(c*x+1)^(1/2)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} d x^{2} + d}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}{c^{6} d^{3} x^{7} - 3 \, c^{4} d^{3} x^{5} + 3 \, c^{2} d^{3} x^{3} - d^{3} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)/(c^6*d^3*x^7 - 3*c^4*d^3*x^5 + 3*c^2*d^3*x^3 - d^3*x), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(c*x))/x/(-c**2*d*x**2+d)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)/((-c^2*d*x^2 + d)^(5/2)*x), x)