3.1.79 \(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \cosh ^{-1}(c x))}{x^{12}} \, dx\) [79]

Optimal. Leaf size=409 \[ -\frac {b c d \sqrt {d-c^2 d x^2}}{110 x^{10} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d \sqrt {d-c^2 d x^2}}{66 x^8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{1386 x^6 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^7 d \sqrt {d-c^2 d x^2}}{770 x^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b c^9 d \sqrt {d-c^2 d x^2}}{1155 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{11 d x^{11}}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{33 d x^9}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{231 d x^7}-\frac {16 c^6 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 d x^5}+\frac {16 b c^{11} d \sqrt {d-c^2 d x^2} \log (x)}{1155 \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

-1/11*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/d/x^11-2/33*c^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/d/x^9-8/
231*c^4*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/d/x^7-16/1155*c^6*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/d/x^
5-1/110*b*c*d*(-c^2*d*x^2+d)^(1/2)/x^10/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/66*b*c^3*d*(-c^2*d*x^2+d)^(1/2)/x^8/(c*x
-1)^(1/2)/(c*x+1)^(1/2)-1/1386*b*c^5*d*(-c^2*d*x^2+d)^(1/2)/x^6/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/770*b*c^7*d*(-c^
2*d*x^2+d)^(1/2)/x^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)-4/1155*b*c^9*d*(-c^2*d*x^2+d)^(1/2)/x^2/(c*x-1)^(1/2)/(c*x+1)
^(1/2)+16/1155*b*c^11*d*ln(x)*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)

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Rubi [A]
time = 0.19, antiderivative size = 409, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {277, 270, 5922, 12, 1813, 1634} \begin {gather*} -\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{11 d x^{11}}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{33 d x^9}-\frac {16 c^6 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{231 d x^7}-\frac {b c d \sqrt {d-c^2 d x^2}}{110 x^{10} \sqrt {c x-1} \sqrt {c x+1}}+\frac {16 b c^{11} d \log (x) \sqrt {d-c^2 d x^2}}{1155 \sqrt {c x-1} \sqrt {c x+1}}-\frac {4 b c^9 d \sqrt {d-c^2 d x^2}}{1155 x^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^7 d \sqrt {d-c^2 d x^2}}{770 x^4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{1386 x^6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d \sqrt {d-c^2 d x^2}}{66 x^8 \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^12,x]

[Out]

-1/110*(b*c*d*Sqrt[d - c^2*d*x^2])/(x^10*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*Sqrt[d - c^2*d*x^2])/(66*x^8
*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d*Sqrt[d - c^2*d*x^2])/(1386*x^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^
7*d*Sqrt[d - c^2*d*x^2])/(770*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (4*b*c^9*d*Sqrt[d - c^2*d*x^2])/(1155*x^2*Sq
rt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(11*d*x^11) - (2*c^2*(d - c^2*d*x^2
)^(5/2)*(a + b*ArcCosh[c*x]))/(33*d*x^9) - (8*c^4*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(231*d*x^7) - (1
6*c^6*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(1155*d*x^5) + (16*b*c^11*d*Sqrt[d - c^2*d*x^2]*Log[x])/(115
5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 12

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 1634

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rule 1813

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 5922

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x
^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 +
c*x])], Int[SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0
] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])

Rubi steps

\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^{12}} \, dx &=-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^{12}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{33 x^9}-\frac {c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^7}-\frac {2 c^6 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{385 x^5}-\frac {8 c^8 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x^3}-\frac {16 c^{10} d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{11 x^{11}}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^2 \left (105+70 c^2 x^2+40 c^4 x^4+16 c^6 x^6\right )}{1155 x^{11}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{33 x^9}-\frac {c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^7}-\frac {2 c^6 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{385 x^5}-\frac {8 c^8 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x^3}-\frac {16 c^{10} d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{11 x^{11}}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^2 \left (105+70 c^2 x^2+40 c^4 x^4+16 c^6 x^6\right )}{x^{11}} \, dx}{1155 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{33 x^9}-\frac {c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^7}-\frac {2 c^6 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{385 x^5}-\frac {8 c^8 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x^3}-\frac {16 c^{10} d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{11 x^{11}}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\left (1-c^2 x\right )^2 \left (105+70 c^2 x+40 c^4 x^2+16 c^6 x^3\right )}{x^6} \, dx,x,x^2\right )}{2310 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{33 x^9}-\frac {c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^7}-\frac {2 c^6 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{385 x^5}-\frac {8 c^8 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x^3}-\frac {16 c^{10} d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{11 x^{11}}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {105}{x^6}-\frac {140 c^2}{x^5}+\frac {5 c^4}{x^4}+\frac {6 c^6}{x^3}+\frac {8 c^8}{x^2}+\frac {16 c^{10}}{x}\right ) \, dx,x,x^2\right )}{2310 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d \sqrt {d-c^2 d x^2}}{110 x^{10} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d \sqrt {d-c^2 d x^2}}{66 x^8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{1386 x^6 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^7 d \sqrt {d-c^2 d x^2}}{770 x^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b c^9 d \sqrt {d-c^2 d x^2}}{1155 x^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{33 x^9}-\frac {c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^7}-\frac {2 c^6 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{385 x^5}-\frac {8 c^8 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x^3}-\frac {16 c^{10} d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{11 x^{11}}+\frac {16 b c^{11} d \sqrt {d-c^2 d x^2} \log (x)}{1155 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]
time = 0.29, size = 170, normalized size = 0.42 \begin {gather*} -\frac {d \sqrt {d-c^2 d x^2} \left (630 (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+12 c^2 x^2 (-1+c x)^{5/2} (1+c x)^{5/2} \left (35+20 c^2 x^2+8 c^4 x^4\right ) \left (a+b \cosh ^{-1}(c x)\right )+b c x \left (63-105 c^2 x^2+5 c^4 x^4+9 c^6 x^6+24 c^8 x^8-96 c^{10} x^{10} \log (x)\right )\right )}{6930 x^{11} \sqrt {-1+c x} \sqrt {1+c x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^12,x]

[Out]

-1/6930*(d*Sqrt[d - c^2*d*x^2]*(630*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x]) + 12*c^2*x^2*(-1 + c
*x)^(5/2)*(1 + c*x)^(5/2)*(35 + 20*c^2*x^2 + 8*c^4*x^4)*(a + b*ArcCosh[c*x]) + b*c*x*(63 - 105*c^2*x^2 + 5*c^4
*x^4 + 9*c^6*x^6 + 24*c^8*x^8 - 96*c^10*x^10*Log[x])))/(x^11*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(5522\) vs. \(2(345)=690\).
time = 9.74, size = 5523, normalized size = 13.50

method	result	size
default	\(\text {Expression too large to display}\)	\(5523\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^12,x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [A]
time = 0.53, size = 287, normalized size = 0.70 \begin {gather*} \frac {1}{6930} \, {\left (96 \, c^{10} \sqrt {-d} d \log \left (x\right ) - \frac {24 \, c^{8} \sqrt {-d} d x^{8} + 9 \, c^{6} \sqrt {-d} d x^{6} + 5 \, c^{4} \sqrt {-d} d x^{4} - 105 \, c^{2} \sqrt {-d} d x^{2} + 63 \, \sqrt {-d} d}{x^{10}}\right )} b c - \frac {1}{1155} \, {\left (\frac {16 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{6}}{d x^{5}} + \frac {40 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{4}}{d x^{7}} + \frac {70 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{9}} + \frac {105 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{11}}\right )} b \operatorname {arcosh}\left (c x\right ) - \frac {1}{1155} \, {\left (\frac {16 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{6}}{d x^{5}} + \frac {40 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{4}}{d x^{7}} + \frac {70 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{9}} + \frac {105 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{11}}\right )} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^12,x, algorithm="maxima")

[Out]

1/6930*(96*c^10*sqrt(-d)*d*log(x) - (24*c^8*sqrt(-d)*d*x^8 + 9*c^6*sqrt(-d)*d*x^6 + 5*c^4*sqrt(-d)*d*x^4 - 105
*c^2*sqrt(-d)*d*x^2 + 63*sqrt(-d)*d)/x^10)*b*c - 1/1155*(16*(-c^2*d*x^2 + d)^(5/2)*c^6/(d*x^5) + 40*(-c^2*d*x^
2 + d)^(5/2)*c^4/(d*x^7) + 70*(-c^2*d*x^2 + d)^(5/2)*c^2/(d*x^9) + 105*(-c^2*d*x^2 + d)^(5/2)/(d*x^11))*b*arcc
osh(c*x) - 1/1155*(16*(-c^2*d*x^2 + d)^(5/2)*c^6/(d*x^5) + 40*(-c^2*d*x^2 + d)^(5/2)*c^4/(d*x^7) + 70*(-c^2*d*
x^2 + d)^(5/2)*c^2/(d*x^9) + 105*(-c^2*d*x^2 + d)^(5/2)/(d*x^11))*a

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Fricas [A]
time = 0.43, size = 792, normalized size = 1.94 \begin {gather*} \left [-\frac {6 \, {\left (16 \, b c^{12} d x^{12} - 8 \, b c^{10} d x^{10} - 2 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 145 \, b c^{4} d x^{4} + 245 \, b c^{2} d x^{2} - 105 \, b d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 48 \, {\left (b c^{13} d x^{13} - b c^{11} d x^{11}\right )} \sqrt {-d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} - \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{4} - 1\right )} \sqrt {-d} - d}{c^{2} x^{4} - x^{2}}\right ) + {\left (24 \, b c^{9} d x^{9} + 9 \, b c^{7} d x^{7} - {\left (24 \, b c^{9} + 9 \, b c^{7} + 5 \, b c^{5} - 105 \, b c^{3} + 63 \, b c\right )} d x^{11} + 5 \, b c^{5} d x^{5} - 105 \, b c^{3} d x^{3} + 63 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 6 \, {\left (16 \, a c^{12} d x^{12} - 8 \, a c^{10} d x^{10} - 2 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 145 \, a c^{4} d x^{4} + 245 \, a c^{2} d x^{2} - 105 \, a d\right )} \sqrt {-c^{2} d x^{2} + d}}{6930 \, {\left (c^{2} x^{13} - x^{11}\right )}}, \frac {96 \, {\left (b c^{13} d x^{13} - b c^{11} d x^{11}\right )} \sqrt {d} \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{2} + 1\right )} \sqrt {d}}{c^{2} d x^{4} - {\left (c^{2} + 1\right )} d x^{2} + d}\right ) - 6 \, {\left (16 \, b c^{12} d x^{12} - 8 \, b c^{10} d x^{10} - 2 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 145 \, b c^{4} d x^{4} + 245 \, b c^{2} d x^{2} - 105 \, b d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (24 \, b c^{9} d x^{9} + 9 \, b c^{7} d x^{7} - {\left (24 \, b c^{9} + 9 \, b c^{7} + 5 \, b c^{5} - 105 \, b c^{3} + 63 \, b c\right )} d x^{11} + 5 \, b c^{5} d x^{5} - 105 \, b c^{3} d x^{3} + 63 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 6 \, {\left (16 \, a c^{12} d x^{12} - 8 \, a c^{10} d x^{10} - 2 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 145 \, a c^{4} d x^{4} + 245 \, a c^{2} d x^{2} - 105 \, a d\right )} \sqrt {-c^{2} d x^{2} + d}}{6930 \, {\left (c^{2} x^{13} - x^{11}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^12,x, algorithm="fricas")

[Out]

[-1/6930*(6*(16*b*c^12*d*x^12 - 8*b*c^10*d*x^10 - 2*b*c^8*d*x^8 - b*c^6*d*x^6 - 145*b*c^4*d*x^4 + 245*b*c^2*d*
x^2 - 105*b*d)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - 48*(b*c^13*d*x^13 - b*c^11*d*x^11)*sqrt(-d)
*log((c^2*d*x^6 + c^2*d*x^2 - d*x^4 - sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*(x^4 - 1)*sqrt(-d) - d)/(c^2*x^4
- x^2)) + (24*b*c^9*d*x^9 + 9*b*c^7*d*x^7 - (24*b*c^9 + 9*b*c^7 + 5*b*c^5 - 105*b*c^3 + 63*b*c)*d*x^11 + 5*b*c
^5*d*x^5 - 105*b*c^3*d*x^3 + 63*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 6*(16*a*c^12*d*x^12 - 8*a*c^
10*d*x^10 - 2*a*c^8*d*x^8 - a*c^6*d*x^6 - 145*a*c^4*d*x^4 + 245*a*c^2*d*x^2 - 105*a*d)*sqrt(-c^2*d*x^2 + d))/(
c^2*x^13 - x^11), 1/6930*(96*(b*c^13*d*x^13 - b*c^11*d*x^11)*sqrt(d)*arctan(sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2
- 1)*(x^2 + 1)*sqrt(d)/(c^2*d*x^4 - (c^2 + 1)*d*x^2 + d)) - 6*(16*b*c^12*d*x^12 - 8*b*c^10*d*x^10 - 2*b*c^8*d*
x^8 - b*c^6*d*x^6 - 145*b*c^4*d*x^4 + 245*b*c^2*d*x^2 - 105*b*d)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 -
1)) - (24*b*c^9*d*x^9 + 9*b*c^7*d*x^7 - (24*b*c^9 + 9*b*c^7 + 5*b*c^5 - 105*b*c^3 + 63*b*c)*d*x^11 + 5*b*c^5*
d*x^5 - 105*b*c^3*d*x^3 + 63*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 6*(16*a*c^12*d*x^12 - 8*a*c^10*
d*x^10 - 2*a*c^8*d*x^8 - a*c^6*d*x^6 - 145*a*c^4*d*x^4 + 245*a*c^2*d*x^2 - 105*a*d)*sqrt(-c^2*d*x^2 + d))/(c^2
*x^13 - x^11)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*d*x**2+d)**(3/2)*(a+b*acosh(c*x))/x**12,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3877 deep

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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)^(3/2)*(a+b*arccosh(c*x))/x^12,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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^{12}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2))/x^12,x)

[Out]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2))/x^12, x)

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