3.1.94 \(\int x^3 (d+e x^2)^2 (a+b \text {csch}^{-1}(c x)) \, dx\) [94]

Optimal. Leaf size=250 \[ -\frac {b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) x \sqrt {-1-c^2 x^2}}{24 c^7 \sqrt {-c^2 x^2}}-\frac {b \left (6 c^4 d^2-16 c^2 d e+9 e^2\right ) x \left (-1-c^2 x^2\right )^{3/2}}{72 c^7 \sqrt {-c^2 x^2}}+\frac {b \left (8 c^2 d-9 e\right ) e x \left (-1-c^2 x^2\right )^{5/2}}{120 c^7 \sqrt {-c^2 x^2}}-\frac {b e^2 x \left (-1-c^2 x^2\right )^{7/2}}{56 c^7 \sqrt {-c^2 x^2}}+\frac {1}{4} d^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {csch}^{-1}(c x)\right ) \]

[Out]

1/4*d^2*x^4*(a+b*arccsch(c*x))+1/3*d*e*x^6*(a+b*arccsch(c*x))+1/8*e^2*x^8*(a+b*arccsch(c*x))-1/72*b*(6*c^4*d^2
-16*c^2*d*e+9*e^2)*x*(-c^2*x^2-1)^(3/2)/c^7/(-c^2*x^2)^(1/2)+1/120*b*(8*c^2*d-9*e)*e*x*(-c^2*x^2-1)^(5/2)/c^7/
(-c^2*x^2)^(1/2)-1/56*b*e^2*x*(-c^2*x^2-1)^(7/2)/c^7/(-c^2*x^2)^(1/2)-1/24*b*(6*c^4*d^2-8*c^2*d*e+3*e^2)*x*(-c
^2*x^2-1)^(1/2)/c^7/(-c^2*x^2)^(1/2)

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Rubi [A]
time = 0.17, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {272, 45, 6437, 12, 1265, 785} \begin {gather*} \frac {1}{4} d^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b e x \left (-c^2 x^2-1\right )^{5/2} \left (8 c^2 d-9 e\right )}{120 c^7 \sqrt {-c^2 x^2}}-\frac {b e^2 x \left (-c^2 x^2-1\right )^{7/2}}{56 c^7 \sqrt {-c^2 x^2}}-\frac {b x \left (-c^2 x^2-1\right )^{3/2} \left (6 c^4 d^2-16 c^2 d e+9 e^2\right )}{72 c^7 \sqrt {-c^2 x^2}}-\frac {b x \sqrt {-c^2 x^2-1} \left (6 c^4 d^2-8 c^2 d e+3 e^2\right )}{24 c^7 \sqrt {-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/24*(b*(6*c^4*d^2 - 8*c^2*d*e + 3*e^2)*x*Sqrt[-1 - c^2*x^2])/(c^7*Sqrt[-(c^2*x^2)]) - (b*(6*c^4*d^2 - 16*c^2
*d*e + 9*e^2)*x*(-1 - c^2*x^2)^(3/2))/(72*c^7*Sqrt[-(c^2*x^2)]) + (b*(8*c^2*d - 9*e)*e*x*(-1 - c^2*x^2)^(5/2))
/(120*c^7*Sqrt[-(c^2*x^2)]) - (b*e^2*x*(-1 - c^2*x^2)^(7/2))/(56*c^7*Sqrt[-(c^2*x^2)]) + (d^2*x^4*(a + b*ArcCs
ch[c*x]))/4 + (d*e*x^6*(a + b*ArcCsch[c*x]))/3 + (e^2*x^8*(a + b*ArcCsch[c*x]))/8

Rule 12

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

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

Rule 785

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rule 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 6437

Int[((a_.) + ArcCsch[(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*ArcCsch[c*x], u, x] - Dist[b*c*(x/Sqrt[(-c^2)*x^2]), Int[Simp
lifyIntegrand[u/(x*Sqrt[-1 - c^2*x^2]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] &&
!(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0]))
|| (ILtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rubi steps

\begin {align*} \int x^3 \left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx &=\frac {1}{4} d^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )}{24 \sqrt {-1-c^2 x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=\frac {1}{4} d^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )}{\sqrt {-1-c^2 x^2}} \, dx}{24 \sqrt {-c^2 x^2}}\\ &=\frac {1}{4} d^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \text {Subst}\left (\int \frac {x \left (6 d^2+8 d e x+3 e^2 x^2\right )}{\sqrt {-1-c^2 x}} \, dx,x,x^2\right )}{48 \sqrt {-c^2 x^2}}\\ &=\frac {1}{4} d^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \text {Subst}\left (\int \left (\frac {-6 c^4 d^2+8 c^2 d e-3 e^2}{c^6 \sqrt {-1-c^2 x}}+\frac {\left (-6 c^4 d^2+16 c^2 d e-9 e^2\right ) \sqrt {-1-c^2 x}}{c^6}+\frac {\left (8 c^2 d-9 e\right ) e \left (-1-c^2 x\right )^{3/2}}{c^6}-\frac {3 e^2 \left (-1-c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right )}{48 \sqrt {-c^2 x^2}}\\ &=-\frac {b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) x \sqrt {-1-c^2 x^2}}{24 c^7 \sqrt {-c^2 x^2}}-\frac {b \left (6 c^4 d^2-16 c^2 d e+9 e^2\right ) x \left (-1-c^2 x^2\right )^{3/2}}{72 c^7 \sqrt {-c^2 x^2}}+\frac {b \left (8 c^2 d-9 e\right ) e x \left (-1-c^2 x^2\right )^{5/2}}{120 c^7 \sqrt {-c^2 x^2}}-\frac {b e^2 x \left (-1-c^2 x^2\right )^{7/2}}{56 c^7 \sqrt {-c^2 x^2}}+\frac {1}{4} d^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {csch}^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]
time = 0.23, size = 159, normalized size = 0.64 \begin {gather*} \frac {x \left (105 a x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )+\frac {b \sqrt {1+\frac {1}{c^2 x^2}} \left (-144 e^2+8 c^2 e \left (56 d+9 e x^2\right )-2 c^4 \left (210 d^2+112 d e x^2+27 e^2 x^4\right )+3 c^6 \left (70 d^2 x^2+56 d e x^4+15 e^2 x^6\right )\right )}{c^7}+105 b x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right ) \text {csch}^{-1}(c x)\right )}{2520} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(105*a*x^3*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4) + (b*Sqrt[1 + 1/(c^2*x^2)]*(-144*e^2 + 8*c^2*e*(56*d + 9*e*x^2)
- 2*c^4*(210*d^2 + 112*d*e*x^2 + 27*e^2*x^4) + 3*c^6*(70*d^2*x^2 + 56*d*e*x^4 + 15*e^2*x^6)))/c^7 + 105*b*x^3*
(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4)*ArcCsch[c*x]))/2520

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Maple [A]
time = 0.57, size = 377, normalized size = 1.51

method	result	size
derivativedivides	\(\frac {-\frac {a \left (\frac {c^{2} d \left (c^{2} e \,x^{2}+c^{2} d \right )^{3}}{3}-\frac {\left (c^{2} e \,x^{2}+c^{2} d \right )^{4}}{4}\right )}{2 c^{4} e^{2}}+\frac {b \left (-\frac {\mathrm {arccsch}\left (c x \right ) c^{8} d^{4}}{24 e^{2}}+\frac {\mathrm {arccsch}\left (c x \right ) c^{8} d^{2} x^{4}}{4}+\frac {e \,\mathrm {arccsch}\left (c x \right ) c^{8} d \,x^{6}}{3}+\frac {e^{2} \mathrm {arccsch}\left (c x \right ) c^{8} x^{8}}{8}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (105 c^{8} d^{4} \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )+210 c^{6} d^{2} e^{2} x^{2} \sqrt {c^{2} x^{2}+1}+168 c^{6} d \,e^{3} x^{4} \sqrt {c^{2} x^{2}+1}+45 e^{4} c^{6} x^{6} \sqrt {c^{2} x^{2}+1}-420 c^{4} d^{2} e^{2} \sqrt {c^{2} x^{2}+1}-224 c^{4} d \,e^{3} x^{2} \sqrt {c^{2} x^{2}+1}-54 e^{4} c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+448 c^{2} d \,e^{3} \sqrt {c^{2} x^{2}+1}+72 e^{4} c^{2} x^{2} \sqrt {c^{2} x^{2}+1}-144 e^{4} \sqrt {c^{2} x^{2}+1}\right )}{2520 e^{2} \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}}}{c^{4}}\)	\(377\)
default	\(\frac {-\frac {a \left (\frac {c^{2} d \left (c^{2} e \,x^{2}+c^{2} d \right )^{3}}{3}-\frac {\left (c^{2} e \,x^{2}+c^{2} d \right )^{4}}{4}\right )}{2 c^{4} e^{2}}+\frac {b \left (-\frac {\mathrm {arccsch}\left (c x \right ) c^{8} d^{4}}{24 e^{2}}+\frac {\mathrm {arccsch}\left (c x \right ) c^{8} d^{2} x^{4}}{4}+\frac {e \,\mathrm {arccsch}\left (c x \right ) c^{8} d \,x^{6}}{3}+\frac {e^{2} \mathrm {arccsch}\left (c x \right ) c^{8} x^{8}}{8}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (105 c^{8} d^{4} \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )+210 c^{6} d^{2} e^{2} x^{2} \sqrt {c^{2} x^{2}+1}+168 c^{6} d \,e^{3} x^{4} \sqrt {c^{2} x^{2}+1}+45 e^{4} c^{6} x^{6} \sqrt {c^{2} x^{2}+1}-420 c^{4} d^{2} e^{2} \sqrt {c^{2} x^{2}+1}-224 c^{4} d \,e^{3} x^{2} \sqrt {c^{2} x^{2}+1}-54 e^{4} c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+448 c^{2} d \,e^{3} \sqrt {c^{2} x^{2}+1}+72 e^{4} c^{2} x^{2} \sqrt {c^{2} x^{2}+1}-144 e^{4} \sqrt {c^{2} x^{2}+1}\right )}{2520 e^{2} \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}}}{c^{4}}\)	\(377\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^4*(-1/2*a/c^4/e^2*(1/3*c^2*d*(c^2*e*x^2+c^2*d)^3-1/4*(c^2*e*x^2+c^2*d)^4)+b/c^4*(-1/24/e^2*arccsch(c*x)*c^
8*d^4+1/4*arccsch(c*x)*c^8*d^2*x^4+1/3*e*arccsch(c*x)*c^8*d*x^6+1/8*e^2*arccsch(c*x)*c^8*x^8+1/2520/e^2*(c^2*x
^2+1)^(1/2)*(105*c^8*d^4*arctanh(1/(c^2*x^2+1)^(1/2))+210*c^6*d^2*e^2*x^2*(c^2*x^2+1)^(1/2)+168*c^6*d*e^3*x^4*
(c^2*x^2+1)^(1/2)+45*e^4*c^6*x^6*(c^2*x^2+1)^(1/2)-420*c^4*d^2*e^2*(c^2*x^2+1)^(1/2)-224*c^4*d*e^3*x^2*(c^2*x^
2+1)^(1/2)-54*e^4*c^4*x^4*(c^2*x^2+1)^(1/2)+448*c^2*d*e^3*(c^2*x^2+1)^(1/2)+72*e^4*c^2*x^2*(c^2*x^2+1)^(1/2)-1
44*e^4*(c^2*x^2+1)^(1/2))/((c^2*x^2+1)/c^2/x^2)^(1/2)/c/x))

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Maxima [A]
time = 0.27, size = 244, normalized size = 0.98 \begin {gather*} \frac {1}{8} \, a x^{8} e^{2} + \frac {1}{3} \, a d x^{6} e + \frac {1}{4} \, a d^{2} x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arcsch}\left (c x\right ) + \frac {c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b d^{2} + \frac {1}{45} \, {\left (15 \, x^{6} \operatorname {arcsch}\left (c x\right ) + \frac {3 \, c^{4} x^{5} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 10 \, c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b d e + \frac {1}{280} \, {\left (35 \, x^{8} \operatorname {arcsch}\left (c x\right ) + \frac {5 \, c^{6} x^{7} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {7}{2}} - 21 \, c^{4} x^{5} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 35 \, c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 35 \, x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{7}}\right )} b e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a*x^8*e^2 + 1/3*a*d*x^6*e + 1/4*a*d^2*x^4 + 1/12*(3*x^4*arccsch(c*x) + (c^2*x^3*(1/(c^2*x^2) + 1)^(3/2) -
3*x*sqrt(1/(c^2*x^2) + 1))/c^3)*b*d^2 + 1/45*(15*x^6*arccsch(c*x) + (3*c^4*x^5*(1/(c^2*x^2) + 1)^(5/2) - 10*c^
2*x^3*(1/(c^2*x^2) + 1)^(3/2) + 15*x*sqrt(1/(c^2*x^2) + 1))/c^5)*b*d*e + 1/280*(35*x^8*arccsch(c*x) + (5*c^6*x
^7*(1/(c^2*x^2) + 1)^(7/2) - 21*c^4*x^5*(1/(c^2*x^2) + 1)^(5/2) + 35*c^2*x^3*(1/(c^2*x^2) + 1)^(3/2) - 35*x*sq
rt(1/(c^2*x^2) + 1))/c^7)*b*e^2

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Fricas [A]
time = 0.38, size = 412, normalized size = 1.65 \begin {gather*} \frac {315 \, a c^{7} x^{8} \cosh \left (1\right )^{2} + 315 \, a c^{7} x^{8} \sinh \left (1\right )^{2} + 840 \, a c^{7} d x^{6} \cosh \left (1\right ) + 630 \, a c^{7} d^{2} x^{4} + 105 \, {\left (3 \, b c^{7} x^{8} \cosh \left (1\right )^{2} + 3 \, b c^{7} x^{8} \sinh \left (1\right )^{2} + 8 \, b c^{7} d x^{6} \cosh \left (1\right ) + 6 \, b c^{7} d^{2} x^{4} + 2 \, {\left (3 \, b c^{7} x^{8} \cosh \left (1\right ) + 4 \, b c^{7} d x^{6}\right )} \sinh \left (1\right )\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 210 \, {\left (3 \, a c^{7} x^{8} \cosh \left (1\right ) + 4 \, a c^{7} d x^{6}\right )} \sinh \left (1\right ) + {\left (210 \, b c^{6} d^{2} x^{3} - 420 \, b c^{4} d^{2} x + 9 \, {\left (5 \, b c^{6} x^{7} - 6 \, b c^{4} x^{5} + 8 \, b c^{2} x^{3} - 16 \, b x\right )} \cosh \left (1\right )^{2} + 9 \, {\left (5 \, b c^{6} x^{7} - 6 \, b c^{4} x^{5} + 8 \, b c^{2} x^{3} - 16 \, b x\right )} \sinh \left (1\right )^{2} + 56 \, {\left (3 \, b c^{6} d x^{5} - 4 \, b c^{4} d x^{3} + 8 \, b c^{2} d x\right )} \cosh \left (1\right ) + 2 \, {\left (84 \, b c^{6} d x^{5} - 112 \, b c^{4} d x^{3} + 224 \, b c^{2} d x + 9 \, {\left (5 \, b c^{6} x^{7} - 6 \, b c^{4} x^{5} + 8 \, b c^{2} x^{3} - 16 \, b x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{2520 \, c^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2520*(315*a*c^7*x^8*cosh(1)^2 + 315*a*c^7*x^8*sinh(1)^2 + 840*a*c^7*d*x^6*cosh(1) + 630*a*c^7*d^2*x^4 + 105*
(3*b*c^7*x^8*cosh(1)^2 + 3*b*c^7*x^8*sinh(1)^2 + 8*b*c^7*d*x^6*cosh(1) + 6*b*c^7*d^2*x^4 + 2*(3*b*c^7*x^8*cosh
(1) + 4*b*c^7*d*x^6)*sinh(1))*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + 210*(3*a*c^7*x^8*cosh(1) +
4*a*c^7*d*x^6)*sinh(1) + (210*b*c^6*d^2*x^3 - 420*b*c^4*d^2*x + 9*(5*b*c^6*x^7 - 6*b*c^4*x^5 + 8*b*c^2*x^3 - 1
6*b*x)*cosh(1)^2 + 9*(5*b*c^6*x^7 - 6*b*c^4*x^5 + 8*b*c^2*x^3 - 16*b*x)*sinh(1)^2 + 56*(3*b*c^6*d*x^5 - 4*b*c^
4*d*x^3 + 8*b*c^2*d*x)*cosh(1) + 2*(84*b*c^6*d*x^5 - 112*b*c^4*d*x^3 + 224*b*c^2*d*x + 9*(5*b*c^6*x^7 - 6*b*c^
4*x^5 + 8*b*c^2*x^3 - 16*b*x)*cosh(1))*sinh(1))*sqrt((c^2*x^2 + 1)/(c^2*x^2)))/c^7

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(e*x**2+d)**2*(a+b*acsch(c*x)),x)

[Out]

Integral(x**3*(a + b*acsch(c*x))*(d + e*x**2)**2, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x^2 + d)^2*(b*arccsch(c*x) + a)*x^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(d + e*x^2)^2*(a + b*asinh(1/(c*x))),x)

[Out]

int(x^3*(d + e*x^2)^2*(a + b*asinh(1/(c*x))), x)

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