3.1.88 \(\int x^2 (d+e x^2)^2 (a+b \text {csch}^{-1}(c x)) \, dx\) [88]

Optimal. Leaf size=260 \[ \frac {b \left (280 c^4 d^2-252 c^2 d e+75 e^2\right ) x^2 \sqrt {-1-c^2 x^2}}{1680 c^5 \sqrt {-c^2 x^2}}+\frac {b \left (84 c^2 d-25 e\right ) e x^4 \sqrt {-1-c^2 x^2}}{840 c^3 \sqrt {-c^2 x^2}}+\frac {b e^2 x^6 \sqrt {-1-c^2 x^2}}{42 c \sqrt {-c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b \left (280 c^4 d^2-252 c^2 d e+75 e^2\right ) x \text {ArcTan}\left (\frac {c x}{\sqrt {-1-c^2 x^2}}\right )}{1680 c^6 \sqrt {-c^2 x^2}} \]

[Out]

1/3*d^2*x^3*(a+b*arccsch(c*x))+2/5*d*e*x^5*(a+b*arccsch(c*x))+1/7*e^2*x^7*(a+b*arccsch(c*x))+1/1680*b*(280*c^4
*d^2-252*c^2*d*e+75*e^2)*x*arctan(c*x/(-c^2*x^2-1)^(1/2))/c^6/(-c^2*x^2)^(1/2)+1/1680*b*(280*c^4*d^2-252*c^2*d
*e+75*e^2)*x^2*(-c^2*x^2-1)^(1/2)/c^5/(-c^2*x^2)^(1/2)+1/840*b*(84*c^2*d-25*e)*e*x^4*(-c^2*x^2-1)^(1/2)/c^3/(-
c^2*x^2)^(1/2)+1/42*b*e^2*x^6*(-c^2*x^2-1)^(1/2)/c/(-c^2*x^2)^(1/2)

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Rubi [A]
time = 0.17, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {276, 6437, 12, 1281, 470, 327, 223, 209} \begin {gather*} \frac {1}{3} d^2 x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b x \text {ArcTan}\left (\frac {c x}{\sqrt {-c^2 x^2-1}}\right ) \left (280 c^4 d^2-252 c^2 d e+75 e^2\right )}{1680 c^6 \sqrt {-c^2 x^2}}+\frac {b e^2 x^6 \sqrt {-c^2 x^2-1}}{42 c \sqrt {-c^2 x^2}}+\frac {b e x^4 \sqrt {-c^2 x^2-1} \left (84 c^2 d-25 e\right )}{840 c^3 \sqrt {-c^2 x^2}}+\frac {b x^2 \sqrt {-c^2 x^2-1} \left (280 c^4 d^2-252 c^2 d e+75 e^2\right )}{1680 c^5 \sqrt {-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(280*c^4*d^2 - 252*c^2*d*e + 75*e^2)*x^2*Sqrt[-1 - c^2*x^2])/(1680*c^5*Sqrt[-(c^2*x^2)]) + (b*(84*c^2*d - 2
5*e)*e*x^4*Sqrt[-1 - c^2*x^2])/(840*c^3*Sqrt[-(c^2*x^2)]) + (b*e^2*x^6*Sqrt[-1 - c^2*x^2])/(42*c*Sqrt[-(c^2*x^
2)]) + (d^2*x^3*(a + b*ArcCsch[c*x]))/3 + (2*d*e*x^5*(a + b*ArcCsch[c*x]))/5 + (e^2*x^7*(a + b*ArcCsch[c*x]))/
7 + (b*(280*c^4*d^2 - 252*c^2*d*e + 75*e^2)*x*ArcTan[(c*x)/Sqrt[-1 - c^2*x^2]])/(1680*c^6*Sqrt[-(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 209

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

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 276

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

Rule 327

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

Rule 470

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

Rule 1281

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Si
mp[c^p*(f*x)^(m + 4*p - 1)*((d + e*x^2)^(q + 1)/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1))), x] + Dist[1/(e*(m + 4*p
+ 2*q + 1)), Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + b*x^2 + c*x^4)^p - c^p*x^(4*p))
- d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] &&
IGtQ[p, 0] &&  !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]

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^2 \left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx &=\frac {1}{3} d^2 x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^2 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{105 \sqrt {-1-c^2 x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=\frac {1}{3} d^2 x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^2 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{\sqrt {-1-c^2 x^2}} \, dx}{105 \sqrt {-c^2 x^2}}\\ &=\frac {b e^2 x^6 \sqrt {-1-c^2 x^2}}{42 c \sqrt {-c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {(b x) \int \frac {x^2 \left (-210 c^2 d^2-3 \left (84 c^2 d-25 e\right ) e x^2\right )}{\sqrt {-1-c^2 x^2}} \, dx}{630 c \sqrt {-c^2 x^2}}\\ &=\frac {b \left (84 c^2 d-25 e\right ) e x^4 \sqrt {-1-c^2 x^2}}{840 c^3 \sqrt {-c^2 x^2}}+\frac {b e^2 x^6 \sqrt {-1-c^2 x^2}}{42 c \sqrt {-c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {\left (b \left (-840 c^4 d^2+9 \left (84 c^2 d-25 e\right ) e\right ) x\right ) \int \frac {x^2}{\sqrt {-1-c^2 x^2}} \, dx}{2520 c^3 \sqrt {-c^2 x^2}}\\ &=\frac {b \left (280 c^4 d^2-252 c^2 d e+75 e^2\right ) x^2 \sqrt {-1-c^2 x^2}}{1680 c^5 \sqrt {-c^2 x^2}}+\frac {b \left (84 c^2 d-25 e\right ) e x^4 \sqrt {-1-c^2 x^2}}{840 c^3 \sqrt {-c^2 x^2}}+\frac {b e^2 x^6 \sqrt {-1-c^2 x^2}}{42 c \sqrt {-c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {\left (b \left (-840 c^4 d^2+9 \left (84 c^2 d-25 e\right ) e\right ) x\right ) \int \frac {1}{\sqrt {-1-c^2 x^2}} \, dx}{5040 c^5 \sqrt {-c^2 x^2}}\\ &=\frac {b \left (280 c^4 d^2-252 c^2 d e+75 e^2\right ) x^2 \sqrt {-1-c^2 x^2}}{1680 c^5 \sqrt {-c^2 x^2}}+\frac {b \left (84 c^2 d-25 e\right ) e x^4 \sqrt {-1-c^2 x^2}}{840 c^3 \sqrt {-c^2 x^2}}+\frac {b e^2 x^6 \sqrt {-1-c^2 x^2}}{42 c \sqrt {-c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {\left (b \left (-840 c^4 d^2+9 \left (84 c^2 d-25 e\right ) e\right ) x\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1-c^2 x^2}}\right )}{5040 c^5 \sqrt {-c^2 x^2}}\\ &=\frac {b \left (280 c^4 d^2-252 c^2 d e+75 e^2\right ) x^2 \sqrt {-1-c^2 x^2}}{1680 c^5 \sqrt {-c^2 x^2}}+\frac {b \left (84 c^2 d-25 e\right ) e x^4 \sqrt {-1-c^2 x^2}}{840 c^3 \sqrt {-c^2 x^2}}+\frac {b e^2 x^6 \sqrt {-1-c^2 x^2}}{42 c \sqrt {-c^2 x^2}}+\frac {1}{3} d^2 x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b \left (280 c^4 d^2-252 c^2 d e+75 e^2\right ) x \tan ^{-1}\left (\frac {c x}{\sqrt {-1-c^2 x^2}}\right )}{1680 c^6 \sqrt {-c^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.45, size = 182, normalized size = 0.70 \begin {gather*} \frac {c^2 x^2 \left (16 a c^5 x \left (35 d^2+42 d e x^2+15 e^2 x^4\right )+b \sqrt {1+\frac {1}{c^2 x^2}} \left (75 e^2-2 c^2 e \left (126 d+25 e x^2\right )+8 c^4 \left (35 d^2+21 d e x^2+5 e^2 x^4\right )\right )\right )+16 b c^7 x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right ) \text {csch}^{-1}(c x)+b \left (-280 c^4 d^2+252 c^2 d e-75 e^2\right ) \log \left (\left (1+\sqrt {1+\frac {1}{c^2 x^2}}\right ) x\right )}{1680 c^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*x^2*(16*a*c^5*x*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4) + b*Sqrt[1 + 1/(c^2*x^2)]*(75*e^2 - 2*c^2*e*(126*d + 2
5*e*x^2) + 8*c^4*(35*d^2 + 21*d*e*x^2 + 5*e^2*x^4))) + 16*b*c^7*x^3*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4)*ArcCsch
[c*x] + b*(-280*c^4*d^2 + 252*c^2*d*e - 75*e^2)*Log[(1 + Sqrt[1 + 1/(c^2*x^2)])*x])/(1680*c^7)

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Maple [A]
time = 0.46, size = 286, normalized size = 1.10

method	result	size
derivativedivides	\(\frac {\frac {a \left (\frac {1}{3} d^{2} c^{7} x^{3}+\frac {2}{5} d \,c^{7} e \,x^{5}+\frac {1}{7} e^{2} c^{7} x^{7}\right )}{c^{4}}+\frac {b \left (\frac {\mathrm {arccsch}\left (c x \right ) d^{2} c^{7} x^{3}}{3}+\frac {2 \,\mathrm {arccsch}\left (c x \right ) d \,c^{7} e \,x^{5}}{5}+\frac {\mathrm {arccsch}\left (c x \right ) e^{2} c^{7} x^{7}}{7}-\frac {\sqrt {c^{2} x^{2}+1}\, \left (-280 d^{2} c^{5} x \sqrt {c^{2} x^{2}+1}-168 d \,c^{5} e \,x^{3} \sqrt {c^{2} x^{2}+1}-40 e^{2} c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+280 d^{2} c^{4} \arcsinh \left (c x \right )+252 d \,c^{3} e x \sqrt {c^{2} x^{2}+1}+50 e^{2} c^{3} x^{3} \sqrt {c^{2} x^{2}+1}-252 d \,c^{2} e \arcsinh \left (c x \right )-75 e^{2} c x \sqrt {c^{2} x^{2}+1}+75 e^{2} \arcsinh \left (c x \right )\right )}{1680 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}}}{c^{3}}\)	\(286\)
default	\(\frac {\frac {a \left (\frac {1}{3} d^{2} c^{7} x^{3}+\frac {2}{5} d \,c^{7} e \,x^{5}+\frac {1}{7} e^{2} c^{7} x^{7}\right )}{c^{4}}+\frac {b \left (\frac {\mathrm {arccsch}\left (c x \right ) d^{2} c^{7} x^{3}}{3}+\frac {2 \,\mathrm {arccsch}\left (c x \right ) d \,c^{7} e \,x^{5}}{5}+\frac {\mathrm {arccsch}\left (c x \right ) e^{2} c^{7} x^{7}}{7}-\frac {\sqrt {c^{2} x^{2}+1}\, \left (-280 d^{2} c^{5} x \sqrt {c^{2} x^{2}+1}-168 d \,c^{5} e \,x^{3} \sqrt {c^{2} x^{2}+1}-40 e^{2} c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+280 d^{2} c^{4} \arcsinh \left (c x \right )+252 d \,c^{3} e x \sqrt {c^{2} x^{2}+1}+50 e^{2} c^{3} x^{3} \sqrt {c^{2} x^{2}+1}-252 d \,c^{2} e \arcsinh \left (c x \right )-75 e^{2} c x \sqrt {c^{2} x^{2}+1}+75 e^{2} \arcsinh \left (c x \right )\right )}{1680 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}}}{c^{3}}\)	\(286\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^3*(a/c^4*(1/3*d^2*c^7*x^3+2/5*d*c^7*e*x^5+1/7*e^2*c^7*x^7)+b/c^4*(1/3*arccsch(c*x)*d^2*c^7*x^3+2/5*arccsch
(c*x)*d*c^7*e*x^5+1/7*arccsch(c*x)*e^2*c^7*x^7-1/1680*(c^2*x^2+1)^(1/2)*(-280*d^2*c^5*x*(c^2*x^2+1)^(1/2)-168*
d*c^5*e*x^3*(c^2*x^2+1)^(1/2)-40*e^2*c^5*x^5*(c^2*x^2+1)^(1/2)+280*d^2*c^4*arcsinh(c*x)+252*d*c^3*e*x*(c^2*x^2
+1)^(1/2)+50*e^2*c^3*x^3*(c^2*x^2+1)^(1/2)-252*d*c^2*e*arcsinh(c*x)-75*e^2*c*x*(c^2*x^2+1)^(1/2)+75*e^2*arcsin
h(c*x))/((c^2*x^2+1)/c^2/x^2)^(1/2)/c/x))

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Maxima [A]
time = 0.27, size = 396, normalized size = 1.52 \begin {gather*} \frac {1}{7} \, a x^{7} e^{2} + \frac {2}{5} \, a d x^{5} e + \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{12} \, {\left (4 \, x^{3} \operatorname {arcsch}\left (c x\right ) + \frac {\frac {2 \, \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} - c^{2}} - \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} + \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b d^{2} + \frac {1}{40} \, {\left (16 \, x^{5} \operatorname {arcsch}\left (c x\right ) - \frac {\frac {2 \, {\left (3 \, {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{2} - 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} + c^{4}} - \frac {3 \, \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac {3 \, \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b d e + \frac {1}{672} \, {\left (96 \, x^{7} \operatorname {arcsch}\left (c x\right ) + \frac {\frac {2 \, {\left (15 \, {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 40 \, {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{6} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{3} - 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{2} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} - c^{6}} - \frac {15 \, \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{6}} + \frac {15 \, \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{6}}}{c}\right )} b e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a*x^7*e^2 + 2/5*a*d*x^5*e + 1/3*a*d^2*x^3 + 1/12*(4*x^3*arccsch(c*x) + (2*sqrt(1/(c^2*x^2) + 1)/(c^2*(1/(c
^2*x^2) + 1) - c^2) - log(sqrt(1/(c^2*x^2) + 1) + 1)/c^2 + log(sqrt(1/(c^2*x^2) + 1) - 1)/c^2)/c)*b*d^2 + 1/40
*(16*x^5*arccsch(c*x) - (2*(3*(1/(c^2*x^2) + 1)^(3/2) - 5*sqrt(1/(c^2*x^2) + 1))/(c^4*(1/(c^2*x^2) + 1)^2 - 2*
c^4*(1/(c^2*x^2) + 1) + c^4) - 3*log(sqrt(1/(c^2*x^2) + 1) + 1)/c^4 + 3*log(sqrt(1/(c^2*x^2) + 1) - 1)/c^4)/c)
*b*d*e + 1/672*(96*x^7*arccsch(c*x) + (2*(15*(1/(c^2*x^2) + 1)^(5/2) - 40*(1/(c^2*x^2) + 1)^(3/2) + 33*sqrt(1/
(c^2*x^2) + 1))/(c^6*(1/(c^2*x^2) + 1)^3 - 3*c^6*(1/(c^2*x^2) + 1)^2 + 3*c^6*(1/(c^2*x^2) + 1) - c^6) - 15*log
(sqrt(1/(c^2*x^2) + 1) + 1)/c^6 + 15*log(sqrt(1/(c^2*x^2) + 1) - 1)/c^6)/c)*b*e^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 679 vs. \(2 (231) = 462\).
time = 0.48, size = 679, normalized size = 2.61 \begin {gather*} \frac {240 \, a c^{7} x^{7} \cosh \left (1\right )^{2} + 240 \, a c^{7} x^{7} \sinh \left (1\right )^{2} + 672 \, a c^{7} d x^{5} \cosh \left (1\right ) + 560 \, a c^{7} d^{2} x^{3} + 16 \, {\left (35 \, b c^{7} d^{2} + 42 \, b c^{7} d \cosh \left (1\right ) + 15 \, b c^{7} \cosh \left (1\right )^{2} + 15 \, b c^{7} \sinh \left (1\right )^{2} + 6 \, {\left (7 \, b c^{7} d + 5 \, b c^{7} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) + {\left (280 \, b c^{4} d^{2} - 252 \, b c^{2} d \cosh \left (1\right ) + 75 \, b \cosh \left (1\right )^{2} + 75 \, b \sinh \left (1\right )^{2} - 6 \, {\left (42 \, b c^{2} d - 25 \, b \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) - 16 \, {\left (35 \, b c^{7} d^{2} + 42 \, b c^{7} d \cosh \left (1\right ) + 15 \, b c^{7} \cosh \left (1\right )^{2} + 15 \, b c^{7} \sinh \left (1\right )^{2} + 6 \, {\left (7 \, b c^{7} d + 5 \, b c^{7} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 16 \, {\left (35 \, b c^{7} d^{2} x^{3} - 35 \, b c^{7} d^{2} + 15 \, {\left (b c^{7} x^{7} - b c^{7}\right )} \cosh \left (1\right )^{2} + 15 \, {\left (b c^{7} x^{7} - b c^{7}\right )} \sinh \left (1\right )^{2} + 42 \, {\left (b c^{7} d x^{5} - b c^{7} d\right )} \cosh \left (1\right ) + 6 \, {\left (7 \, b c^{7} d x^{5} - 7 \, b c^{7} d + 5 \, {\left (b c^{7} x^{7} - b c^{7}\right )} \cosh \left (1\right )\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 ) + 96 \, {\left (5 \, a c^{7} x^{7} \cosh \left (1\right ) + 7 \, a c^{7} d x^{5}\right )} \sinh \left (1\right ) + {\left (280 \, b c^{6} d^{2} x^{2} + 5 \, {\left (8 \, b c^{6} x^{6} - 10 \, b c^{4} x^{4} + 15 \, b c^{2} x^{2}\right )} \cosh \left (1\right )^{2} + 5 \, {\left (8 \, b c^{6} x^{6} - 10 \, b c^{4} x^{4} + 15 \, b c^{2} x^{2}\right )} \sinh \left (1\right )^{2} + 84 \, {\left (2 \, b c^{6} d x^{4} - 3 \, b c^{4} d x^{2}\right )} \cosh \left (1\right ) + 2 \, {\left (84 \, b c^{6} d x^{4} - 126 \, b c^{4} d x^{2} + 5 \, {\left (8 \, b c^{6} x^{6} - 10 \, b c^{4} x^{4} + 15 \, b c^{2} x^{2}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{1680 \, c^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1680*(240*a*c^7*x^7*cosh(1)^2 + 240*a*c^7*x^7*sinh(1)^2 + 672*a*c^7*d*x^5*cosh(1) + 560*a*c^7*d^2*x^3 + 16*(
35*b*c^7*d^2 + 42*b*c^7*d*cosh(1) + 15*b*c^7*cosh(1)^2 + 15*b*c^7*sinh(1)^2 + 6*(7*b*c^7*d + 5*b*c^7*cosh(1))*
sinh(1))*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x + 1) + (280*b*c^4*d^2 - 252*b*c^2*d*cosh(1) + 75*b*cosh(1
)^2 + 75*b*sinh(1)^2 - 6*(42*b*c^2*d - 25*b*cosh(1))*sinh(1))*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x) - 1
6*(35*b*c^7*d^2 + 42*b*c^7*d*cosh(1) + 15*b*c^7*cosh(1)^2 + 15*b*c^7*sinh(1)^2 + 6*(7*b*c^7*d + 5*b*c^7*cosh(1
))*sinh(1))*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x - 1) + 16*(35*b*c^7*d^2*x^3 - 35*b*c^7*d^2 + 15*(b*c^7
*x^7 - b*c^7)*cosh(1)^2 + 15*(b*c^7*x^7 - b*c^7)*sinh(1)^2 + 42*(b*c^7*d*x^5 - b*c^7*d)*cosh(1) + 6*(7*b*c^7*d
*x^5 - 7*b*c^7*d + 5*(b*c^7*x^7 - b*c^7)*cosh(1))*sinh(1))*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x))
+ 96*(5*a*c^7*x^7*cosh(1) + 7*a*c^7*d*x^5)*sinh(1) + (280*b*c^6*d^2*x^2 + 5*(8*b*c^6*x^6 - 10*b*c^4*x^4 + 15*b
*c^2*x^2)*cosh(1)^2 + 5*(8*b*c^6*x^6 - 10*b*c^4*x^4 + 15*b*c^2*x^2)*sinh(1)^2 + 84*(2*b*c^6*d*x^4 - 3*b*c^4*d*
x^2)*cosh(1) + 2*(84*b*c^6*d*x^4 - 126*b*c^4*d*x^2 + 5*(8*b*c^6*x^6 - 10*b*c^4*x^4 + 15*b*c^2*x^2)*cosh(1))*si
nh(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^{2} \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**2*(e*x**2+d)**2*(a+b*acsch(c*x)),x)

[Out]

Integral(x**2*(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^2*(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^2, x)

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

[Out]

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

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