3.91 \(\int \frac {(d+e x^2)^2 (a+b \csc ^{-1}(c x))}{x^4} \, dx\)
Optimal. Leaf size=157 \[ -\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \csc ^{-1}(c x)\right )-\frac {b c d^2 \sqrt {c^2 x^2-1}}{9 x^2 \sqrt {c^2 x^2}}-\frac {2 b c d \sqrt {c^2 x^2-1} \left (c^2 d+9 e\right )}{9 \sqrt {c^2 x^2}}+\frac {b e^2 x \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{\sqrt {c^2 x^2}} \]
[Out]
-1/3*d^2*(a+b*arccsc(c*x))/x^3-2*d*e*(a+b*arccsc(c*x))/x+e^2*x*(a+b*arccsc(c*x))+b*e^2*x*arctanh(c*x/(c^2*x^2-
1)^(1/2))/(c^2*x^2)^(1/2)-2/9*b*c*d*(c^2*d+9*e)*(c^2*x^2-1)^(1/2)/(c^2*x^2)^(1/2)-1/9*b*c*d^2*(c^2*x^2-1)^(1/2
)/x^2/(c^2*x^2)^(1/2)
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Rubi [A] time = 0.13, antiderivative size = 157, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used =
{270, 5239, 12, 1265, 451, 217, 206} \[ -\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \csc ^{-1}(c x)\right )-\frac {b c d^2 \sqrt {c^2 x^2-1}}{9 x^2 \sqrt {c^2 x^2}}-\frac {2 b c d \sqrt {c^2 x^2-1} \left (c^2 d+9 e\right )}{9 \sqrt {c^2 x^2}}+\frac {b e^2 x \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{\sqrt {c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
Int[((d + e*x^2)^2*(a + b*ArcCsc[c*x]))/x^4,x]
[Out]
(-2*b*c*d*(c^2*d + 9*e)*Sqrt[-1 + c^2*x^2])/(9*Sqrt[c^2*x^2]) - (b*c*d^2*Sqrt[-1 + c^2*x^2])/(9*x^2*Sqrt[c^2*x
^2]) - (d^2*(a + b*ArcCsc[c*x]))/(3*x^3) - (2*d*e*(a + b*ArcCsc[c*x]))/x + e^2*x*(a + b*ArcCsc[c*x]) + (b*e^2*
x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/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 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 217
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 270
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 451
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[d/e^n, Int[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a,
b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n*(p + 1) + 1, 0] && (IntegerQ[n] || GtQ[e, 0]) && (
(GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1]))
Rule 1265
Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Wit
h[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, f*x, x], R = PolynomialRemainder[(a + b*x^2 + c*x^4)^p, f*x,
x]}, Simp[(R*(f*x)^(m + 1)*(d + e*x^2)^(q + 1))/(d*f*(m + 1)), x] + Dist[1/(d*f^2*(m + 1)), Int[(f*x)^(m + 2)
*(d + e*x^2)^q*ExpandToSum[(d*f*(m + 1)*Qx)/x - e*R*(m + 2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q},
x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[m, -1]
Rule 5239
Int[((a_.) + ArcCsc[(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*ArcCsc[c*x], u, x] + Dist[(b*c*x)/Sqrt[c^2*x^2], Int[SimplifyI
ntegrand[u/(x*Sqrt[c^2*x^2 - 1]), 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])) || (I
LtQ[(m + 2*p + 1)/2, 0] && !ILtQ[(m - 1)/2, 0]))
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^4} \, dx &=-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {-d^2-6 d e x^2+3 e^2 x^4}{3 x^4 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {-d^2-6 d e x^2+3 e^2 x^4}{x^4 \sqrt {-1+c^2 x^2}} \, dx}{3 \sqrt {c^2 x^2}}\\ &=-\frac {b c d^2 \sqrt {-1+c^2 x^2}}{9 x^2 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {-2 d \left (c^2 d+9 e\right )+9 e^2 x^2}{x^2 \sqrt {-1+c^2 x^2}} \, dx}{9 \sqrt {c^2 x^2}}\\ &=-\frac {2 b c d \left (c^2 d+9 e\right ) \sqrt {-1+c^2 x^2}}{9 \sqrt {c^2 x^2}}-\frac {b c d^2 \sqrt {-1+c^2 x^2}}{9 x^2 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac {\left (b c e^2 x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=-\frac {2 b c d \left (c^2 d+9 e\right ) \sqrt {-1+c^2 x^2}}{9 \sqrt {c^2 x^2}}-\frac {b c d^2 \sqrt {-1+c^2 x^2}}{9 x^2 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac {\left (b c e^2 x\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{\sqrt {c^2 x^2}}\\ &=-\frac {2 b c d \left (c^2 d+9 e\right ) \sqrt {-1+c^2 x^2}}{9 \sqrt {c^2 x^2}}-\frac {b c d^2 \sqrt {-1+c^2 x^2}}{9 x^2 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac {b e^2 x \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{\sqrt {c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 125, normalized size = 0.80 \[ -\frac {3 a \left (d^2+6 d e x^2-3 e^2 x^4\right )+b c d x \sqrt {1-\frac {1}{c^2 x^2}} \left (2 c^2 d x^2+d+18 e x^2\right )}{9 x^3}+\frac {b e^2 \log \left (x \left (\sqrt {1-\frac {1}{c^2 x^2}}+1\right )\right )}{c}-\frac {b \csc ^{-1}(c x) \left (d^2+6 d e x^2-3 e^2 x^4\right )}{3 x^3} \]
Antiderivative was successfully verified.
[In]
Integrate[((d + e*x^2)^2*(a + b*ArcCsc[c*x]))/x^4,x]
[Out]
-1/9*(b*c*d*Sqrt[1 - 1/(c^2*x^2)]*x*(d + 2*c^2*d*x^2 + 18*e*x^2) + 3*a*(d^2 + 6*d*e*x^2 - 3*e^2*x^4))/x^3 - (b
*(d^2 + 6*d*e*x^2 - 3*e^2*x^4)*ArcCsc[c*x])/(3*x^3) + (b*e^2*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/c
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fricas [A] time = 0.51, size = 222, normalized size = 1.41 \[ \frac {9 \, a c e^{2} x^{4} - 9 \, b e^{2} x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 18 \, a c d e x^{2} + 6 \, {\left (b c d^{2} + 6 \, b c d e - 3 \, b c e^{2}\right )} x^{3} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 3 \, a c d^{2} - 2 \, {\left (b c^{4} d^{2} + 9 \, b c^{2} d e\right )} x^{3} + 3 \, {\left (3 \, b c e^{2} x^{4} - 6 \, b c d e x^{2} - b c d^{2} + {\left (b c d^{2} + 6 \, b c d e - 3 \, b c e^{2}\right )} x^{3}\right )} \operatorname {arccsc}\left (c x\right ) - {\left (b c d^{2} + 2 \, {\left (b c^{3} d^{2} + 9 \, b c d e\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{9 \, c x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x^2+d)^2*(a+b*arccsc(c*x))/x^4,x, algorithm="fricas")
[Out]
1/9*(9*a*c*e^2*x^4 - 9*b*e^2*x^3*log(-c*x + sqrt(c^2*x^2 - 1)) - 18*a*c*d*e*x^2 + 6*(b*c*d^2 + 6*b*c*d*e - 3*b
*c*e^2)*x^3*arctan(-c*x + sqrt(c^2*x^2 - 1)) - 3*a*c*d^2 - 2*(b*c^4*d^2 + 9*b*c^2*d*e)*x^3 + 3*(3*b*c*e^2*x^4
- 6*b*c*d*e*x^2 - b*c*d^2 + (b*c*d^2 + 6*b*c*d*e - 3*b*c*e^2)*x^3)*arccsc(c*x) - (b*c*d^2 + 2*(b*c^3*d^2 + 9*b
*c*d*e)*x^2)*sqrt(c^2*x^2 - 1))/(c*x^3)
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giac [B] time = 14.42, size = 4280, normalized size = 27.26 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x^2+d)^2*(a+b*arccsc(c*x))/x^4,x, algorithm="giac")
[Out]
-1/18*(4*b*c^3*d^2/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^
2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7))
) - 9*b*arcsin(1/(c*x))*e^2/(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3
/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7)) - 9*a*e^2/(c/(x*(sqrt(
-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5
) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7)) + 36*b*c*d*e/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)*(c/(x*(sqrt(-1/(c
^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1
/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7))) + 72*b*d*arcsin(1/(c*x))*e/(x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*(c/
(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) +
1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7))) - 18*b*e^2*log(sqrt(-1/(c^2*x^2) + 1) + 1)/(c*x*(sqr
t(-1/(c^2*x^2) + 1) + 1)*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c
^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7))) + 18*b*e^2*log(1/(abs(c)
*abs(x)))/(c*x*(sqrt(-1/(c^2*x^2) + 1) + 1)*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2)
+ 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7))) + 72*
a*d*e/(x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2) +
1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7))) + 12*b*
c*d^2/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2) +
1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7))) - 36*b*
arcsin(1/(c*x))*e^2/(c^2*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sq
rt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1)
+ 1)^7))) + 48*b*d^2*arcsin(1/(c*x))/(x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1))
+ 3/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/
(c^2*x^2) + 1) + 1)^7))) - 36*a*e^2/(c^2*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)
) + 3/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-
1/(c^2*x^2) + 1) + 1)^7))) + 36*b*d*e/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)
) + 3/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-
1/(c^2*x^2) + 1) + 1)^7))) + 48*a*d^2/(x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1))
+ 3/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/
(c^2*x^2) + 1) + 1)^7))) + 144*b*d*arcsin(1/(c*x))*e/(c^2*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*(c/(x*(sqrt(-1/(c
^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1
/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7))) - 54*b*e^2*log(sqrt(-1/(c^2*x^2) + 1) + 1)/(c^3*x^3*(sqrt(-1/(c^2*
x^2) + 1) + 1)^3*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(
sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7))) + 54*b*e^2*log(1/(abs(c)*abs(x))
)/(c^3*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2) +
1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7))) + 144*a
*d*e/(c^2*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2)
+ 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7))) - 12
*b*d^2/(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2)
+ 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7))) - 54
*b*arcsin(1/(c*x))*e^2/(c^4*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*
(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) +
1) + 1)^7))) - 54*a*e^2/(c^4*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^
3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2)
+ 1) + 1)^7))) - 36*b*d*e/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*
x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^
2) + 1) + 1)^7))) + 72*b*d*arcsin(1/(c*x))*e/(c^4*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6*(c/(x*(sqrt(-1/(c^2*x^2)
+ 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^
7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7))) - 54*b*e^2*log(sqrt(-1/(c^2*x^2) + 1) + 1)/(c^5*x^5*(sqrt(-1/(c^2*x^2) + 1
) + 1)^5*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/
(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7))) + 54*b*e^2*log(1/(abs(c)*abs(x)))/(c^5*x
^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^
3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7))) + 72*a*d*e/(c^4
*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1
)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7))) - 4*b*d^2/(c^
3*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) +
1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7))) - 36*b*arcsi
n(1/(c*x))*e^2/(c^6*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(-1
/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)
^7))) - 36*a*e^2/(c^6*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(
-1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) +
1)^7))) - 36*b*d*e/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqr
t(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1)
+ 1)^7))) - 18*b*e^2*log(sqrt(-1/(c^2*x^2) + 1) + 1)/(c^7*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7*(c/(x*(sqrt(-1/(c
^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1
/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7))) + 18*b*e^2*log(1/(abs(c)*abs(x)))/(c^7*x^7*(sqrt(-1/(c^2*x^2) + 1)
+ 1)^7*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c^3*x^5*(sqrt(-1/(
c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7))) - 9*b*arcsin(1/(c*x))*e^2/(c^8*x^8*(sqrt(
-1/(c^2*x^2) + 1) + 1)^8*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(c
^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7))) - 9*a*e^2/(c^8*x^8*(sqrt
(-1/(c^2*x^2) + 1) + 1)^8*(c/(x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3/(
c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5) + 1/(c^5*x^7*(sqrt(-1/(c^2*x^2) + 1) + 1)^7))))*c
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maple [A] time = 0.07, size = 254, normalized size = 1.62 \[ a x \,e^{2}-\frac {a \,d^{2}}{3 x^{3}}-\frac {2 a e d}{x}+b \,\mathrm {arccsc}\left (c x \right ) x \,e^{2}-\frac {b \,\mathrm {arccsc}\left (c x \right ) d^{2}}{3 x^{3}}-\frac {2 b \,\mathrm {arccsc}\left (c x \right ) e d}{x}-\frac {2 c^{3} b \,d^{2}}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {c b \,d^{2}}{9 x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {2 c b e d}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {2 b e d}{c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{2}}+\frac {b \,d^{2}}{9 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{4}}+\frac {b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x^2+d)^2*(a+b*arccsc(c*x))/x^4,x)
[Out]
a*x*e^2-1/3*a*d^2/x^3-2*a*e*d/x+b*arccsc(c*x)*x*e^2-1/3*b*arccsc(c*x)*d^2/x^3-2*b*arccsc(c*x)*e*d/x-2/9*c^3*b/
((c^2*x^2-1)/c^2/x^2)^(1/2)*d^2+1/9*c*b/x^2/((c^2*x^2-1)/c^2/x^2)^(1/2)*d^2-2*c*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*
e*d+2/c*b/((c^2*x^2-1)/c^2/x^2)^(1/2)/x^2*e*d+1/9/c*b/((c^2*x^2-1)/c^2/x^2)^(1/2)/x^4*d^2+1/c^2*b*(c^2*x^2-1)^
(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*e^2*ln(c*x+(c^2*x^2-1)^(1/2))
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maxima [A] time = 0.33, size = 158, normalized size = 1.01 \[ -2 \, {\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {\operatorname {arccsc}\left (c x\right )}{x}\right )} b d e + a e^{2} x + \frac {1}{9} \, b d^{2} {\left (\frac {c^{4} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {3 \, \operatorname {arccsc}\left (c x\right )}{x^{3}}\right )} + \frac {{\left (2 \, c x \operatorname {arccsc}\left (c x\right ) + \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )\right )} b e^{2}}{2 \, c} - \frac {2 \, a d e}{x} - \frac {a d^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x^2+d)^2*(a+b*arccsc(c*x))/x^4,x, algorithm="maxima")
[Out]
-2*(c*sqrt(-1/(c^2*x^2) + 1) + arccsc(c*x)/x)*b*d*e + a*e^2*x + 1/9*b*d^2*((c^4*(-1/(c^2*x^2) + 1)^(3/2) - 3*c
^4*sqrt(-1/(c^2*x^2) + 1))/c - 3*arccsc(c*x)/x^3) + 1/2*(2*c*x*arccsc(c*x) + log(sqrt(-1/(c^2*x^2) + 1) + 1) -
log(-sqrt(-1/(c^2*x^2) + 1) + 1))*b*e^2/c - 2*a*d*e/x - 1/3*a*d^2/x^3
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((d + e*x^2)^2*(a + b*asin(1/(c*x))))/x^4,x)
[Out]
int(((d + e*x^2)^2*(a + b*asin(1/(c*x))))/x^4, x)
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sympy [A] time = 8.43, size = 211, normalized size = 1.34 \[ - \frac {a d^{2}}{3 x^{3}} - \frac {2 a d e}{x} + a e^{2} x - 2 b c d e \sqrt {1 - \frac {1}{c^{2} x^{2}}} - \frac {b d^{2} \operatorname {acsc}{\left (c x \right )}}{3 x^{3}} - \frac {2 b d e \operatorname {acsc}{\left (c x \right )}}{x} + b e^{2} x \operatorname {acsc}{\left (c x \right )} - \frac {b d^{2} \left (\begin {cases} \frac {2 c^{3} \sqrt {c^{2} x^{2} - 1}}{3 x} + \frac {c \sqrt {c^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {2 i c^{3} \sqrt {- c^{2} x^{2} + 1}}{3 x} + \frac {i c \sqrt {- c^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right )}{3 c} + \frac {b e^{2} \left (\begin {cases} \operatorname {acosh}{\left (c x \right )} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- i \operatorname {asin}{\left (c x \right )} & \text {otherwise} \end {cases}\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x**2+d)**2*(a+b*acsc(c*x))/x**4,x)
[Out]
-a*d**2/(3*x**3) - 2*a*d*e/x + a*e**2*x - 2*b*c*d*e*sqrt(1 - 1/(c**2*x**2)) - b*d**2*acsc(c*x)/(3*x**3) - 2*b*
d*e*acsc(c*x)/x + b*e**2*x*acsc(c*x) - b*d**2*Piecewise((2*c**3*sqrt(c**2*x**2 - 1)/(3*x) + c*sqrt(c**2*x**2 -
1)/(3*x**3), Abs(c**2*x**2) > 1), (2*I*c**3*sqrt(-c**2*x**2 + 1)/(3*x) + I*c*sqrt(-c**2*x**2 + 1)/(3*x**3), T
rue))/(3*c) + b*e**2*Piecewise((acosh(c*x), Abs(c**2*x**2) > 1), (-I*asin(c*x), True))/c
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