3.1.90 \(\int \frac {(d+e x^2)^2 (a+b \csc ^{-1}(c x))}{x^2} \, dx\) [90]

Optimal. Leaf size=163 \[ -\frac {b c d^2 \sqrt {-1+c^2 x^2}}{\sqrt {c^2 x^2}}+\frac {b e^2 x^2 \sqrt {-1+c^2 x^2}}{6 c \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {b e \left (12 c^2 d+e\right ) x \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{6 c^2 \sqrt {c^2 x^2}} \]

[Out]

-d^2*(a+b*arccsc(c*x))/x+2*d*e*x*(a+b*arccsc(c*x))+1/3*e^2*x^3*(a+b*arccsc(c*x))+1/6*b*e*(12*c^2*d+e)*x*arctan
h(c*x/(c^2*x^2-1)^(1/2))/c^2/(c^2*x^2)^(1/2)-b*c*d^2*(c^2*x^2-1)^(1/2)/(c^2*x^2)^(1/2)+1/6*b*e^2*x^2*(c^2*x^2-
1)^(1/2)/c/(c^2*x^2)^(1/2)

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Rubi [A]
time = 0.09, antiderivative size = 163, 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 = {276, 5347, 12, 1279, 396, 223, 212} \begin {gather*} -\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \csc ^{-1}(c x)\right )-\frac {b c d^2 \sqrt {c^2 x^2-1}}{\sqrt {c^2 x^2}}+\frac {b e x \left (12 c^2 d+e\right ) \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{6 c^2 \sqrt {c^2 x^2}}+\frac {b e^2 x^2 \sqrt {c^2 x^2-1}}{6 c \sqrt {c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*c*d^2*Sqrt[-1 + c^2*x^2])/Sqrt[c^2*x^2]) + (b*e^2*x^2*Sqrt[-1 + c^2*x^2])/(6*c*Sqrt[c^2*x^2]) - (d^2*(a +
b*ArcCsc[c*x]))/x + 2*d*e*x*(a + b*ArcCsc[c*x]) + (e^2*x^3*(a + b*ArcCsc[c*x]))/3 + (b*e*(12*c^2*d + e)*x*Arc
Tanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(6*c^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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[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 396

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

Rule 1279

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 5347

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^2} \, dx &=-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {-3 d^2+6 d e x^2+e^2 x^4}{3 x^2 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {-3 d^2+6 d e x^2+e^2 x^4}{x^2 \sqrt {-1+c^2 x^2}} \, dx}{3 \sqrt {c^2 x^2}}\\ &=-\frac {b c d^2 \sqrt {-1+c^2 x^2}}{\sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {6 d e+e^2 x^2}{\sqrt {-1+c^2 x^2}} \, dx}{3 \sqrt {c^2 x^2}}\\ &=-\frac {b c d^2 \sqrt {-1+c^2 x^2}}{\sqrt {c^2 x^2}}+\frac {b e^2 x^2 \sqrt {-1+c^2 x^2}}{6 c \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+-\frac {\left (b \left (-12 c^2 d e-e^2\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{6 c \sqrt {c^2 x^2}}\\ &=-\frac {b c d^2 \sqrt {-1+c^2 x^2}}{\sqrt {c^2 x^2}}+\frac {b e^2 x^2 \sqrt {-1+c^2 x^2}}{6 c \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+-\frac {\left (b \left (-12 c^2 d e-e^2\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 )}{6 c \sqrt {c^2 x^2}}\\ &=-\frac {b c d^2 \sqrt {-1+c^2 x^2}}{\sqrt {c^2 x^2}}+\frac {b e^2 x^2 \sqrt {-1+c^2 x^2}}{6 c \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {b e \left (12 c^2 d+e\right ) x \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{6 c^2 \sqrt {c^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 134, normalized size = 0.82 \begin {gather*} \frac {c^2 \left (b \sqrt {1-\frac {1}{c^2 x^2}} x \left (-6 c^2 d^2+e^2 x^2\right )+2 a c \left (-3 d^2+6 d e x^2+e^2 x^4\right )\right )+2 b c^3 \left (-3 d^2+6 d e x^2+e^2 x^4\right ) \csc ^{-1}(c x)+b e \left (12 c^2 d+e\right ) x \log \left (\left (1+\sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{6 c^3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*(b*Sqrt[1 - 1/(c^2*x^2)]*x*(-6*c^2*d^2 + e^2*x^2) + 2*a*c*(-3*d^2 + 6*d*e*x^2 + e^2*x^4)) + 2*b*c^3*(-3*d
^2 + 6*d*e*x^2 + e^2*x^4)*ArcCsc[c*x] + b*e*(12*c^2*d + e)*x*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/(6*c^3*x)

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Maple [A]
time = 0.37, size = 273, normalized size = 1.67

method	result	size
derivativedivides	\(c \left (\frac {a \left (2 c^{3} d e x +\frac {e^{2} c^{3} x^{3}}{3}-\frac {c^{3} d^{2}}{x}\right )}{c^{4}}+\frac {2 b \,\mathrm {arccsc}\left (c x \right ) d e x}{c}+\frac {b \,\mathrm {arccsc}\left (c x \right ) e^{2} x^{3}}{3 c}-\frac {b \,\mathrm {arccsc}\left (c x \right ) d^{2}}{c x}-\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {2 b \sqrt {c^{2} x^{2}-1}\, d e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \left (c^{2} x^{2}-1\right ) e^{2}}{6 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\right )\)	\(273\)
default	\(c \left (\frac {a \left (2 c^{3} d e x +\frac {e^{2} c^{3} x^{3}}{3}-\frac {c^{3} d^{2}}{x}\right )}{c^{4}}+\frac {2 b \,\mathrm {arccsc}\left (c x \right ) d e x}{c}+\frac {b \,\mathrm {arccsc}\left (c x \right ) e^{2} x^{3}}{3 c}-\frac {b \,\mathrm {arccsc}\left (c x \right ) d^{2}}{c x}-\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {2 b \sqrt {c^{2} x^{2}-1}\, d e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \left (c^{2} x^{2}-1\right ) e^{2}}{6 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\right )\)	\(273\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*(a/c^4*(2*c^3*d*e*x+1/3*e^2*c^3*x^3-c^3*d^2/x)+2*b/c*arccsc(c*x)*d*e*x+1/3*b/c*arccsc(c*x)*e^2*x^3-b*arccsc(
c*x)*d^2/c/x-b*(c^2*x^2-1)/c^2/x^2/((c^2*x^2-1)/c^2/x^2)^(1/2)*d^2+2*b/c^3*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/
x^2)^(1/2)/x*d*e*ln(c*x+(c^2*x^2-1)^(1/2))+1/6*b/c^4*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)*e^2+1/6*b/c^5*(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 = 0.27, size = 197, normalized size = 1.21 \begin {gather*} \frac {1}{3} \, a x^{3} e^{2} - {\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {\operatorname {arccsc}\left (c x\right )}{x}\right )} b d^{2} + 2 \, a d x e + \frac {1}{12} \, {\left (4 \, x^{3} \operatorname {arccsc}\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 e^{2} + \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 d e}{c} - \frac {a d^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*x^3*e^2 - (c*sqrt(-1/(c^2*x^2) + 1) + arccsc(c*x)/x)*b*d^2 + 2*a*d*x*e + 1/12*(4*x^3*arccsc(c*x) + (2*sq
rt(-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*e^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*d*e/c - a*d^2/x

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Fricas [A]
time = 0.46, size = 234, normalized size = 1.44 \begin {gather*} \frac {2 \, a c^{3} x^{4} e^{2} - 6 \, b c^{4} d^{2} x + 12 \, a c^{3} d x^{2} e - 6 \, a c^{3} d^{2} + 2 \, {\left (3 \, b c^{3} d^{2} x - 3 \, b c^{3} d^{2} + {\left (b c^{3} x^{4} - b c^{3} x\right )} e^{2} + 6 \, {\left (b c^{3} d x^{2} - b c^{3} d x\right )} e\right )} \operatorname {arccsc}\left (c x\right ) + 4 \, {\left (3 \, b c^{3} d^{2} x - 6 \, b c^{3} d x e - b c^{3} x e^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (12 \, b c^{2} d x e + b x e^{2}\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (6 \, b c^{3} d^{2} - b c x^{2} e^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{6 \, c^{3} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*a*c^3*x^4*e^2 - 6*b*c^4*d^2*x + 12*a*c^3*d*x^2*e - 6*a*c^3*d^2 + 2*(3*b*c^3*d^2*x - 3*b*c^3*d^2 + (b*c^
3*x^4 - b*c^3*x)*e^2 + 6*(b*c^3*d*x^2 - b*c^3*d*x)*e)*arccsc(c*x) + 4*(3*b*c^3*d^2*x - 6*b*c^3*d*x*e - b*c^3*x
*e^2)*arctan(-c*x + sqrt(c^2*x^2 - 1)) - (12*b*c^2*d*x*e + b*x*e^2)*log(-c*x + sqrt(c^2*x^2 - 1)) - (6*b*c^3*d
^2 - b*c*x^2*e^2)*sqrt(c^2*x^2 - 1))/(c^3*x)

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Sympy [A]
time = 5.55, size = 207, normalized size = 1.27 \begin {gather*} - \frac {a d^{2}}{x} + 2 a d e x + \frac {a e^{2} x^{3}}{3} - b c d^{2} \sqrt {1 - \frac {1}{c^{2} x^{2}}} - \frac {b d^{2} \operatorname {acsc}{\left (c x \right )}}{x} + 2 b d e x \operatorname {acsc}{\left (c x \right )} + \frac {b e^{2} x^{3} \operatorname {acsc}{\left (c x \right )}}{3} + \frac {2 b d e \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} + \frac {b e^{2} \left (\begin {cases} \frac {x \sqrt {c^{2} x^{2} - 1}}{2 c} + \frac {\operatorname {acosh}{\left (c x \right )}}{2 c^{2}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{3}}{2 \sqrt {- c^{2} x^{2} + 1}} + \frac {i x}{2 c \sqrt {- c^{2} x^{2} + 1}} - \frac {i \operatorname {asin}{\left (c x \right )}}{2 c^{2}} & \text {otherwise} \end {cases}\right )}{3 c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**2*(a+b*acsc(c*x))/x**2,x)

[Out]

-a*d**2/x + 2*a*d*e*x + a*e**2*x**3/3 - b*c*d**2*sqrt(1 - 1/(c**2*x**2)) - b*d**2*acsc(c*x)/x + 2*b*d*e*x*acsc
(c*x) + b*e**2*x**3*acsc(c*x)/3 + 2*b*d*e*Piecewise((acosh(c*x), Abs(c**2*x**2) > 1), (-I*asin(c*x), True))/c
+ b*e**2*Piecewise((x*sqrt(c**2*x**2 - 1)/(2*c) + acosh(c*x)/(2*c**2), Abs(c**2*x**2) > 1), (-I*c*x**3/(2*sqrt
(-c**2*x**2 + 1)) + I*x/(2*c*sqrt(-c**2*x**2 + 1)) - I*asin(c*x)/(2*c**2), True))/(3*c)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2502 vs. \(2 (145) = 290\).
time = 2.56, size = 2502, normalized size = 15.35 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(b*e^2*arcsin(1/(c*x))/(c/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 1/(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5)
) + a*e^2/(c/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 1/(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5)) + b*e^2/(c*x*(sq
rt(-1/(c^2*x^2) + 1) + 1)*(c/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 1/(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5)))
+ 24*b*d*e*arcsin(1/(c*x))/(x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*(c/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 1/(c
*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5))) + 24*a*d*e/(x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*(c/(x^3*(sqrt(-1/(c^2*x^
2) + 1) + 1)^3) + 1/(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5))) - 24*b*c*d^2/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3*
(c/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 1/(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5))) + 4*b*e^2*arcsin(1/(c*x))
/(c^2*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*(c/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 1/(c*x^5*(sqrt(-1/(c^2*x^2)
+ 1) + 1)^5))) + 4*a*e^2/(c^2*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*(c/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 1/
(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5))) + 48*b*d*e*log(sqrt(-1/(c^2*x^2) + 1) + 1)/(c*x^3*(sqrt(-1/(c^2*x^2)
+ 1) + 1)^3*(c/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 1/(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5))) - 48*b*d*e*lo
g(1/(abs(c)*abs(x)))/(c*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3*(c/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 1/(c*x^5*
(sqrt(-1/(c^2*x^2) + 1) + 1)^5))) - 48*b*d^2*arcsin(1/(c*x))/(x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*(c/(x^3*(sqrt
(-1/(c^2*x^2) + 1) + 1)^3) + 1/(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5))) - 48*a*d^2/(x^4*(sqrt(-1/(c^2*x^2) + 1
) + 1)^4*(c/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 1/(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5))) + 4*b*e^2*log(sq
rt(-1/(c^2*x^2) + 1) + 1)/(c^3*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3*(c/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 1/
(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5))) - 4*b*e^2*log(1/(abs(c)*abs(x)))/(c^3*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1
)^3*(c/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 1/(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5))) + b*e^2/(c^3*x^3*(sqr
t(-1/(c^2*x^2) + 1) + 1)^3*(c/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 1/(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5))
) + 48*b*d*e*arcsin(1/(c*x))/(c^2*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*(c/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) +
1/(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5))) + 48*a*d*e/(c^2*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*(c/(x^3*(sqrt(-
1/(c^2*x^2) + 1) + 1)^3) + 1/(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5))) + 24*b*d^2/(c*x^5*(sqrt(-1/(c^2*x^2) + 1
) + 1)^5*(c/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 1/(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5))) + 6*b*e^2*arcsin
(1/(c*x))/(c^4*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*(c/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 1/(c*x^5*(sqrt(-1/
(c^2*x^2) + 1) + 1)^5))) + 6*a*e^2/(c^4*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*(c/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1
)^3) + 1/(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5))) + 48*b*d*e*log(sqrt(-1/(c^2*x^2) + 1) + 1)/(c^3*x^5*(sqrt(-1
/(c^2*x^2) + 1) + 1)^5*(c/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 1/(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5))) -
48*b*d*e*log(1/(abs(c)*abs(x)))/(c^3*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5*(c/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3
) + 1/(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5))) + 4*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^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 1/(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5))) - 4*b*
e^2*log(1/(abs(c)*abs(x)))/(c^5*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5*(c/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 1
/(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5))) - b*e^2/(c^5*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5*(c/(x^3*(sqrt(-1/(c^
2*x^2) + 1) + 1)^3) + 1/(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5))) + 24*b*d*e*arcsin(1/(c*x))/(c^4*x^6*(sqrt(-1/
(c^2*x^2) + 1) + 1)^6*(c/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 1/(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5))) + 2
4*a*d*e/(c^4*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6*(c/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 1/(c*x^5*(sqrt(-1/(c
^2*x^2) + 1) + 1)^5))) + 4*b*e^2*arcsin(1/(c*x))/(c^6*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1)^6*(c/(x^3*(sqrt(-1/(c^2
*x^2) + 1) + 1)^3) + 1/(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5))) + 4*a*e^2/(c^6*x^6*(sqrt(-1/(c^2*x^2) + 1) + 1
)^6*(c/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 1/(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5))) - b*e^2/(c^7*x^7*(sqr
t(-1/(c^2*x^2) + 1) + 1)^7*(c/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 1/(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5))
) + b*e^2*arcsin(1/(c*x))/(c^8*x^8*(sqrt(-1/(c^2*x^2) + 1) + 1)^8*(c/(x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 1/
(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5))) + a*e^2/(c^8*x^8*(sqrt(-1/(c^2*x^2) + 1) + 1)^8*(c/(x^3*(sqrt(-1/(c^2
*x^2) + 1) + 1)^3) + 1/(c*x^5*(sqrt(-1/(c^2*x^2) + 1) + 1)^5))))*c

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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