3.45 \(\int (d+e x)^2 (a+b \csc ^{-1}(c x)) \, dx\)

Optimal. Leaf size=123 \[ \frac {(d+e x)^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {b d e x \sqrt {1-\frac {1}{c^2 x^2}}}{c}+\frac {b e^2 x^2 \sqrt {1-\frac {1}{c^2 x^2}}}{6 c}+\frac {b \left (6 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{6 c^3}-\frac {b d^3 \csc ^{-1}(c x)}{3 e} \]

[Out]

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

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Rubi [A]  time = 0.28, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {5227, 1568, 1475, 1807, 844, 216, 266, 63, 208} \[ \frac {(d+e x)^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {b \left (6 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{6 c^3}+\frac {b d e x \sqrt {1-\frac {1}{c^2 x^2}}}{c}+\frac {b e^2 x^2 \sqrt {1-\frac {1}{c^2 x^2}}}{6 c}-\frac {b d^3 \csc ^{-1}(c x)}{3 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rule 216

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

Rule 266

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 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 1475

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x
] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1568

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^(mn_.))^(q_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> Int[x^(m + mn*q
)*(e + d/x^mn)^q*(a + c*x^n2)^p, x] /; FreeQ[{a, c, d, e, m, mn, p}, x] && EqQ[n2, -2*mn] && IntegerQ[q] && (P
osQ[n2] ||  !IntegerQ[p])

Rule 1807

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, c*x, x],
R = PolynomialRemainder[Pq, c*x, x]}, Simp[(R*(c*x)^(m + 1)*(a + b*x^2)^(p + 1))/(a*c*(m + 1)), x] + Dist[1/(
a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

Rule 5227

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(a + b
*ArcCsc[c*x]))/(e*(m + 1)), x] + Dist[b/(c*e*(m + 1)), Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x],
x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.19, size = 122, normalized size = 0.99 \[ \frac {c^2 x \left (2 a c \left (3 d^2+3 d e x+e^2 x^2\right )+b e \sqrt {1-\frac {1}{c^2 x^2}} (6 d+e x)\right )+2 b c^3 x \csc ^{-1}(c x) \left (3 d^2+3 d e x+e^2 x^2\right )+b \left (6 c^2 d^2+e^2\right ) \log \left (x \left (\sqrt {1-\frac {1}{c^2 x^2}}+1\right )\right )}{6 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.55, size = 209, normalized size = 1.70 \[ \frac {2 \, a c^{3} e^{2} x^{3} + 6 \, a c^{3} d e x^{2} + 6 \, a c^{3} d^{2} x + 2 \, {\left (b c^{3} e^{2} x^{3} + 3 \, b c^{3} d e x^{2} + 3 \, b c^{3} d^{2} x - 3 \, b c^{3} d^{2} - 3 \, b c^{3} d e - b c^{3} e^{2}\right )} \operatorname {arccsc}\left (c x\right ) - 4 \, {\left (3 \, b c^{3} d^{2} + 3 \, b c^{3} d e + b c^{3} e^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (6 \, b c^{2} d^{2} + b e^{2}\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (b c e^{2} x + 6 \, b c d e\right )} \sqrt {c^{2} x^{2} - 1}}{6 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 2.93, size = 595, normalized size = 4.84 \[ \frac {1}{24} \, {\left (\frac {b x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3} \arcsin \left (\frac {1}{c x}\right ) e^{2}}{c} + \frac {a x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3} e^{2}}{c} - \frac {24 \, b d x^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right ) e}{c} + \frac {12 \, b d^{2} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c} - \frac {24 \, a d x^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} e}{c} + \frac {12 \, a d^{2} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c} + \frac {b x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2} e^{2}}{c^{2}} + \frac {24 \, b d x \sqrt {-\frac {1}{c^{2} x^{2}} + 1} e}{c^{2}} + \frac {3 \, b x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} \arcsin \left (\frac {1}{c x}\right ) e^{2}}{c^{3}} + \frac {24 \, b d^{2} \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac {24 \, b d^{2} \log \left (\frac {1}{{\left | c \right |} {\left | x \right |}}\right )}{c^{2}} + \frac {3 \, a x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} e^{2}}{c^{3}} + \frac {24 \, b d \arcsin \left (\frac {1}{c x}\right ) e}{c^{3}} + \frac {24 \, a d e}{c^{3}} + \frac {12 \, b d^{2} \arcsin \left (\frac {1}{c x}\right )}{c^{3} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {4 \, b e^{2} \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} - \frac {4 \, b e^{2} \log \left (\frac {1}{{\left | c \right |} {\left | x \right |}}\right )}{c^{4}} + \frac {12 \, a d^{2}}{c^{3} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {3 \, b \arcsin \left (\frac {1}{c x}\right ) e^{2}}{c^{5} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {3 \, a e^{2}}{c^{5} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} - \frac {b e^{2}}{c^{6} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {b \arcsin \left (\frac {1}{c x}\right ) e^{2}}{c^{7} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} + \frac {a e^{2}}{c^{7} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}\right )} c \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(b*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3*arcsin(1/(c*x))*e^2/c + a*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3*e^2/c
- 24*b*d*x^2*(1/(c^2*x^2) - 1)*arcsin(1/(c*x))*e/c + 12*b*d^2*x*(sqrt(-1/(c^2*x^2) + 1) + 1)*arcsin(1/(c*x))/c
- 24*a*d*x^2*(1/(c^2*x^2) - 1)*e/c + 12*a*d^2*x*(sqrt(-1/(c^2*x^2) + 1) + 1)/c + b*x^2*(sqrt(-1/(c^2*x^2) + 1
) + 1)^2*e^2/c^2 + 24*b*d*x*sqrt(-1/(c^2*x^2) + 1)*e/c^2 + 3*b*x*(sqrt(-1/(c^2*x^2) + 1) + 1)*arcsin(1/(c*x))*
e^2/c^3 + 24*b*d^2*log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^2 - 24*b*d^2*log(1/(abs(c)*abs(x)))/c^2 + 3*a*x*(sqrt(-1/
(c^2*x^2) + 1) + 1)*e^2/c^3 + 24*b*d*arcsin(1/(c*x))*e/c^3 + 24*a*d*e/c^3 + 12*b*d^2*arcsin(1/(c*x))/(c^3*x*(s
qrt(-1/(c^2*x^2) + 1) + 1)) + 4*b*e^2*log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^4 - 4*b*e^2*log(1/(abs(c)*abs(x)))/c^4
+ 12*a*d^2/(c^3*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 3*b*arcsin(1/(c*x))*e^2/(c^5*x*(sqrt(-1/(c^2*x^2) + 1) + 1)
) + 3*a*e^2/(c^5*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) - b*e^2/(c^6*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) + b*arcsin(1
/(c*x))*e^2/(c^7*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + a*e^2/(c^7*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3))*c

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maple [B]  time = 0.05, size = 361, normalized size = 2.93 \[ \frac {a \,e^{2} x^{3}}{3}+a e d \,x^{2}+a x \,d^{2}+\frac {a \,d^{3}}{3 e}+\frac {b \,e^{2} \mathrm {arccsc}\left (c x \right ) x^{3}}{3}+b e \,\mathrm {arccsc}\left (c x \right ) x^{2} d +b \,\mathrm {arccsc}\left (c x \right ) x \,d^{2}+\frac {b \,d^{3} \mathrm {arccsc}\left (c x \right )}{3 e}-\frac {b \sqrt {c^{2} x^{2}-1}\, d^{3} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{3 c e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \sqrt {c^{2} x^{2}-1}\, d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \,e^{2} x^{2}}{6 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \,e^{2}}{6 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b e x d}{c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b e d}{c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \,e^{2} \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(a+b*arccsc(c*x)),x)

[Out]

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

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maxima [A]  time = 0.37, size = 198, normalized size = 1.61 \[ \frac {1}{3} \, a e^{2} x^{3} + a d e x^{2} + {\left (x^{2} \operatorname {arccsc}\left (c x\right ) + \frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} b d e + \frac {1}{12} \, {\left (4 \, x^{3} \operatorname {arccsc}\left (c x\right ) + \frac {\frac {2 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b e^{2} + a d^{2} x + \frac {{\left (2 \, c x \operatorname {arccsc}\left (c x\right ) + \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )\right )} b d^{2}}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 7.03, size = 228, normalized size = 1.85 \[ a d^{2} x + a d e x^{2} + \frac {a e^{2} x^{3}}{3} + b d^{2} x \operatorname {acsc}{\left (c x \right )} + b d e x^{2} \operatorname {acsc}{\left (c x \right )} + \frac {b e^{2} x^{3} \operatorname {acsc}{\left (c x \right )}}{3} + \frac {b d^{2} \left (\begin {cases} \operatorname {acosh}{\left (c x \right )} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- i \operatorname {asin}{\left (c x \right )} & \text {otherwise} \end {cases}\right )}{c} + \frac {b d e \left (\begin {cases} \frac {\sqrt {c^{2} x^{2} - 1}}{c} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i \sqrt {- c^{2} x^{2} + 1}}{c} & \text {otherwise} \end {cases}\right )}{c} + \frac {b e^{2} \left (\begin {cases} \frac {x \sqrt {c^{2} x^{2} - 1}}{2 c} + \frac {\operatorname {acosh}{\left (c x \right )}}{2 c^{2}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{3}}{2 \sqrt {- c^{2} x^{2} + 1}} + \frac {i x}{2 c \sqrt {- c^{2} x^{2} + 1}} - \frac {i \operatorname {asin}{\left (c x \right )}}{2 c^{2}} & \text {otherwise} \end {cases}\right )}{3 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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