3.1.44 \(\int (d+e x)^3 (a+b \csc ^{-1}(c x)) \, dx\) [44]

Optimal. Leaf size=167 \[ \frac {b e \left (9 c^2 d^2+e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b d e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{2 c}+\frac {b e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}-\frac {b d^4 \csc ^{-1}(c x)}{4 e}+\frac {(d+e x)^4 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac {b d \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{2 c^3} \]

[Out]

-1/4*b*d^4*arccsc(c*x)/e+1/4*(e*x+d)^4*(a+b*arccsc(c*x))/e+1/2*b*d*(2*c^2*d^2+e^2)*arctanh((1-1/c^2/x^2)^(1/2)
)/c^3+1/6*b*e*(9*c^2*d^2+e^2)*x*(1-1/c^2/x^2)^(1/2)/c^3+1/2*b*d*e^2*x^2*(1-1/c^2/x^2)^(1/2)/c+1/12*b*e^3*x^3*(
1-1/c^2/x^2)^(1/2)/c

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Rubi [A]
time = 0.30, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {5335, 1582, 1489, 1821, 858, 222, 272, 65, 214} \begin {gather*} \frac {(d+e x)^4 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac {b d e^2 x^2 \sqrt {1-\frac {1}{c^2 x^2}}}{2 c}+\frac {b e^3 x^3 \sqrt {1-\frac {1}{c^2 x^2}}}{12 c}+\frac {b e x \sqrt {1-\frac {1}{c^2 x^2}} \left (9 c^2 d^2+e^2\right )}{6 c^3}+\frac {b d \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{2 c^3}-\frac {b d^4 \csc ^{-1}(c x)}{4 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*e*(9*c^2*d^2 + e^2)*Sqrt[1 - 1/(c^2*x^2)]*x)/(6*c^3) + (b*d*e^2*Sqrt[1 - 1/(c^2*x^2)]*x^2)/(2*c) + (b*e^3*S
qrt[1 - 1/(c^2*x^2)]*x^3)/(12*c) - (b*d^4*ArcCsc[c*x])/(4*e) + ((d + e*x)^4*(a + b*ArcCsc[c*x]))/(4*e) + (b*d*
(2*c^2*d^2 + e^2)*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/(2*c^3)

Rule 65

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 214

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

Rule 222

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 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 858

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 1489

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 1582

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 1821

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 5335

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)^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac {(d+e x)^4 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac {b \int \frac {(d+e x)^4}{\sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{4 c e}\\ &=\frac {(d+e x)^4 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac {b \int \frac {\left (e+\frac {d}{x}\right )^4 x^2}{\sqrt {1-\frac {1}{c^2 x^2}}} \, dx}{4 c e}\\ &=\frac {(d+e x)^4 \left (a+b \csc ^{-1}(c x)\right )}{4 e}-\frac {b \text {Subst}\left (\int \frac {(e+d x)^4}{x^4 \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{4 c e}\\ &=\frac {b e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {(d+e x)^4 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac {b \text {Subst}\left (\int \frac {-12 d e^3-2 e^2 \left (9 d^2+\frac {e^2}{c^2}\right ) x-12 d^3 e x^2-3 d^4 x^3}{x^3 \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{12 c e}\\ &=\frac {b d e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{2 c}+\frac {b e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {(d+e x)^4 \left (a+b \csc ^{-1}(c x)\right )}{4 e}-\frac {b \text {Subst}\left (\int \frac {4 e^2 \left (9 d^2+\frac {e^2}{c^2}\right )+12 d e \left (2 d^2+\frac {e^2}{c^2}\right ) x+6 d^4 x^2}{x^2 \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{24 c e}\\ &=\frac {b e \left (9 c^2 d^2+e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b d e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{2 c}+\frac {b e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {(d+e x)^4 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac {b \text {Subst}\left (\int \frac {-12 d e \left (2 d^2+\frac {e^2}{c^2}\right )-6 d^4 x}{x \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{24 c e}\\ &=\frac {b e \left (9 c^2 d^2+e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b d e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{2 c}+\frac {b e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {(d+e x)^4 \left (a+b \csc ^{-1}(c x)\right )}{4 e}-\frac {\left (b d^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{4 c e}-\frac {\left (b d \left (2 c^2 d^2+e^2\right )\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c^3}\\ &=\frac {b e \left (9 c^2 d^2+e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b d e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{2 c}+\frac {b e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}-\frac {b d^4 \csc ^{-1}(c x)}{4 e}+\frac {(d+e x)^4 \left (a+b \csc ^{-1}(c x)\right )}{4 e}-\frac {\left (b d \left (2 c^2 d^2+e^2\right )\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{4 c^3}\\ &=\frac {b e \left (9 c^2 d^2+e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b d e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{2 c}+\frac {b e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}-\frac {b d^4 \csc ^{-1}(c x)}{4 e}+\frac {(d+e x)^4 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac {\left (b d \left (2 c^2 d^2+e^2\right )\right ) \text {Subst}\left (\int \frac {1}{c^2-c^2 x^2} \, dx,x,\sqrt {1-\frac {1}{c^2 x^2}}\right )}{2 c}\\ &=\frac {b e \left (9 c^2 d^2+e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b d e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{2 c}+\frac {b e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}-\frac {b d^4 \csc ^{-1}(c x)}{4 e}+\frac {(d+e x)^4 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac {b d \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{2 c^3}\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 165, normalized size = 0.99 \begin {gather*} \frac {3 a c^3 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+b e \sqrt {1-\frac {1}{c^2 x^2}} x \left (2 e^2+c^2 \left (18 d^2+6 d e x+e^2 x^2\right )\right )+3 b c^3 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right ) \csc ^{-1}(c x)+6 b d \left (2 c^2 d^2+e^2\right ) \log \left (\left (1+\sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{12 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a*c^3*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + b*e*Sqrt[1 - 1/(c^2*x^2)]*x*(2*e^2 + c^2*(18*d^2 + 6*
d*e*x + e^2*x^2)) + 3*b*c^3*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)*ArcCsc[c*x] + 6*b*d*(2*c^2*d^2 + e^2
)*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/(12*c^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(412\) vs. \(2(147)=294\).
time = 0.23, size = 413, normalized size = 2.47

method	result	size
derivativedivides	\(\frac {\frac {\left (e c x +c d \right )^{4} a}{4 c^{3} e}+\frac {b c \,\mathrm {arccsc}\left (c x \right ) d^{4}}{4 e}+b \,\mathrm {arccsc}\left (c x \right ) d^{3} c x +\frac {3 b c e \,\mathrm {arccsc}\left (c x \right ) d^{2} x^{2}}{2}+b c \,e^{2} \mathrm {arccsc}\left (c x \right ) d \,x^{3}+\frac {b c \,e^{3} \mathrm {arccsc}\left (c x \right ) x^{4}}{4}-\frac {b \sqrt {c^{2} x^{2}-1}\, d^{4} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{4 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \sqrt {c^{2} x^{2}-1}\, d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {3 b e \left (c^{2} x^{2}-1\right ) d^{2}}{2 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \,e^{2} \left (c^{2} x^{2}-1\right ) d}{2 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \,e^{3} \left (c^{2} x^{2}-1\right ) x}{12 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \,e^{2} \sqrt {c^{2} x^{2}-1}\, d \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{2 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \,e^{3} \left (c^{2} x^{2}-1\right )}{6 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c}\)	\(413\)
default	\(\frac {\frac {\left (e c x +c d \right )^{4} a}{4 c^{3} e}+\frac {b c \,\mathrm {arccsc}\left (c x \right ) d^{4}}{4 e}+b \,\mathrm {arccsc}\left (c x \right ) d^{3} c x +\frac {3 b c e \,\mathrm {arccsc}\left (c x \right ) d^{2} x^{2}}{2}+b c \,e^{2} \mathrm {arccsc}\left (c x \right ) d \,x^{3}+\frac {b c \,e^{3} \mathrm {arccsc}\left (c x \right ) x^{4}}{4}-\frac {b \sqrt {c^{2} x^{2}-1}\, d^{4} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{4 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \sqrt {c^{2} x^{2}-1}\, d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {3 b e \left (c^{2} x^{2}-1\right ) d^{2}}{2 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \,e^{2} \left (c^{2} x^{2}-1\right ) d}{2 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \,e^{3} \left (c^{2} x^{2}-1\right ) x}{12 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \,e^{2} \sqrt {c^{2} x^{2}-1}\, d \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{2 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \,e^{3} \left (c^{2} x^{2}-1\right )}{6 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c}\)	\(413\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 267, normalized size = 1.60 \begin {gather*} \frac {1}{4} \, a x^{4} e^{3} + a d x^{3} e^{2} + \frac {3}{2} \, a d^{2} x^{2} e + a d^{3} x + \frac {3}{2} \, {\left (x^{2} \operatorname {arccsc}\left (c x\right ) + \frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} b d^{2} e + \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^{3}}{2 \, c} + \frac {1}{4} \, {\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 d e^{2} + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arccsc}\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 e^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.47, size = 283, normalized size = 1.69 \begin {gather*} \frac {3 \, a c^{4} x^{4} e^{3} + 12 \, a c^{4} d x^{3} e^{2} + 18 \, a c^{4} d^{2} x^{2} e + 12 \, a c^{4} d^{3} x + 3 \, {\left (4 \, b c^{4} d^{3} x - 4 \, b c^{4} d^{3} + {\left (b c^{4} x^{4} - b c^{4}\right )} e^{3} + 4 \, {\left (b c^{4} d x^{3} - b c^{4} d\right )} e^{2} + 6 \, {\left (b c^{4} d^{2} x^{2} - b c^{4} d^{2}\right )} e\right )} \operatorname {arccsc}\left (c x\right ) - 6 \, {\left (4 \, b c^{4} d^{3} + 6 \, b c^{4} d^{2} e + 4 \, b c^{4} d e^{2} + b c^{4} e^{3}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 6 \, {\left (2 \, b c^{3} d^{3} + b c d e^{2}\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (6 \, b c^{2} d x e^{2} + 18 \, b c^{2} d^{2} e + {\left (b c^{2} x^{2} + 2 \, b\right )} e^{3}\right )} \sqrt {c^{2} x^{2} - 1}}{12 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*a*c^4*x^4*e^3 + 12*a*c^4*d*x^3*e^2 + 18*a*c^4*d^2*x^2*e + 12*a*c^4*d^3*x + 3*(4*b*c^4*d^3*x - 4*b*c^4*
d^3 + (b*c^4*x^4 - b*c^4)*e^3 + 4*(b*c^4*d*x^3 - b*c^4*d)*e^2 + 6*(b*c^4*d^2*x^2 - b*c^4*d^2)*e)*arccsc(c*x) -
6*(4*b*c^4*d^3 + 6*b*c^4*d^2*e + 4*b*c^4*d*e^2 + b*c^4*e^3)*arctan(-c*x + sqrt(c^2*x^2 - 1)) - 6*(2*b*c^3*d^3
+ b*c*d*e^2)*log(-c*x + sqrt(c^2*x^2 - 1)) + (6*b*c^2*d*x*e^2 + 18*b*c^2*d^2*e + (b*c^2*x^2 + 2*b)*e^3)*sqrt(
c^2*x^2 - 1))/c^4

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Sympy [A]
time = 5.99, size = 362, normalized size = 2.17 \begin {gather*} a d^{3} x + \frac {3 a d^{2} e x^{2}}{2} + a d e^{2} x^{3} + \frac {a e^{3} x^{4}}{4} + b d^{3} x \operatorname {acsc}{\left (c x \right )} + \frac {3 b d^{2} e x^{2} \operatorname {acsc}{\left (c x \right )}}{2} + b d e^{2} x^{3} \operatorname {acsc}{\left (c x \right )} + \frac {b e^{3} x^{4} \operatorname {acsc}{\left (c x \right )}}{4} + \frac {b d^{3} \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 {3 b d^{2} 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 )}{2 c} + \frac {b d 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 )}{c} + \frac {b e^{3} \left (\begin {cases} \frac {x^{2} \sqrt {c^{2} x^{2} - 1}}{3 c} + \frac {2 \sqrt {c^{2} x^{2} - 1}}{3 c^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {2 i \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} & \text {otherwise} \end {cases}\right )}{4 c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d**3*x + 3*a*d**2*e*x**2/2 + a*d*e**2*x**3 + a*e**3*x**4/4 + b*d**3*x*acsc(c*x) + 3*b*d**2*e*x**2*acsc(c*x)/
2 + b*d*e**2*x**3*acsc(c*x) + b*e**3*x**4*acsc(c*x)/4 + b*d**3*Piecewise((acosh(c*x), Abs(c**2*x**2) > 1), (-I
*asin(c*x), True))/c + 3*b*d**2*e*Piecewise((sqrt(c**2*x**2 - 1)/c, Abs(c**2*x**2) > 1), (I*sqrt(-c**2*x**2 +
1)/c, True))/(2*c) + b*d*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))/c + b*e
**3*Piecewise((x**2*sqrt(c**2*x**2 - 1)/(3*c) + 2*sqrt(c**2*x**2 - 1)/(3*c**3), Abs(c**2*x**2) > 1), (I*x**2*s
qrt(-c**2*x**2 + 1)/(3*c) + 2*I*sqrt(-c**2*x**2 + 1)/(3*c**3), True))/(4*c)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1130 vs. \(2 (147) = 294\).
time = 2.53, size = 1130, normalized size = 6.77 \begin {gather*} \frac {1}{192} \, {\left (\frac {3 \, b e^{3} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {3 \, a e^{3} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}}{c} + \frac {24 \, b d e^{2} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {24 \, a d e^{2} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}{c} + \frac {2 \, b e^{3} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}{c^{2}} + \frac {72 \, b d^{2} e x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {72 \, a d^{2} e x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c} + \frac {24 \, b d e^{2} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c^{2}} + \frac {96 \, b d^{3} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {12 \, b e^{3} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {96 \, a d^{3} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c} + \frac {12 \, a e^{3} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c^{3}} + \frac {144 \, b d^{2} e x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{2}} + \frac {72 \, b d e^{2} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {72 \, a d e^{2} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{3}} + \frac {192 \, b d^{3} \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac {192 \, b d^{3} \log \left (\frac {1}{{\left | c \right |} {\left | x \right |}}\right )}{c^{2}} + \frac {18 \, b e^{3} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{4}} + \frac {144 \, b d^{2} e \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {144 \, a d^{2} e}{c^{3}} + \frac {96 \, b d e^{2} \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} - \frac {96 \, b d e^{2} \log \left (\frac {1}{{\left | c \right |} {\left | x \right |}}\right )}{c^{4}} + \frac {18 \, b e^{3} \arcsin \left (\frac {1}{c x}\right )}{c^{5}} + \frac {96 \, b d^{3} \arcsin \left (\frac {1}{c x}\right )}{c^{3} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {18 \, a e^{3}}{c^{5}} + \frac {96 \, a d^{3}}{c^{3} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} - \frac {144 \, b d^{2} e}{c^{4} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {72 \, b d e^{2} \arcsin \left (\frac {1}{c x}\right )}{c^{5} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {72 \, a d e^{2}}{c^{5} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} - \frac {18 \, b e^{3}}{c^{6} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {72 \, b d^{2} e \arcsin \left (\frac {1}{c x}\right )}{c^{5} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {72 \, a d^{2} e}{c^{5} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} - \frac {24 \, b d e^{2}}{c^{6} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {12 \, b e^{3} \arcsin \left (\frac {1}{c x}\right )}{c^{7} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {12 \, a e^{3}}{c^{7} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {24 \, b d e^{2} \arcsin \left (\frac {1}{c x}\right )}{c^{7} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} + \frac {24 \, a d e^{2}}{c^{7} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} - \frac {2 \, b e^{3}}{c^{8} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} + \frac {3 \, b e^{3} \arcsin \left (\frac {1}{c x}\right )}{c^{9} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}} + \frac {3 \, a e^{3}}{c^{9} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}}\right )} c \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(3*b*e^3*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*arcsin(1/(c*x))/c + 3*a*e^3*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)
^4/c + 24*b*d*e^2*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3*arcsin(1/(c*x))/c + 24*a*d*e^2*x^3*(sqrt(-1/(c^2*x^2) + 1
) + 1)^3/c + 2*b*e^3*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3/c^2 + 72*b*d^2*e*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*ar
csin(1/(c*x))/c + 72*a*d^2*e*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2/c + 24*b*d*e^2*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1
)^2/c^2 + 96*b*d^3*x*(sqrt(-1/(c^2*x^2) + 1) + 1)*arcsin(1/(c*x))/c + 12*b*e^3*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1
)^2*arcsin(1/(c*x))/c^3 + 96*a*d^3*x*(sqrt(-1/(c^2*x^2) + 1) + 1)/c + 12*a*e^3*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1
)^2/c^3 + 144*b*d^2*e*x*(sqrt(-1/(c^2*x^2) + 1) + 1)/c^2 + 72*b*d*e^2*x*(sqrt(-1/(c^2*x^2) + 1) + 1)*arcsin(1/
(c*x))/c^3 + 72*a*d*e^2*x*(sqrt(-1/(c^2*x^2) + 1) + 1)/c^3 + 192*b*d^3*log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^2 - 1
92*b*d^3*log(1/(abs(c)*abs(x)))/c^2 + 18*b*e^3*x*(sqrt(-1/(c^2*x^2) + 1) + 1)/c^4 + 144*b*d^2*e*arcsin(1/(c*x)
)/c^3 + 144*a*d^2*e/c^3 + 96*b*d*e^2*log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^4 - 96*b*d*e^2*log(1/(abs(c)*abs(x)))/c
^4 + 18*b*e^3*arcsin(1/(c*x))/c^5 + 96*b*d^3*arcsin(1/(c*x))/(c^3*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 18*a*e^3/c
^5 + 96*a*d^3/(c^3*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) - 144*b*d^2*e/(c^4*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 72*b*d
*e^2*arcsin(1/(c*x))/(c^5*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 72*a*d*e^2/(c^5*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) -
18*b*e^3/(c^6*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 72*b*d^2*e*arcsin(1/(c*x))/(c^5*x^2*(sqrt(-1/(c^2*x^2) + 1) +
1)^2) + 72*a*d^2*e/(c^5*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) - 24*b*d*e^2/(c^6*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)
^2) + 12*b*e^3*arcsin(1/(c*x))/(c^7*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) + 12*a*e^3/(c^7*x^2*(sqrt(-1/(c^2*x^2)
+ 1) + 1)^2) + 24*b*d*e^2*arcsin(1/(c*x))/(c^7*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 24*a*d*e^2/(c^7*x^3*(sqr
t(-1/(c^2*x^2) + 1) + 1)^3) - 2*b*e^3/(c^8*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3*b*e^3*arcsin(1/(c*x))/(c^9*
x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4) + 3*a*e^3/(c^9*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4))*c

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )\,{\left (d+e\,x\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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