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

Optimal. Leaf size=191 \[ d^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac{b x \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{120 c^4 \sqrt{c^2 x^2}}+\frac{b e x^2 \sqrt{c^2 x^2-1} \left (40 c^2 d+9 e\right )}{120 c^3 \sqrt{c^2 x^2}}+\frac{b e^2 x^4 \sqrt{c^2 x^2-1}}{20 c \sqrt{c^2 x^2}} \]

[Out]

(b*e*(40*c^2*d + 9*e)*x^2*Sqrt[-1 + c^2*x^2])/(120*c^3*Sqrt[c^2*x^2]) + (b*e^2*x^4*Sqrt[-1 + c^2*x^2])/(20*c*S
qrt[c^2*x^2]) + d^2*x*(a + b*ArcCsc[c*x]) + (2*d*e*x^3*(a + b*ArcCsc[c*x]))/3 + (e^2*x^5*(a + b*ArcCsc[c*x]))/
5 + (b*(120*c^4*d^2 + 40*c^2*d*e + 9*e^2)*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(120*c^4*Sqrt[c^2*x^2])

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Rubi [A]  time = 0.114112, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {194, 5229, 12, 1159, 388, 217, 206} \[ d^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac{b x \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{120 c^4 \sqrt{c^2 x^2}}+\frac{b e x^2 \sqrt{c^2 x^2-1} \left (40 c^2 d+9 e\right )}{120 c^3 \sqrt{c^2 x^2}}+\frac{b e^2 x^4 \sqrt{c^2 x^2-1}}{20 c \sqrt{c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*e*(40*c^2*d + 9*e)*x^2*Sqrt[-1 + c^2*x^2])/(120*c^3*Sqrt[c^2*x^2]) + (b*e^2*x^4*Sqrt[-1 + c^2*x^2])/(20*c*S
qrt[c^2*x^2]) + d^2*x*(a + b*ArcCsc[c*x]) + (2*d*e*x^3*(a + b*ArcCsc[c*x]))/3 + (e^2*x^5*(a + b*ArcCsc[c*x]))/
5 + (b*(120*c^4*d^2 + 40*c^2*d*e + 9*e^2)*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(120*c^4*Sqrt[c^2*x^2])

Rule 194

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

Rule 5229

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^p, x]}, Dist[a + b*ArcCsc[c*x], u, x] + Dist[(b*c*x)/Sqrt[c^2*x^2], Int[SimplifyIntegrand[u/(x*Sqrt[c^2*x^2
- 1]), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1159

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

Rule 388

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 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 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])

Rubi steps

\begin{align*} \int \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx &=d^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac{(b c x) \int \frac{15 d^2+10 d e x^2+3 e^2 x^4}{15 \sqrt{-1+c^2 x^2}} \, dx}{\sqrt{c^2 x^2}}\\ &=d^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac{(b c x) \int \frac{15 d^2+10 d e x^2+3 e^2 x^4}{\sqrt{-1+c^2 x^2}} \, dx}{15 \sqrt{c^2 x^2}}\\ &=\frac{b e^2 x^4 \sqrt{-1+c^2 x^2}}{20 c \sqrt{c^2 x^2}}+d^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac{(b x) \int \frac{60 c^2 d^2+e \left (40 c^2 d+9 e\right ) x^2}{\sqrt{-1+c^2 x^2}} \, dx}{60 c \sqrt{c^2 x^2}}\\ &=\frac{b e \left (40 c^2 d+9 e\right ) x^2 \sqrt{-1+c^2 x^2}}{120 c^3 \sqrt{c^2 x^2}}+\frac{b e^2 x^4 \sqrt{-1+c^2 x^2}}{20 c \sqrt{c^2 x^2}}+d^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \csc ^{-1}(c x)\right )-\frac{\left (b \left (-120 c^4 d^2-e \left (40 c^2 d+9 e\right )\right ) x\right ) \int \frac{1}{\sqrt{-1+c^2 x^2}} \, dx}{120 c^3 \sqrt{c^2 x^2}}\\ &=\frac{b e \left (40 c^2 d+9 e\right ) x^2 \sqrt{-1+c^2 x^2}}{120 c^3 \sqrt{c^2 x^2}}+\frac{b e^2 x^4 \sqrt{-1+c^2 x^2}}{20 c \sqrt{c^2 x^2}}+d^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \csc ^{-1}(c x)\right )-\frac{\left (b \left (-120 c^4 d^2-e \left (40 c^2 d+9 e\right )\right ) x\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x^2} \, dx,x,\frac{x}{\sqrt{-1+c^2 x^2}}\right )}{120 c^3 \sqrt{c^2 x^2}}\\ &=\frac{b e \left (40 c^2 d+9 e\right ) x^2 \sqrt{-1+c^2 x^2}}{120 c^3 \sqrt{c^2 x^2}}+\frac{b e^2 x^4 \sqrt{-1+c^2 x^2}}{20 c \sqrt{c^2 x^2}}+d^2 x \left (a+b \csc ^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac{b \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) x \tanh ^{-1}\left (\frac{c x}{\sqrt{-1+c^2 x^2}}\right )}{120 c^4 \sqrt{c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.236115, size = 151, normalized size = 0.79 \[ \frac{c^2 x \left (8 a c^3 \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+b e x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 \left (40 d+6 e x^2\right )+9 e\right )\right )+b \left (120 c^4 d^2+40 c^2 d e+9 e^2\right ) \log \left (x \left (\sqrt{1-\frac{1}{c^2 x^2}}+1\right )\right )+8 b c^5 x \csc ^{-1}(c x) \left (15 d^2+10 d e x^2+3 e^2 x^4\right )}{120 c^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*x*(8*a*c^3*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) + b*e*Sqrt[1 - 1/(c^2*x^2)]*x*(9*e + c^2*(40*d + 6*e*x^2)))
+ 8*b*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4)*ArcCsc[c*x] + b*(120*c^4*d^2 + 40*c^2*d*e + 9*e^2)*Log[(1 + Sqrt
[1 - 1/(c^2*x^2)])*x])/(120*c^5)

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Maple [B]  time = 0.183, size = 371, normalized size = 1.9 \begin{align*}{\frac{a{e}^{2}{x}^{5}}{5}}+{\frac{2\,a{x}^{3}de}{3}}+ax{d}^{2}+{\frac{b{\rm arccsc} \left (cx\right ){e}^{2}{x}^{5}}{5}}+{\frac{2\,b{\rm arccsc} \left (cx\right ){x}^{3}de}{3}}+b{\rm arccsc} \left (cx\right )x{d}^{2}+{\frac{b{d}^{2}}{{c}^{2}x}\sqrt{{c}^{2}{x}^{2}-1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{b{x}^{4}{e}^{2}}{20\,c}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{b{x}^{2}{e}^{2}}{40\,{c}^{3}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{bed{x}^{2}}{3\,c}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{bed}{3\,{c}^{3}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{bed}{3\,{c}^{4}x}\sqrt{{c}^{2}{x}^{2}-1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{3\,b{e}^{2}}{40\,{c}^{5}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{3\,b{e}^{2}}{40\,{c}^{6}x}\sqrt{{c}^{2}{x}^{2}-1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a*e^2*x^5+2/3*a*x^3*d*e+a*x*d^2+1/5*b*arccsc(c*x)*e^2*x^5+2/3*b*arccsc(c*x)*x^3*d*e+b*arccsc(c*x)*x*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*d^2*ln(c*x+(c^2*x^2-1)^(1/2))+1/20/c*b/((c^2*x^2-1)/c^2/
x^2)^(1/2)*x^4*e^2+1/40/c^3*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*x^2*e^2+1/3/c*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*e*d*x^2-
1/3/c^3*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*e*d+1/3/c^4*b*(c^2*x^2-1)^(1/2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x*e*d*ln(c*x
+(c^2*x^2-1)^(1/2))-3/40/c^5*b/((c^2*x^2-1)/c^2/x^2)^(1/2)*e^2+3/40/c^6*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 = 0.968983, size = 400, normalized size = 2.09 \begin{align*} \frac{1}{5} \, a e^{2} x^{5} + \frac{2}{3} \, a d e x^{3} + \frac{1}{6} \,{\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 + \frac{1}{80} \,{\left (16 \, x^{5} \operatorname{arccsc}\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 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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a*e^2*x^5 + 2/3*a*d*e*x^3 + 1/6*(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 + 1/80*(16*x^5*arccsc
(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*e^2 + a*d
^2*x + 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^2/c

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Fricas [A]  time = 4.94779, size = 547, normalized size = 2.86 \begin{align*} \frac{24 \, a c^{5} e^{2} x^{5} + 80 \, a c^{5} d e x^{3} + 120 \, a c^{5} d^{2} x + 8 \,{\left (3 \, b c^{5} e^{2} x^{5} + 10 \, b c^{5} d e x^{3} + 15 \, b c^{5} d^{2} x - 15 \, b c^{5} d^{2} - 10 \, b c^{5} d e - 3 \, b c^{5} e^{2}\right )} \operatorname{arccsc}\left (c x\right ) - 16 \,{\left (15 \, b c^{5} d^{2} + 10 \, b c^{5} d e + 3 \, b c^{5} e^{2}\right )} \arctan \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (120 \, b c^{4} d^{2} + 40 \, b c^{2} d e + 9 \, b e^{2}\right )} \log \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) +{\left (6 \, b c^{3} e^{2} x^{3} +{\left (40 \, b c^{3} d e + 9 \, b c e^{2}\right )} x\right )} \sqrt{c^{2} x^{2} - 1}}{120 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(24*a*c^5*e^2*x^5 + 80*a*c^5*d*e*x^3 + 120*a*c^5*d^2*x + 8*(3*b*c^5*e^2*x^5 + 10*b*c^5*d*e*x^3 + 15*b*c^
5*d^2*x - 15*b*c^5*d^2 - 10*b*c^5*d*e - 3*b*c^5*e^2)*arccsc(c*x) - 16*(15*b*c^5*d^2 + 10*b*c^5*d*e + 3*b*c^5*e
^2)*arctan(-c*x + sqrt(c^2*x^2 - 1)) - (120*b*c^4*d^2 + 40*b*c^2*d*e + 9*b*e^2)*log(-c*x + sqrt(c^2*x^2 - 1))
+ (6*b*c^3*e^2*x^3 + (40*b*c^3*d*e + 9*b*c*e^2)*x)*sqrt(c^2*x^2 - 1))/c^5

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{acsc}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x^{2} + d\right )}^{2}{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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