3.1.78 \(\int (d+e x^2) (a+b \csc ^{-1}(c x)) \, dx\) [78]

Optimal. Leaf size=109 \[ \frac {b e x^2 \sqrt {-1+c^2 x^2}}{6 c \sqrt {c^2 x^2}}+d x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {b \left (6 c^2 d+e\right ) x \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{6 c^2 \sqrt {c^2 x^2}} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5337, 12, 396, 223, 212} \begin {gather*} d x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {b x \left (6 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 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)*(a + b*ArcCsc[c*x]),x]

[Out]

(b*e*x^2*Sqrt[-1 + c^2*x^2])/(6*c*Sqrt[c^2*x^2]) + d*x*(a + b*ArcCsc[c*x]) + (e*x^3*(a + b*ArcCsc[c*x]))/3 + (
b*(6*c^2*d + e)*x*ArcTanh[(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 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 5337

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])

Rubi steps

\begin {align*} \int \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx &=d x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {3 d+e x^2}{3 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=d x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {3 d+e x^2}{\sqrt {-1+c^2 x^2}} \, dx}{3 \sqrt {c^2 x^2}}\\ &=\frac {b e x^2 \sqrt {-1+c^2 x^2}}{6 c \sqrt {c^2 x^2}}+d x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \csc ^{-1}(c x)\right )-\frac {\left (b \left (-6 c^2 d-e\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{6 c \sqrt {c^2 x^2}}\\ &=\frac {b e x^2 \sqrt {-1+c^2 x^2}}{6 c \sqrt {c^2 x^2}}+d x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \csc ^{-1}(c x)\right )-\frac {\left (b \left (-6 c^2 d-e\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 e x^2 \sqrt {-1+c^2 x^2}}{6 c \sqrt {c^2 x^2}}+d x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {b \left (6 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 = 0.19, size = 149, normalized size = 1.37 \begin {gather*} a d x+\frac {1}{3} a e x^3+\frac {b e x^2 \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}}{6 c}+b d x \csc ^{-1}(c x)+\frac {1}{3} b e x^3 \csc ^{-1}(c x)+\frac {b d \sqrt {1-\frac {1}{c^2 x^2}} x \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{\sqrt {-1+c^2 x^2}}+\frac {b e \log \left (x \left (1+\sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}\right )\right )}{6 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.27, size = 191, normalized size = 1.75

method	result	size
derivativedivides	\(\frac {\frac {a \left (d \,c^{3} x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+b \,\mathrm {arccsc}\left (c x \right ) d c x +\frac {b c \,\mathrm {arccsc}\left (c x \right ) e \,x^{3}}{3}+\frac {b \sqrt {c^{2} x^{2}-1}\, d \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {b \left (c^{2} x^{2}-1\right ) e}{6 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \sqrt {c^{2} x^{2}-1}\, e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c}\)	\(191\)
default	\(\frac {\frac {a \left (d \,c^{3} x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+b \,\mathrm {arccsc}\left (c x \right ) d c x +\frac {b c \,\mathrm {arccsc}\left (c x \right ) e \,x^{3}}{3}+\frac {b \sqrt {c^{2} x^{2}-1}\, d \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {b \left (c^{2} x^{2}-1\right ) e}{6 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \sqrt {c^{2} x^{2}-1}\, e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c}\)	\(191\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 155, normalized size = 1.42 \begin {gather*} \frac {1}{3} \, a x^{3} e + a d x + \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 + \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 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.43, size = 147, normalized size = 1.35 \begin {gather*} \frac {2 \, a c^{3} x^{3} e + 6 \, a c^{3} d x + \sqrt {c^{2} x^{2} - 1} b c x e + 2 \, {\left (3 \, b c^{3} d x - 3 \, b c^{3} d + {\left (b c^{3} x^{3} - b c^{3}\right )} e\right )} \operatorname {arccsc}\left (c x\right ) - 4 \, {\left (3 \, b c^{3} d + b c^{3} e\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (6 \, b c^{2} d + b e\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{6 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 4.56, size = 153, normalized size = 1.40 \begin {gather*} a d x + \frac {a e x^{3}}{3} + b d x \operatorname {acsc}{\left (c x \right )} + \frac {b e x^{3} \operatorname {acsc}{\left (c x \right )}}{3} + \frac {b d \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 \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)*(a+b*acsc(c*x)),x)

[Out]

a*d*x + a*e*x**3/3 + b*d*x*acsc(c*x) + b*e*x**3*acsc(c*x)/3 + b*d*Piecewise((acosh(c*x), Abs(c**2*x**2) > 1),
(-I*asin(c*x), True))/c + b*e*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. 473 vs. \(2 (95) = 190\).
time = 1.09, size = 473, normalized size = 4.34 \begin {gather*} \frac {1}{24} \, {\left (\frac {b e x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {a e x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}{c} + \frac {b e x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c^{2}} + \frac {12 \, b d x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {12 \, a d x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c} + \frac {3 \, b e x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {3 \, a e x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{3}} + \frac {24 \, b d \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac {24 \, b d \log \left (\frac {1}{{\left | c \right |} {\left | x \right |}}\right )}{c^{2}} + \frac {4 \, b e \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} - \frac {4 \, b e \log \left (\frac {1}{{\left | c \right |} {\left | x \right |}}\right )}{c^{4}} + \frac {12 \, b d \arcsin \left (\frac {1}{c x}\right )}{c^{3} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {12 \, a d}{c^{3} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {3 \, b e \arcsin \left (\frac {1}{c x}\right )}{c^{5} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {3 \, a e}{c^{5} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} - \frac {b e}{c^{6} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {b e \arcsin \left (\frac {1}{c x}\right )}{c^{7} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} + \frac {a e}{c^{7} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}\right )} c \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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