3.1.6 \(\int \frac {a+b \text {ArcSin}(c x)}{(d+e x)^2} \, dx\) [6]

Optimal. Leaf size=85 \[ -\frac {a+b \text {ArcSin}(c x)}{e (d+e x)}+\frac {b c \text {ArcTan}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e \sqrt {c^2 d^2-e^2}} \]

[Out]

(-a-b*arcsin(c*x))/e/(e*x+d)+b*c*arctan((c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2*x^2+1)^(1/2))/e/(c^2*d^2-e^2)^(1
/2)

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Rubi [A]
time = 0.04, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {4827, 739, 210} \begin {gather*} \frac {b c \text {ArcTan}\left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {a+b \text {ArcSin}(c x)}{e (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + b*ArcSin[c*x])/(e*(d + e*x))) + (b*c*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(e*
Sqrt[c^2*d^2 - e^2])

Rule 210

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

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 4827

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*ArcSin[c*x])^n/(e*(m + 1))), x] - Dist[b*c*(n/(e*(m + 1))), Int[(d + e*x)^(m + 1)*((a + b*ArcSin[c*x])^(
n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {a+b \sin ^{-1}(c x)}{(d+e x)^2} \, dx &=-\frac {a+b \sin ^{-1}(c x)}{e (d+e x)}+\frac {(b c) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{e}\\ &=-\frac {a+b \sin ^{-1}(c x)}{e (d+e x)}-\frac {(b c) \text {Subst}\left (\int \frac {1}{-c^2 d^2+e^2-x^2} \, dx,x,\frac {e+c^2 d x}{\sqrt {1-c^2 x^2}}\right )}{e}\\ &=-\frac {a+b \sin ^{-1}(c x)}{e (d+e x)}+\frac {b c \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e \sqrt {c^2 d^2-e^2}}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 83, normalized size = 0.98 \begin {gather*} \frac {-\frac {a+b \text {ArcSin}(c x)}{d+e x}+\frac {b c \text {ArcTan}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d^2-e^2}}}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-((a + b*ArcSin[c*x])/(d + e*x)) + (b*c*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/Sqrt[c
^2*d^2 - e^2])/e

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(200\) vs. \(2(81)=162\).
time = 0.82, size = 201, normalized size = 2.36

method	result	size
derivativedivides	\(\frac {-\frac {a \,c^{2}}{\left (c e x +d c \right ) e}-\frac {b \,c^{2} \arcsin \left (c x \right )}{\left (c e x +d c \right ) e}-\frac {b \,c^{2} \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e^{2} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}}{c}\)	\(201\)
default	\(\frac {-\frac {a \,c^{2}}{\left (c e x +d c \right ) e}-\frac {b \,c^{2} \arcsin \left (c x \right )}{\left (c e x +d c \right ) e}-\frac {b \,c^{2} \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e^{2} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}}{c}\)	\(201\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c*d-%e>0)', see `assume?` for
more details

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 166 vs. \(2 (77) = 154\).
time = 2.48, size = 359, normalized size = 4.22 \begin {gather*} \left [-\frac {2 \, a c^{2} d^{2} + \sqrt {-c^{2} d^{2} + e^{2}} {\left (b c x e + b c d\right )} \log \left (\frac {2 \, c^{4} d^{2} x^{2} + 2 \, c^{2} d x e - c^{2} d^{2} - 2 \, \sqrt {-c^{2} d^{2} + e^{2}} {\left (c^{2} d x + e\right )} \sqrt {-c^{2} x^{2} + 1} - {\left (c^{2} x^{2} - 2\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + 2 \, {\left (b c^{2} d^{2} - b e^{2}\right )} \arcsin \left (c x\right ) - 2 \, a e^{2}}{2 \, {\left (c^{2} d^{2} x e^{2} + c^{2} d^{3} e - x e^{4} - d e^{3}\right )}}, -\frac {a c^{2} d^{2} - \sqrt {c^{2} d^{2} - e^{2}} {\left (b c x e + b c d\right )} \arctan \left (-\frac {\sqrt {c^{2} d^{2} - e^{2}} {\left (c^{2} d x + e\right )} \sqrt {-c^{2} x^{2} + 1}}{c^{4} d^{2} x^{2} - c^{2} d^{2} - {\left (c^{2} x^{2} - 1\right )} e^{2}}\right ) + {\left (b c^{2} d^{2} - b e^{2}\right )} \arcsin \left (c x\right ) - a e^{2}}{c^{2} d^{2} x e^{2} + c^{2} d^{3} e - x e^{4} - d e^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(2*a*c^2*d^2 + sqrt(-c^2*d^2 + e^2)*(b*c*x*e + b*c*d)*log((2*c^4*d^2*x^2 + 2*c^2*d*x*e - c^2*d^2 - 2*sqr
t(-c^2*d^2 + e^2)*(c^2*d*x + e)*sqrt(-c^2*x^2 + 1) - (c^2*x^2 - 2)*e^2)/(x^2*e^2 + 2*d*x*e + d^2)) + 2*(b*c^2*
d^2 - b*e^2)*arcsin(c*x) - 2*a*e^2)/(c^2*d^2*x*e^2 + c^2*d^3*e - x*e^4 - d*e^3), -(a*c^2*d^2 - sqrt(c^2*d^2 -
e^2)*(b*c*x*e + b*c*d)*arctan(-sqrt(c^2*d^2 - e^2)*(c^2*d*x + e)*sqrt(-c^2*x^2 + 1)/(c^4*d^2*x^2 - c^2*d^2 - (
c^2*x^2 - 1)*e^2)) + (b*c^2*d^2 - b*e^2)*arcsin(c*x) - a*e^2)/(c^2*d^2*x*e^2 + c^2*d^3*e - x*e^4 - d*e^3)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \operatorname {asin}{\left (c x \right )}}{\left (d + e x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (79) = 158\).
time = 0.39, size = 200, normalized size = 2.35 \begin {gather*} -\frac {b e^{2} {\left (\frac {2 \, c^{2} \arctan \left (\frac {\frac {c d e {\left (\sqrt {-\frac {{\left (e x + d\right )}^{2} {\left (c - \frac {c d}{e x + d}\right )}^{2}}{e^{2}} + 1} - 1\right )}}{{\left (e x + d\right )} {\left (c - \frac {c d}{e x + d}\right )}} - e}{\sqrt {c^{2} d^{2} - e^{2}}}\right )}{\sqrt {c^{2} d^{2} - e^{2}} e^{3}} + \frac {c^{2} \arcsin \left (-\frac {c {\left (d - \frac {\frac {{\left (e x + d\right )} {\left (c - \frac {c d}{e x + d}\right )} e}{c} + d e}{e}\right )}}{e}\right )}{{\left ({\left (e x + d\right )} {\left (c - \frac {c d}{e x + d}\right )} + c d\right )} e^{3}}\right )}}{c} - \frac {a}{{\left (e x + d\right )} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*e^2*(2*c^2*arctan((c*d*e*(sqrt(-(e*x + d)^2*(c - c*d/(e*x + d))^2/e^2 + 1) - 1)/((e*x + d)*(c - c*d/(e*x +
d))) - e)/sqrt(c^2*d^2 - e^2))/(sqrt(c^2*d^2 - e^2)*e^3) + c^2*arcsin(-c*(d - ((e*x + d)*(c - c*d/(e*x + d))*e
/c + d*e)/e)/e)/(((e*x + d)*(c - c*d/(e*x + d)) + c*d)*e^3))/c - a/((e*x + d)*e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c*x))/(d + e*x)^2,x)

[Out]

int((a + b*asin(c*x))/(d + e*x)^2, x)

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