3.6 \(\int \frac{a+b \sin ^{-1}(c x)}{(d+e x)^2} \, dx\)

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

[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])

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Rubi [A]  time = 0.0534458, antiderivative size = 85, normalized size of antiderivative = 1., 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 = {4743, 725, 204} \[ \frac{b c \tan ^{-1}\left (\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right )}{e \sqrt{c^2 d^2-e^2}}-\frac{a+b \sin ^{-1}(c x)}{e (d+e x)} \]

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 4743

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]

Rule 725

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 204

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

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) \operatorname{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*}

Mathematica [A]  time = 0.150491, size = 83, normalized size = 0.98 \[ \frac{\frac{b c \tan ^{-1}\left (\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right )}{\sqrt{c^2 d^2-e^2}}-\frac{a+b \sin ^{-1}(c x)}{d+e x}}{e} \]

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]  time = 0.018, size = 191, normalized size = 2.3 \begin{align*} -{\frac{ca}{ \left ( ecx+dc \right ) e}}-{\frac{bc\arcsin \left ( cx \right ) }{ \left ( ecx+dc \right ) e}}-{\frac{bc}{{e}^{2}}\ln \left ({ \left ( -2\,{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}+2\,{\frac{dc}{e} \left ( cx+{\frac{dc}{e}} \right ) }+2\,\sqrt{-{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}}\sqrt{- \left ( cx+{\frac{dc}{e}} \right ) ^{2}+2\,{\frac{dc}{e} \left ( cx+{\frac{dc}{e}} \right ) }-{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}} \right ) \left ( cx+{\frac{dc}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c*a/(c*e*x+c*d)/e-c*b/(c*e*x+c*d)/e*arcsin(c*x)-c*b/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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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

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Fricas [B]  time = 2.62081, size = 738, normalized size = 8.68 \begin{align*} \left [-\frac{2 \, a c^{2} d^{2} - 2 \, a e^{2} + \sqrt{-c^{2} d^{2} + e^{2}}{\left (b c e x + b c d\right )} \log \left (\frac{2 \, c^{2} d e x - c^{2} d^{2} +{\left (2 \, c^{4} d^{2} - c^{2} e^{2}\right )} x^{2} - 2 \, \sqrt{-c^{2} d^{2} + e^{2}}{\left (c^{2} d x + e\right )} \sqrt{-c^{2} x^{2} + 1} + 2 \, e^{2}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \,{\left (b c^{2} d^{2} - b e^{2}\right )} \arcsin \left (c x\right )}{2 \,{\left (c^{2} d^{3} e - d e^{3} +{\left (c^{2} d^{2} e^{2} - e^{4}\right )} x\right )}}, -\frac{a c^{2} d^{2} - a e^{2} - \sqrt{c^{2} d^{2} - e^{2}}{\left (b c e x + 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^{2} d^{2} -{\left (c^{4} d^{2} - c^{2} e^{2}\right )} x^{2} - e^{2}}\right ) +{\left (b c^{2} d^{2} - b e^{2}\right )} \arcsin \left (c x\right )}{c^{2} d^{3} e - d e^{3} +{\left (c^{2} d^{2} e^{2} - e^{4}\right )} x}\right ] \end{align*}

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

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

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

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]

sage0*x