3.2.95 \(\int \frac {(a+b \text {ArcSin}(c+d x))^2}{(c e+d e x)^3} \, dx\) [195]

Optimal. Leaf size=87 \[ -\frac {b \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))}{d e^3 (c+d x)}-\frac {(a+b \text {ArcSin}(c+d x))^2}{2 d e^3 (c+d x)^2}+\frac {b^2 \log (c+d x)}{d e^3} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4889, 12, 4723, 4771, 29} \begin {gather*} -\frac {b \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))}{d e^3 (c+d x)}-\frac {(a+b \text {ArcSin}(c+d x))^2}{2 d e^3 (c+d x)^2}+\frac {b^2 \log (c+d x)}{d e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c + d*x])^2/(c*e + d*e*x)^3,x]

[Out]

-((b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/(d*e^3*(c + d*x))) - (a + b*ArcSin[c + d*x])^2/(2*d*e^3*(c
+ d*x)^2) + (b^2*Log[c + d*x])/(d*e^3)

Rule 12

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 4723

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

Rule 4771

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(d*f*(m + 1))), x] - Dist[b*c*(n/(f*(m + 1)))*Simp[(d
+ e*x^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /;
FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]

Rule 4889

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]

Rubi steps

\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c+d x)\right )^2}{(c e+d e x)^3} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^2}{e^3 x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^2}{x^3} \, dx,x,c+d x\right )}{d e^3}\\ &=-\frac {\left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)^2}+\frac {b \text {Subst}\left (\int \frac {a+b \sin ^{-1}(x)}{x^2 \sqrt {1-x^2}} \, dx,x,c+d x\right )}{d e^3}\\ &=-\frac {b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{d e^3 (c+d x)}-\frac {\left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)^2}+\frac {b^2 \text {Subst}\left (\int \frac {1}{x} \, dx,x,c+d x\right )}{d e^3}\\ &=-\frac {b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{d e^3 (c+d x)}-\frac {\left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)^2}+\frac {b^2 \log (c+d x)}{d e^3}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 126, normalized size = 1.45 \begin {gather*} -\frac {a \left (a+2 b (c+d x) \sqrt {1-c^2-2 c d x-d^2 x^2}\right )+2 b \left (a+b (c+d x) \sqrt {1-c^2-2 c d x-d^2 x^2}\right ) \text {ArcSin}(c+d x)+b^2 \text {ArcSin}(c+d x)^2-2 b^2 (c+d x)^2 \log (c+d x)}{2 d e^3 (c+d x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSin[c + d*x])^2/(c*e + d*e*x)^3,x]

[Out]

-1/2*(a*(a + 2*b*(c + d*x)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]) + 2*b*(a + b*(c + d*x)*Sqrt[1 - c^2 - 2*c*d*x -
d^2*x^2])*ArcSin[c + d*x] + b^2*ArcSin[c + d*x]^2 - 2*b^2*(c + d*x)^2*Log[c + d*x])/(d*e^3*(c + d*x)^2)

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Maple [A]
time = 0.17, size = 136, normalized size = 1.56

method	result	size
derivativedivides	\(\frac {-\frac {a^{2}}{2 e^{3} \left (d x +c \right )^{2}}-\frac {b^{2} \arcsin \left (d x +c \right )^{2}}{2 e^{3} \left (d x +c \right )^{2}}-\frac {b^{2} \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{e^{3} \left (d x +c \right )}+\frac {b^{2} \ln \left (d x +c \right )}{e^{3}}+\frac {2 a b \left (-\frac {\arcsin \left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{2 \left (d x +c \right )}\right )}{e^{3}}}{d}\)	\(136\)
default	\(\frac {-\frac {a^{2}}{2 e^{3} \left (d x +c \right )^{2}}-\frac {b^{2} \arcsin \left (d x +c \right )^{2}}{2 e^{3} \left (d x +c \right )^{2}}-\frac {b^{2} \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{e^{3} \left (d x +c \right )}+\frac {b^{2} \ln \left (d x +c \right )}{e^{3}}+\frac {2 a b \left (-\frac {\arcsin \left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{2 \left (d x +c \right )}\right )}{e^{3}}}{d}\)	\(136\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(d*x+c))^2/(d*e*x+c*e)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (80) = 160\).
time = 0.50, size = 222, normalized size = 2.55 \begin {gather*} -{\left (\frac {\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} d \arcsin \left (d x + c\right )}{d^{3} x e^{3} + c d^{2} e^{3}} - \frac {e^{\left (-3\right )} \log \left (d x + c\right )}{d}\right )} b^{2} - a b {\left (\frac {\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} d}{d^{3} x e^{3} + c d^{2} e^{3}} + \frac {\arcsin \left (d x + c\right )}{d^{3} x^{2} e^{3} + 2 \, c d^{2} x e^{3} + c^{2} d e^{3}}\right )} - \frac {b^{2} \arcsin \left (d x + c\right )^{2}}{2 \, {\left (d^{3} x^{2} e^{3} + 2 \, c d^{2} x e^{3} + c^{2} d e^{3}\right )}} - \frac {a^{2}}{2 \, {\left (d^{3} x^{2} e^{3} + 2 \, c d^{2} x e^{3} + c^{2} d e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(d*x+c))^2/(d*e*x+c*e)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.62, size = 139, normalized size = 1.60 \begin {gather*} -\frac {{\left (b^{2} \arcsin \left (d x + c\right )^{2} + 2 \, a b \arcsin \left (d x + c\right ) + a^{2} - 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (d x + c\right ) + 2 \, {\left (a b d x + a b c + {\left (b^{2} d x + b^{2} c\right )} \arcsin \left (d x + c\right )\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}\right )} e^{\left (-3\right )}}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(d*x+c))^2/(d*e*x+c*e)^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a^{2}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx}{e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(d*x+c))**2/(d*e*x+c*e)**3,x)

[Out]

(Integral(a**2/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral(b**2*asin(c + d*x)**2/(c**3 + 3*
c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral(2*a*b*asin(c + d*x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 +
d**3*x**3), x))/e**3

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 510 vs. \(2 (83) = 166\).
time = 0.49, size = 510, normalized size = 5.86 \begin {gather*} -\frac {b^{2} \arcsin \left (d x + c\right )^{2}}{4 \, d e^{3}} - \frac {{\left (d x + c\right )}^{2} b^{2} \arcsin \left (d x + c\right )^{2}}{8 \, d e^{3} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2}} - \frac {b^{2} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2} \arcsin \left (d x + c\right )^{2}}{8 \, {\left (d x + c\right )}^{2} d e^{3}} - \frac {a b \arcsin \left (d x + c\right )}{2 \, d e^{3}} - \frac {{\left (d x + c\right )}^{2} a b \arcsin \left (d x + c\right )}{4 \, d e^{3} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2}} + \frac {{\left (d x + c\right )} b^{2} \arcsin \left (d x + c\right )}{2 \, d e^{3} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}} - \frac {b^{2} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )} \arcsin \left (d x + c\right )}{2 \, {\left (d x + c\right )} d e^{3}} - \frac {a b {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2} \arcsin \left (d x + c\right )}{4 \, {\left (d x + c\right )}^{2} d e^{3}} + \frac {2 \, b^{2} \log \left (2\right )}{d e^{3}} - \frac {b^{2} \log \left (2 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} + 2\right )}{d e^{3}} + \frac {b^{2} \log \left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}{d e^{3}} + \frac {b^{2} \log \left ({\left | d x + c \right |}\right )}{d e^{3}} - \frac {a^{2}}{4 \, d e^{3}} - \frac {{\left (d x + c\right )}^{2} a^{2}}{8 \, d e^{3} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2}} + \frac {{\left (d x + c\right )} a b}{2 \, d e^{3} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}} - \frac {a b {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}}{2 \, {\left (d x + c\right )} d e^{3}} - \frac {a^{2} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2}}{8 \, {\left (d x + c\right )}^{2} d e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(d*x+c))^2/(d*e*x+c*e)^3,x, algorithm="giac")

[Out]

-1/4*b^2*arcsin(d*x + c)^2/(d*e^3) - 1/8*(d*x + c)^2*b^2*arcsin(d*x + c)^2/(d*e^3*(sqrt(-(d*x + c)^2 + 1) + 1)
^2) - 1/8*b^2*(sqrt(-(d*x + c)^2 + 1) + 1)^2*arcsin(d*x + c)^2/((d*x + c)^2*d*e^3) - 1/2*a*b*arcsin(d*x + c)/(
d*e^3) - 1/4*(d*x + c)^2*a*b*arcsin(d*x + c)/(d*e^3*(sqrt(-(d*x + c)^2 + 1) + 1)^2) + 1/2*(d*x + c)*b^2*arcsin
(d*x + c)/(d*e^3*(sqrt(-(d*x + c)^2 + 1) + 1)) - 1/2*b^2*(sqrt(-(d*x + c)^2 + 1) + 1)*arcsin(d*x + c)/((d*x +
c)*d*e^3) - 1/4*a*b*(sqrt(-(d*x + c)^2 + 1) + 1)^2*arcsin(d*x + c)/((d*x + c)^2*d*e^3) + 2*b^2*log(2)/(d*e^3)
- b^2*log(2*sqrt(-(d*x + c)^2 + 1) + 2)/(d*e^3) + b^2*log(sqrt(-(d*x + c)^2 + 1) + 1)/(d*e^3) + b^2*log(abs(d*
x + c))/(d*e^3) - 1/4*a^2/(d*e^3) - 1/8*(d*x + c)^2*a^2/(d*e^3*(sqrt(-(d*x + c)^2 + 1) + 1)^2) + 1/2*(d*x + c)
*a*b/(d*e^3*(sqrt(-(d*x + c)^2 + 1) + 1)) - 1/2*a*b*(sqrt(-(d*x + c)^2 + 1) + 1)/((d*x + c)*d*e^3) - 1/8*a^2*(
sqrt(-(d*x + c)^2 + 1) + 1)^2/((d*x + c)^2*d*e^3)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^2}{{\left (c\,e+d\,e\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c + d*x))^2/(c*e + d*e*x)^3,x)

[Out]

int((a + b*asin(c + d*x))^2/(c*e + d*e*x)^3, x)

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