3.1.91 \(\int \frac {(f+g x) (a+b \text {ArcSin}(c x))}{d+e x} \, dx\) [91]

Optimal. Leaf size=344 \[ \frac {b g \sqrt {1-c^2 x^2}}{c e}-\frac {i b (e f-d g) \text {ArcSin}(c x)^2}{2 e^2}+\frac {g x (a+b \text {ArcSin}(c x))}{e}+\frac {b (e f-d g) \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}+\frac {b (e f-d g) \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2}-\frac {b (e f-d g) \text {ArcSin}(c x) \log (d+e x)}{e^2}+\frac {(e f-d g) (a+b \text {ArcSin}(c x)) \log (d+e x)}{e^2}-\frac {i b (e f-d g) \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}-\frac {i b (e f-d g) \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2} \]

[Out]

-1/2*I*b*(-d*g+e*f)*arcsin(c*x)^2/e^2+g*x*(a+b*arcsin(c*x))/e-b*(-d*g+e*f)*arcsin(c*x)*ln(e*x+d)/e^2+(-d*g+e*f
)*(a+b*arcsin(c*x))*ln(e*x+d)/e^2+b*(-d*g+e*f)*arcsin(c*x)*ln(1-I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d-(c^2*d^2-e
^2)^(1/2)))/e^2+b*(-d*g+e*f)*arcsin(c*x)*ln(1-I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e^2-I*
b*(-d*g+e*f)*polylog(2,I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e^2-I*b*(-d*g+e*f)*polylog(2,
I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e^2+b*g*(-c^2*x^2+1)^(1/2)/c/e

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Rubi [A]
time = 0.48, antiderivative size = 344, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 12, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {45, 4837, 12, 6874, 267, 222, 2451, 4825, 4615, 2221, 2317, 2438} \begin {gather*} \frac {(e f-d g) \log (d+e x) (a+b \text {ArcSin}(c x))}{e^2}+\frac {g x (a+b \text {ArcSin}(c x))}{e}-\frac {i b (e f-d g) \text {Li}_2\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}-\frac {i b (e f-d g) \text {Li}_2\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2}+\frac {b \text {ArcSin}(c x) (e f-d g) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}+\frac {b \text {ArcSin}(c x) (e f-d g) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e^2}-\frac {i b \text {ArcSin}(c x)^2 (e f-d g)}{2 e^2}-\frac {b \text {ArcSin}(c x) (e f-d g) \log (d+e x)}{e^2}+\frac {b g \sqrt {1-c^2 x^2}}{c e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*g*Sqrt[1 - c^2*x^2])/(c*e) - ((I/2)*b*(e*f - d*g)*ArcSin[c*x]^2)/e^2 + (g*x*(a + b*ArcSin[c*x]))/e + (b*(e*
f - d*g)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^2 + (b*(e*f - d*g)*ArcSin
[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^2 - (b*(e*f - d*g)*ArcSin[c*x]*Log[d + e
*x])/e^2 + ((e*f - d*g)*(a + b*ArcSin[c*x])*Log[d + e*x])/e^2 - (I*b*(e*f - d*g)*PolyLog[2, (I*e*E^(I*ArcSin[c
*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^2 - (I*b*(e*f - d*g)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*
d^2 - e^2])])/e^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 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2451

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/Sqrt[(f_) + (g_.)*(x_)^2], x_Symbol] :> With[{u = Int
Hide[1/Sqrt[f + g*x^2], x]}, Simp[u*(a + b*Log[c*(d + e*x)^n]), x] - Dist[b*e*n, Int[SimplifyIntegrand[u/(d +
e*x), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && GtQ[f, 0]

Rule 4615

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
Simp[(-I)*((e + f*x)^(m + 1)/(b*f*(m + 1))), x] + (Int[(e + f*x)^m*(E^(I*(c + d*x))/(a - Rt[a^2 - b^2, 2] - I*
b*E^(I*(c + d*x)))), x] + Int[(e + f*x)^m*(E^(I*(c + d*x))/(a + Rt[a^2 - b^2, 2] - I*b*E^(I*(c + d*x)))), x])
/; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && PosQ[a^2 - b^2]

Rule 4825

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Subst[Int[(a + b*x)^n*(Cos[x]/(
c*d + e*Sin[x])), x], x, ArcSin[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]

Rule 4837

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> With[{u = IntHide[Px*(d
+ e*x)^m, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]
] /; FreeQ[{a, b, c, d, e, m}, x] && PolynomialQ[Px, x]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {(f+g x) \left (a+b \sin ^{-1}(c x)\right )}{d+e x} \, dx &=\frac {g x \left (a+b \sin ^{-1}(c x)\right )}{e}+\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}-(b c) \int \frac {e g x+(e f-d g) \log (d+e x)}{e^2 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {g x \left (a+b \sin ^{-1}(c x)\right )}{e}+\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}-\frac {(b c) \int \frac {e g x+(e f-d g) \log (d+e x)}{\sqrt {1-c^2 x^2}} \, dx}{e^2}\\ &=\frac {g x \left (a+b \sin ^{-1}(c x)\right )}{e}+\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}-\frac {(b c) \int \left (\frac {e g x}{\sqrt {1-c^2 x^2}}+\frac {(e f-d g) \log (d+e x)}{\sqrt {1-c^2 x^2}}\right ) \, dx}{e^2}\\ &=\frac {g x \left (a+b \sin ^{-1}(c x)\right )}{e}+\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}-\frac {(b c g) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{e}-\frac {(b c (e f-d g)) \int \frac {\log (d+e x)}{\sqrt {1-c^2 x^2}} \, dx}{e^2}\\ &=\frac {b g \sqrt {1-c^2 x^2}}{c e}+\frac {g x \left (a+b \sin ^{-1}(c x)\right )}{e}-\frac {b (e f-d g) \sin ^{-1}(c x) \log (d+e x)}{e^2}+\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}+\frac {(b c (e f-d g)) \int \frac {\sin ^{-1}(c x)}{c d+c e x} \, dx}{e}\\ &=\frac {b g \sqrt {1-c^2 x^2}}{c e}+\frac {g x \left (a+b \sin ^{-1}(c x)\right )}{e}-\frac {b (e f-d g) \sin ^{-1}(c x) \log (d+e x)}{e^2}+\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}+\frac {(b c (e f-d g)) \text {Subst}\left (\int \frac {x \cos (x)}{c^2 d+c e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e}\\ &=\frac {b g \sqrt {1-c^2 x^2}}{c e}-\frac {i b (e f-d g) \sin ^{-1}(c x)^2}{2 e^2}+\frac {g x \left (a+b \sin ^{-1}(c x)\right )}{e}-\frac {b (e f-d g) \sin ^{-1}(c x) \log (d+e x)}{e^2}+\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}+\frac {(b c (e f-d g)) \text {Subst}\left (\int \frac {e^{i x} x}{c^2 d-c \sqrt {c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e}+\frac {(b c (e f-d g)) \text {Subst}\left (\int \frac {e^{i x} x}{c^2 d+c \sqrt {c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e}\\ &=\frac {b g \sqrt {1-c^2 x^2}}{c e}-\frac {i b (e f-d g) \sin ^{-1}(c x)^2}{2 e^2}+\frac {g x \left (a+b \sin ^{-1}(c x)\right )}{e}+\frac {b (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}+\frac {b (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2}-\frac {b (e f-d g) \sin ^{-1}(c x) \log (d+e x)}{e^2}+\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}-\frac {(b (e f-d g)) \text {Subst}\left (\int \log \left (1-\frac {i c e e^{i x}}{c^2 d-c \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^2}-\frac {(b (e f-d g)) \text {Subst}\left (\int \log \left (1-\frac {i c e e^{i x}}{c^2 d+c \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^2}\\ &=\frac {b g \sqrt {1-c^2 x^2}}{c e}-\frac {i b (e f-d g) \sin ^{-1}(c x)^2}{2 e^2}+\frac {g x \left (a+b \sin ^{-1}(c x)\right )}{e}+\frac {b (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}+\frac {b (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2}-\frac {b (e f-d g) \sin ^{-1}(c x) \log (d+e x)}{e^2}+\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}+\frac {(i b (e f-d g)) \text {Subst}\left (\int \frac {\log \left (1-\frac {i c e x}{c^2 d-c \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^2}+\frac {(i b (e f-d g)) \text {Subst}\left (\int \frac {\log \left (1-\frac {i c e x}{c^2 d+c \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^2}\\ &=\frac {b g \sqrt {1-c^2 x^2}}{c e}-\frac {i b (e f-d g) \sin ^{-1}(c x)^2}{2 e^2}+\frac {g x \left (a+b \sin ^{-1}(c x)\right )}{e}+\frac {b (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}+\frac {b (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2}-\frac {b (e f-d g) \sin ^{-1}(c x) \log (d+e x)}{e^2}+\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}-\frac {i b (e f-d g) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}-\frac {i b (e f-d g) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2}\\ \end {align*}

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Mathematica [A]
time = 0.87, size = 602, normalized size = 1.75 \begin {gather*} \frac {8 a c e g x+8 a c (e f-d g) \log (d+e x)+b g \left (8 e \sqrt {1-c^2 x^2}+8 c e x \text {ArcSin}(c x)-c d \left (i (\pi -2 \text {ArcSin}(c x))^2-32 i \text {ArcSin}\left (\frac {\sqrt {1+\frac {c d}{e}}}{\sqrt {2}}\right ) \text {ArcTan}\left (\frac {(c d-e) \cot \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )}{\sqrt {c^2 d^2-e^2}}\right )-4 \left (\pi +4 \text {ArcSin}\left (\frac {\sqrt {1+\frac {c d}{e}}}{\sqrt {2}}\right )-2 \text {ArcSin}(c x)\right ) \log \left (1-\frac {i \left (-c d+\sqrt {c^2 d^2-e^2}\right ) e^{-i \text {ArcSin}(c x)}}{e}\right )-4 \left (\pi -4 \text {ArcSin}\left (\frac {\sqrt {1+\frac {c d}{e}}}{\sqrt {2}}\right )-2 \text {ArcSin}(c x)\right ) \log \left (1+\frac {i \left (c d+\sqrt {c^2 d^2-e^2}\right ) e^{-i \text {ArcSin}(c x)}}{e}\right )+4 (\pi -2 \text {ArcSin}(c x)) \log (c (d+e x))+8 \text {ArcSin}(c x) \log (c (d+e x))+8 i \left (\text {PolyLog}\left (2,\frac {i \left (-c d+\sqrt {c^2 d^2-e^2}\right ) e^{-i \text {ArcSin}(c x)}}{e}\right )+\text {PolyLog}\left (2,-\frac {i \left (c d+\sqrt {c^2 d^2-e^2}\right ) e^{-i \text {ArcSin}(c x)}}{e}\right )\right )\right )\right )-4 i b c e f \left (\text {ArcSin}(c x) \left (\text {ArcSin}(c x)+2 i \left (\log \left (1+\frac {i e e^{i \text {ArcSin}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+\log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )\right )+2 \text {PolyLog}\left (2,-\frac {i e e^{i \text {ArcSin}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+2 \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{8 c e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(8*a*c*e*g*x + 8*a*c*(e*f - d*g)*Log[d + e*x] + b*g*(8*e*Sqrt[1 - c^2*x^2] + 8*c*e*x*ArcSin[c*x] - c*d*(I*(Pi
- 2*ArcSin[c*x])^2 - (32*I)*ArcSin[Sqrt[1 + (c*d)/e]/Sqrt[2]]*ArcTan[((c*d - e)*Cot[(Pi + 2*ArcSin[c*x])/4])/S
qrt[c^2*d^2 - e^2]] - 4*(Pi + 4*ArcSin[Sqrt[1 + (c*d)/e]/Sqrt[2]] - 2*ArcSin[c*x])*Log[1 - (I*(-(c*d) + Sqrt[c
^2*d^2 - e^2]))/(e*E^(I*ArcSin[c*x]))] - 4*(Pi - 4*ArcSin[Sqrt[1 + (c*d)/e]/Sqrt[2]] - 2*ArcSin[c*x])*Log[1 +
(I*(c*d + Sqrt[c^2*d^2 - e^2]))/(e*E^(I*ArcSin[c*x]))] + 4*(Pi - 2*ArcSin[c*x])*Log[c*(d + e*x)] + 8*ArcSin[c*
x]*Log[c*(d + e*x)] + (8*I)*(PolyLog[2, (I*(-(c*d) + Sqrt[c^2*d^2 - e^2]))/(e*E^(I*ArcSin[c*x]))] + PolyLog[2,
((-I)*(c*d + Sqrt[c^2*d^2 - e^2]))/(e*E^(I*ArcSin[c*x]))]))) - (4*I)*b*c*e*f*(ArcSin[c*x]*(ArcSin[c*x] + (2*I
)*(Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] + Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sq
rt[c^2*d^2 - e^2])])) + 2*PolyLog[2, ((-I)*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] + 2*PolyLog[2,
(I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]))/(8*c*e^2)

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1592 vs. \(2 (357 ) = 714\).
time = 1.32, size = 1593, normalized size = 4.63

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1593\)
default	\(\text {Expression too large to display}\)	\(1593\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(a*g/e*c*x-a*c/e^2*ln(c*e*x+c*d)*d*g+a*c/e*ln(c*e*x+c*d)*f-b*c^3/e^2*d^3*g*arcsin(c*x)/(c^2*d^2-e^2)*ln((I
*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))+b*c^3/e*d^2*f*arcsin(c*x
)/(c^2*d^2-e^2)*ln((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))+b*c
^3/e*d^2*f*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2
*d^2+e^2)^(1/2)))+I*b*c*e*f/(c^2*d^2-e^2)*dilog((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))+(-c^2*d^2+e^2)^(1/2))/(I*d
*c+(-c^2*d^2+e^2)^(1/2)))+b*g/e*(-c^2*x^2+1)^(1/2)-I*b*c^3/e*f/(c^2*d^2-e^2)*dilog((I*d*c+e*(I*c*x+(-c^2*x^2+1
)^(1/2))+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))*d^2+b*arcsin(c*x)*g/e*c*x-1/2*I*b*c*arcsin(c*x)^2
/e*f-I*b*c*d*g/(c^2*d^2-e^2)*dilog((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+
e^2)^(1/2)))-b*c*e*f*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))+(-c^2*d^2+e^2)^(1/2))/(I
*d*c+(-c^2*d^2+e^2)^(1/2)))+I*b*c^3/e^2*d^3*g/(c^2*d^2-e^2)*dilog((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))-(-c^2*d^
2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))-I*b*c*d*g/(c^2*d^2-e^2)*dilog((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))+
(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))-b*c^3/e^2*d^3*g*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+e*(I*c
*x+(-c^2*x^2+1)^(1/2))+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))+I*b*c^3/e^2*d^3*g/(c^2*d^2-e^2)*dil
og((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))+b*c*d*g*arcsin(c*x)
/(c^2*d^2-e^2)*ln((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))-b*c*
e*f*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^
2)^(1/2)))+I*b*c*e*f/(c^2*d^2-e^2)*dilog((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^
2*d^2+e^2)^(1/2)))+b*c*d*g*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+e*(I*c*x+(-c^2*x^2+1)^(1/2))-(-c^2*d^2+e^2)^(1/
2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))+1/2*I*b*c*arcsin(c*x)^2/e^2*d*g-I*b*c^3/e*f/(c^2*d^2-e^2)*dilog((I*d*c+e*(I*
c*x+(-c^2*x^2+1)^(1/2))-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))*d^2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*f*e^(-1)*log(x*e + d) - (d*e^(-2)*log(x*e + d) - x*e^(-1))*a*g + integrate((b*g*x + b*f)*arctan2(c*x, sqrt(c
*x + 1)*sqrt(-c*x + 1))/(x*e + d), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a*g*x + a*f + (b*g*x + b*f)*arcsin(c*x))/(x*e + d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((g*x + f)*(b*arcsin(c*x) + a)/(e*x + d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)*(a + b*asin(c*x)))/(d + e*x),x)

[Out]

int(((f + g*x)*(a + b*asin(c*x)))/(d + e*x), x)

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