3.4.24 \(\int (a-i b \text {ArcSin}(1+i d x^2)) \, dx\) [324]

Optimal. Leaf size=50 \[ a x-\frac {2 b \sqrt {-2 i d x^2+d^2 x^4}}{d x}-i b x \text {ArcSin}\left (1+i d x^2\right ) \]

[Out]

a*x-I*b*x*arcsin(1+I*d*x^2)-2*b*(-2*I*d*x^2+d^2*x^4)^(1/2)/d/x

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Rubi [A]
time = 0.03, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4924, 12, 1602} \begin {gather*} a x-i b x \text {ArcSin}\left (1+i d x^2\right )-\frac {2 b \sqrt {d^2 x^4-2 i d x^2}}{d x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[a - I*b*ArcSin[1 + I*d*x^2],x]

[Out]

a*x - (2*b*Sqrt[(-2*I)*d*x^2 + d^2*x^4])/(d*x) - I*b*x*ArcSin[1 + I*d*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 1602

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[Coeff[Pp, x, p]*x^(p - q +
1)*(Qq^(m + 1)/((p + m*q + 1)*Coeff[Qq, x, q])), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rule 4924

Int[ArcSin[u_], x_Symbol] :> Simp[x*ArcSin[u], x] - Int[SimplifyIntegrand[x*(D[u, x]/Sqrt[1 - u^2]), x], x] /;
InverseFunctionFreeQ[u, x] &&  !FunctionOfExponentialQ[u, x]

Rubi steps

\begin {align*} \int \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right ) \, dx &=a x-(i b) \int \sin ^{-1}\left (1+i d x^2\right ) \, dx\\ &=a x-i b x \sin ^{-1}\left (1+i d x^2\right )+(i b) \int \frac {2 i d x^2}{\sqrt {-2 i d x^2+d^2 x^4}} \, dx\\ &=a x-i b x \sin ^{-1}\left (1+i d x^2\right )-(2 b d) \int \frac {x^2}{\sqrt {-2 i d x^2+d^2 x^4}} \, dx\\ &=a x-\frac {2 b \sqrt {-2 i d x^2+d^2 x^4}}{d x}-i b x \sin ^{-1}\left (1+i d x^2\right )\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 48, normalized size = 0.96 \begin {gather*} a x-\frac {2 b \sqrt {d x^2 \left (-2 i+d x^2\right )}}{d x}-i b x \text {ArcSin}\left (1+i d x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[a - I*b*ArcSin[1 + I*d*x^2],x]

[Out]

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

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Maple [A]
time = 0.25, size = 48, normalized size = 0.96

method	result	size
default	\(a x +b \left (x \arcsinh \left (d \,x^{2}-i\right )+\frac {2 x \left (-d \,x^{2}+2 i\right )}{\sqrt {d^{2} x^{4}-2 i d \,x^{2}}}\right )\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+b*arcsinh(-I+d*x^2),x,method=_RETURNVERBOSE)

[Out]

a*x+b*(x*arcsinh(-I+d*x^2)+2/(-2*I*d*x^2+d^2*x^4)^(1/2)*x*(-d*x^2+2*I))

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Maxima [A]
time = 0.27, size = 44, normalized size = 0.88 \begin {gather*} {\left (x \operatorname {arsinh}\left (d x^{2} - i\right ) - \frac {2 \, {\left (d^{\frac {3}{2}} x^{2} - 2 i \, \sqrt {d}\right )}}{\sqrt {d x^{2} - 2 i} d}\right )} b + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*arcsinh(-I+d*x^2),x, algorithm="maxima")

[Out]

(x*arcsinh(d*x^2 - I) - 2*(d^(3/2)*x^2 - 2*I*sqrt(d))/(sqrt(d*x^2 - 2*I)*d))*b + a*x

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Fricas [A]
time = 0.38, size = 52, normalized size = 1.04 \begin {gather*} \frac {b d x \log \left (d x^{2} + \sqrt {d^{2} x^{2} - 2 i \, d} x - i\right ) + a d x - 2 \, \sqrt {d^{2} x^{2} - 2 i \, d} b}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*arcsinh(-I+d*x^2),x, algorithm="fricas")

[Out]

(b*d*x*log(d*x^2 + sqrt(d^2*x^2 - 2*I*d)*x - I) + a*d*x - 2*sqrt(d^2*x^2 - 2*I*d)*b)/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*asinh(-I+d*x**2),x)

[Out]

Exception raised: TypeError >> Invalid comparison of non-real -I

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*arcsinh(-I+d*x^2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

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Mupad [B]
time = 0.45, size = 39, normalized size = 0.78 \begin {gather*} a\,x+b\,x\,\mathrm {asinh}\left (d\,x^2-\mathrm {i}\right )-\frac {2\,b\,\sqrt {{\left (d\,x^2-\mathrm {i}\right )}^2+1}}{d\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*asinh(d*x^2 - 1i),x)

[Out]

a*x + b*x*asinh(d*x^2 - 1i) - (2*b*((d*x^2 - 1i)^2 + 1)^(1/2))/(d*x)

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