3.324 \(\int (a-i b \sin ^{-1}(1+i d x^2)) \, dx\)

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

[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]

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Rubi [A]  time = 0.0387746, antiderivative size = 50, normalized size of antiderivative = 1., 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 = {4840, 12, 1588} \[ a x-\frac{2 b \sqrt{d^2 x^4-2 i d x^2}}{d x}-i b x \sin ^{-1}\left (1+i d x^2\right ) \]

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 4840

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]

Rule 12

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

Rule 1588

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]

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*}

Mathematica [A]  time = 0.0267625, size = 48, normalized size = 0.96 \[ a x-\frac{2 b \sqrt{d x^2 \left (d x^2-2 i\right )}}{d x}-i b x \sin ^{-1}\left (1+i d x^2\right ) \]

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 = 0.015, size = 48, normalized size = 1. \begin{align*} ax+b \left ( x{\it Arcsinh} \left ( -i+d{x}^{2} \right ) +2\,{\frac{x \left ( -d{x}^{2}+2\,i \right ) }{\sqrt{-2\,id{x}^{2}+{d}^{2}{x}^{4}}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 1.19206, size = 59, normalized size = 1.18 \begin{align*}{\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{align*}

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 = 2.64874, size = 138, normalized size = 2.76 \begin{align*} \frac{b d x^{2} \log \left (d x^{2} + \sqrt{d^{2} x^{4} - 2 i \, d x^{2}} - i\right ) + a d x^{2} - 2 \, \sqrt{d^{2} x^{4} - 2 i \, d x^{2}} b}{d x} \end{align*}

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^2*log(d*x^2 + sqrt(d^2*x^4 - 2*I*d*x^2) - I) + a*d*x^2 - 2*sqrt(d^2*x^4 - 2*I*d*x^2)*b)/(d*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int b \operatorname{arsinh}\left (d x^{2} - i\right ) + a\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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