3.536 \(\int \sqrt{d+i c d x} \sqrt{f-i c f x} (a+b \sinh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=147 \[ \frac{\sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c \sqrt{c^2 x^2+1}}+\frac{1}{2} x \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c x^2 \sqrt{d+i c d x} \sqrt{f-i c f x}}{4 \sqrt{c^2 x^2+1}} \]

[Out]

-(b*c*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/(4*Sqrt[1 + c^2*x^2]) + (x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*
(a + b*ArcSinh[c*x]))/2 + (Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2)/(4*b*c*Sqrt[1 + c^2*x^2
])

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Rubi [A]  time = 0.195725, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {5712, 5682, 5675, 30} \[ \frac{\sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c \sqrt{c^2 x^2+1}}+\frac{1}{2} x \sqrt{d+i c d x} \sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c x^2 \sqrt{d+i c d x} \sqrt{f-i c f x}}{4 \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]),x]

[Out]

-(b*c*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/(4*Sqrt[1 + c^2*x^2]) + (x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*
(a + b*ArcSinh[c*x]))/2 + (Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2)/(4*b*c*Sqrt[1 + c^2*x^2
])

Rule 5712

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(q_), x_Symbol] :>
Dist[((d + e*x)^q*(f + g*x)^q)/(1 + c^2*x^2)^q, Int[(d + e*x)^(p - q)*(1 + c^2*x^2)^q*(a + b*ArcSinh[c*x])^n,
x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 + e^2, 0] && HalfIntegerQ[p,
q] && GeQ[p - q, 0]

Rule 5682

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(x*Sqrt[d + e*x^2]*
(a + b*ArcSinh[c*x])^n)/2, x] + (Dist[Sqrt[d + e*x^2]/(2*Sqrt[1 + c^2*x^2]), Int[(a + b*ArcSinh[c*x])^n/Sqrt[1
+ c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(2*Sqrt[1 + c^2*x^2]), Int[x*(a + b*ArcSinh[c*x])^(n - 1),
x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0]

Rule 5675

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSinh[c*x]
)^(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && GtQ[d, 0] && NeQ[n, -1
]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.567566, size = 233, normalized size = 1.59 \[ \frac{1}{2} a x \sqrt{i d (c x-i)} \sqrt{-i f (c x+i)}+\frac{a \sqrt{d} \sqrt{f} \log \left (c d f x+\sqrt{d} \sqrt{f} \sqrt{i d (c x-i)} \sqrt{-i f (c x+i)}\right )}{2 c}-\frac{b \sqrt{i (c d x-i d)} \sqrt{-i (c f x+i f)} \sqrt{-d f \left (c^2 x^2+1\right )} \left (\cosh \left (2 \sinh ^{-1}(c x)\right )-2 \sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)+\sinh \left (2 \sinh ^{-1}(c x)\right )\right )\right )}{8 c \sqrt{c^2 x^2+1} \sqrt{-(c d x-i d) (c f x+i f)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]),x]

[Out]

(a*x*Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)])/2 + (a*Sqrt[d]*Sqrt[f]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[I*
d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]])/(2*c) - (b*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f
*(1 + c^2*x^2))]*(Cosh[2*ArcSinh[c*x]] - 2*ArcSinh[c*x]*(ArcSinh[c*x] + Sinh[2*ArcSinh[c*x]])))/(8*c*Sqrt[-(((
-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2])

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Maple [F]  time = 0.252, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) \sqrt{d+icdx}\sqrt{f-icfx}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsinh(c*x))*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2),x)

[Out]

int((a+b*arcsinh(c*x))*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2),x)

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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*arcsinh(c*x))*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} b \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} a, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*b*log(c*x + sqrt(c^2*x^2 + 1)) + sqrt(I*c*d*x + d)*sqrt(-I*c*f*x
+ f)*a, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d \left (i c x + 1\right )} \sqrt{- f \left (i c x - 1\right )} \left (a + b \operatorname{asinh}{\left (c x \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asinh(c*x))*(d+I*c*d*x)**(1/2)*(f-I*c*f*x)**(1/2),x)

[Out]

Integral(sqrt(d*(I*c*x + 1))*sqrt(-f*(I*c*x - 1))*(a + b*asinh(c*x)), x)

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Giac [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*arcsinh(c*x))*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError