3.589 \(\int \frac{(d+i c d x)^{3/2} (a+b \sinh ^{-1}(c x))^2}{\sqrt{f-i c f x}} \, dx\)

Optimal. Leaf size=436 \[ \frac{d^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{2 i d^2 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{d^2 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{b c d^2 x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{4 i b d^2 x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{4 i b^2 d^2 \left (c^2 x^2+1\right )}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{b^2 d^2 x \left (c^2 x^2+1\right )}{4 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{b^2 d^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{4 c \sqrt{d+i c d x} \sqrt{f-i c f x}} \]

[Out]

((4*I)*b^2*d^2*(1 + c^2*x^2))/(c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - (b^2*d^2*x*(1 + c^2*x^2))/(4*Sqrt[d +
I*c*d*x]*Sqrt[f - I*c*f*x]) + (b^2*d^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(4*c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x
]) - ((4*I)*b*d^2*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (b*c*d^2*x
^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + ((2*I)*d^2*(1 + c^2*x^2)*
(a + b*ArcSinh[c*x])^2)/(c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - (d^2*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)
/(2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^3)/(2*b*c*Sqrt[d + I*c*
d*x]*Sqrt[f - I*c*f*x])

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Rubi [A]  time = 0.668527, antiderivative size = 436, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.243, Rules used = {5712, 5831, 3317, 3296, 2638, 3311, 32, 2635, 8} \[ \frac{d^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{2 i d^2 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{d^2 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{b c d^2 x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{4 i b d^2 x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{4 i b^2 d^2 \left (c^2 x^2+1\right )}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{b^2 d^2 x \left (c^2 x^2+1\right )}{4 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{b^2 d^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{4 c \sqrt{d+i c d x} \sqrt{f-i c f x}} \]

Antiderivative was successfully verified.

[In]

Int[((d + I*c*d*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/Sqrt[f - I*c*f*x],x]

[Out]

((4*I)*b^2*d^2*(1 + c^2*x^2))/(c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - (b^2*d^2*x*(1 + c^2*x^2))/(4*Sqrt[d +
I*c*d*x]*Sqrt[f - I*c*f*x]) + (b^2*d^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(4*c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x
]) - ((4*I)*b*d^2*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (b*c*d^2*x
^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + ((2*I)*d^2*(1 + c^2*x^2)*
(a + b*ArcSinh[c*x])^2)/(c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - (d^2*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)
/(2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^3)/(2*b*c*Sqrt[d + I*c*
d*x]*Sqrt[f - I*c*f*x])

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 5831

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol]
:> Dist[1/(c^(m + 1)*Sqrt[d]), Subst[Int[(a + b*x)^n*(c*f + g*Sinh[x])^m, x], x, ArcSinh[c*x]], x] /; FreeQ[{
a, b, c, d, e, f, g, n}, x] && EqQ[e, c^2*d] && IntegerQ[m] && GtQ[d, 0] && (GtQ[m, 0] || IGtQ[n, 0])

Rule 3317

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] ||
IGtQ[m, 0] || NeQ[a^2 - b^2, 0])

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3311

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*m*(c + d*x)^(m - 1)*(
b*Sin[e + f*x])^n)/(f^2*n^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[(b*(c + d*x)^m*Cos[e +
f*x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{(d+i c d x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{f-i c f x}} \, dx &=\frac{\sqrt{1+c^2 x^2} \int \frac{(d+i c d x)^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}} \, dx}{\sqrt{d+i c d x} \sqrt{f-i c f x}}\\ &=\frac{\sqrt{1+c^2 x^2} \operatorname{Subst}\left (\int (a+b x)^2 (c d+i c d \sinh (x))^2 \, dx,x,\sinh ^{-1}(c x)\right )}{c^3 \sqrt{d+i c d x} \sqrt{f-i c f x}}\\ &=\frac{\sqrt{1+c^2 x^2} \operatorname{Subst}\left (\int \left (c^2 d^2 (a+b x)^2+2 i c^2 d^2 (a+b x)^2 \sinh (x)-c^2 d^2 (a+b x)^2 \sinh ^2(x)\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^3 \sqrt{d+i c d x} \sqrt{f-i c f x}}\\ &=\frac{d^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{\left (2 i d^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \sinh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{\left (d^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \sinh ^2(x) \, dx,x,\sinh ^{-1}(c x)\right )}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}\\ &=\frac{b c d^2 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{2 i d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{d^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{\left (d^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \, dx,x,\sinh ^{-1}(c x)\right )}{2 c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{\left (4 i b d^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \cosh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{\left (b^2 d^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \sinh ^2(x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 c \sqrt{d+i c d x} \sqrt{f-i c f x}}\\ &=-\frac{b^2 d^2 x \left (1+c^2 x^2\right )}{4 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{4 i b d^2 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{b c d^2 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{2 i d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{d^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{\left (4 i b^2 d^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \sinh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{\left (b^2 d^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int 1 \, dx,x,\sinh ^{-1}(c x)\right )}{4 c \sqrt{d+i c d x} \sqrt{f-i c f x}}\\ &=\frac{4 i b^2 d^2 \left (1+c^2 x^2\right )}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{b^2 d^2 x \left (1+c^2 x^2\right )}{4 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{b^2 d^2 \sqrt{1+c^2 x^2} \sinh ^{-1}(c x)}{4 c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{4 i b d^2 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{b c d^2 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{2 i d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{d^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}\\ \end{align*}

Mathematica [A]  time = 2.14799, size = 529, normalized size = 1.21 \[ \frac{12 a^2 d^{3/2} \sqrt{f} \sqrt{c^2 x^2+1} \log \left (c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right )+16 i a^2 d \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}-4 a^2 c d x \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+2 b d \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2 \left (6 a+8 i b \sqrt{c^2 x^2+1}-b \sinh \left (2 \sinh ^{-1}(c x)\right )\right )+2 b d \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left (-4 a (c x-4 i) \sqrt{c^2 x^2+1}-16 i b c x+b \cosh \left (2 \sinh ^{-1}(c x)\right )\right )-32 i a b c d x \sqrt{d+i c d x} \sqrt{f-i c f x}+2 a b d \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left (2 \sinh ^{-1}(c x)\right )+32 i b^2 d \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+4 b^2 d \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^3-b^2 d \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left (2 \sinh ^{-1}(c x)\right )}{8 c f \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + I*c*d*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/Sqrt[f - I*c*f*x],x]

[Out]

((-32*I)*a*b*c*d*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] + (16*I)*a^2*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt
[1 + c^2*x^2] + (32*I)*b^2*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - 4*a^2*c*d*x*Sqrt[d + I*c*
d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 4*b^2*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 + 2*a*b*
d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] + 2*b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh
[c*x]*((-16*I)*b*c*x - 4*a*(-4*I + c*x)*Sqrt[1 + c^2*x^2] + b*Cosh[2*ArcSinh[c*x]]) + 12*a^2*d^(3/2)*Sqrt[f]*S
qrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] - b^2*d*Sqrt[d + I*c*d*x]*
Sqrt[f - I*c*f*x]*Sinh[2*ArcSinh[c*x]] + 2*b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2*(6*a + (8*I)
*b*Sqrt[1 + c^2*x^2] - b*Sinh[2*ArcSinh[c*x]]))/(8*c*f*Sqrt[1 + c^2*x^2])

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Maple [F]  time = 0.308, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2} \left ( d+icdx \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{f-icfx}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d+I*c*d*x)^(3/2)*(a+b*arcsinh(c*x))^2/(f-I*c*f*x)^(1/2),x)

[Out]

int((d+I*c*d*x)^(3/2)*(a+b*arcsinh(c*x))^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((d+I*c*d*x)^(3/2)*(a+b*arcsinh(c*x))^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 (-\frac{{\left (b^{2} c d x - i \, b^{2} d\right )} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} +{\left (2 \, a b c d x - 2 i \, a b d\right )} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) +{\left (a^{2} c d x - i \, a^{2} d\right )} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f}}{c f x + i \, f}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+I*c*d*x)^(3/2)*(a+b*arcsinh(c*x))^2/(f-I*c*f*x)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+I*c*d*x)**(3/2)*(a+b*asinh(c*x))**2/(f-I*c*f*x)**(1/2),x)

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+I*c*d*x)^(3/2)*(a+b*arcsinh(c*x))^2/(f-I*c*f*x)^(1/2),x, algorithm="giac")

[Out]

Timed out