\(\int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))^2}{\sqrt {f-i c f x}} \, dx\) [590]

Optimal result

Integrand size = 37, antiderivative size = 259 \[ \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))^2}{\sqrt {f-i c f x}} \, dx=-\frac {2 i a b d x \sqrt {1+c^2 x^2}}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {2 i b^2 d \left (1+c^2 x^2\right )}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i b^2 d x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {i d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{3 b c \sqrt {d+i c d x} \sqrt {f-i c f x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {5796, 5838, 5783, 5798, 5772, 267} \[ \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))^2}{\sqrt {f-i c f x}} \, dx=\frac {d \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^3}{3 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {i d \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i a b d x \sqrt {c^2 x^2+1}}{\sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i b^2 d x \sqrt {c^2 x^2+1} \text {arcsinh}(c x)}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {2 i b^2 d \left (c^2 x^2+1\right )}{c \sqrt {d+i c d x} \sqrt {f-i c f x}} \]

[In]

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

[Out]

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

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 5772

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

Rule 5783

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

Rule 5796

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 5798

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

Rule 5838

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

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+c^2 x^2} \int \frac {(d+i c d x) (a+b \text {arcsinh}(c x))^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} \int \left (\frac {d (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}}+\frac {i c d x (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}}\right ) \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}} \\ & = \frac {\left (d \sqrt {1+c^2 x^2}\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {\left (i c d \sqrt {1+c^2 x^2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}} \\ & = \frac {i d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{3 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {\left (2 i b d \sqrt {1+c^2 x^2}\right ) \int (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}} \\ & = -\frac {2 i a b d x \sqrt {1+c^2 x^2}}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {i d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{3 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {\left (2 i b^2 d \sqrt {1+c^2 x^2}\right ) \int \text {arcsinh}(c x) \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}} \\ & = -\frac {2 i a b d x \sqrt {1+c^2 x^2}}{\sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i b^2 d x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {i d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{3 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {\left (2 i b^2 c d \sqrt {1+c^2 x^2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}} \\ & = -\frac {2 i a b d x \sqrt {1+c^2 x^2}}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {2 i b^2 d \left (1+c^2 x^2\right )}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i b^2 d x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {i d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{3 b c \sqrt {d+i c d x} \sqrt {f-i c f x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.64 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))^2}{\sqrt {f-i c f x}} \, dx=\frac {3 i \sqrt {d+i c d x} \sqrt {f-i c f x} \left (-2 a b c x+a^2 \sqrt {1+c^2 x^2}+2 b^2 \sqrt {1+c^2 x^2}\right )-6 i b \sqrt {d+i c d x} \sqrt {f-i c f x} \left (b c x-a \sqrt {1+c^2 x^2}\right ) \text {arcsinh}(c x)+3 b \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+i b \sqrt {1+c^2 x^2}\right ) \text {arcsinh}(c x)^2+b^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^3+3 a^2 \sqrt {d} \sqrt {f} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )}{3 c f \sqrt {1+c^2 x^2}} \]

[In]

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

[Out]

((3*I)*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(-2*a*b*c*x + a^2*Sqrt[1 + c^2*x^2] + 2*b^2*Sqrt[1 + c^2*x^2]) - (6
*I)*b*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(b*c*x - a*Sqrt[1 + c^2*x^2])*ArcSinh[c*x] + 3*b*Sqrt[d + I*c*d*x]*S
qrt[f - I*c*f*x]*(a + I*b*Sqrt[1 + c^2*x^2])*ArcSinh[c*x]^2 + b^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[
c*x]^3 + 3*a^2*Sqrt[d]*Sqrt[f]*Sqrt[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]])/(3*c*f*Sqrt[1 + c^2*x^2])

Maple [F]

\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2} \sqrt {i c d x +d}}{\sqrt {-i c f x +f}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))^2}{\sqrt {f-i c f x}} \, dx=\int { \frac {\sqrt {i \, c d x + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{\sqrt {-i \, c f x + f}} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))^2}{\sqrt {f-i c f x}} \, dx=\int \frac {\sqrt {i d \left (c x - i\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\sqrt {- i f \left (c x + i\right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))^2}{\sqrt {f-i c f x}} \, dx=\int { \frac {\sqrt {i \, c d x + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{\sqrt {-i \, c f x + f}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))^2}{\sqrt {f-i c f x}} \, dx=\int { \frac {\sqrt {i \, c d x + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{\sqrt {-i \, c f x + f}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))^2}{\sqrt {f-i c f x}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {d+c\,d\,x\,1{}\mathrm {i}}}{\sqrt {f-c\,f\,x\,1{}\mathrm {i}}} \,d x \]

[In]

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

[Out]

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