\(\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {d+c^2 d x^2}} \, dx\) [150]

Optimal result

Integrand size = 23, antiderivative size = 47 \[ \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {d+c^2 d x^2}} \, dx=\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{2 b c \sqrt {d+c^2 d x^2}} \]

[Out]

1/2*(a+b*arcsinh(c*x))^2*(c^2*x^2+1)^(1/2)/b/c/(c^2*d*x^2+d)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {5783} \[ \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {d+c^2 d x^2}} \, dx=\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{2 b c \sqrt {c^2 d x^2+d}} \]

[In]

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

[Out]

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

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{2 b c \sqrt {d+c^2 d x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.62 \[ \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {d+c^2 d x^2}} \, dx=\frac {b \sqrt {1+c^2 x^2} \text {arcsinh}(c x)^2}{2 c \sqrt {d \left (1+c^2 x^2\right )}}+\frac {a \text {arctanh}\left (\frac {c \sqrt {d} x}{\sqrt {d+c^2 d x^2}}\right )}{c \sqrt {d}} \]

[In]

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

[Out]

(b*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2)/(2*c*Sqrt[d*(1 + c^2*x^2)]) + (a*ArcTanh[(c*Sqrt[d]*x)/Sqrt[d + c^2*d*x^2
]])/(c*Sqrt[d])

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.64

method	result	size
default	\(\frac {a \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{\sqrt {c^{2} d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2}}{2 \sqrt {c^{2} x^{2}+1}\, c d}\)	\(77\)
parts	\(\frac {a \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{\sqrt {c^{2} d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2}}{2 \sqrt {c^{2} x^{2}+1}\, c d}\)	\(77\)

[In]

int((a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

a*ln(c^2*d*x/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+1/2*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c/
d*arcsinh(c*x)^2

Fricas [F]

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

[In]

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

[Out]

integral((b*arcsinh(c*x) + a)/sqrt(c^2*d*x^2 + d), x)

Sympy [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {d+c^2 d x^2}} \, dx=\int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{\sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \]

[In]

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

[Out]

Integral((a + b*asinh(c*x))/sqrt(d*(c**2*x**2 + 1)), x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.60 \[ \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {d+c^2 d x^2}} \, dx=\frac {b \operatorname {arsinh}\left (c x\right )^{2}}{2 \, c \sqrt {d}} + \frac {a \operatorname {arsinh}\left (c x\right )}{c \sqrt {d}} \]

[In]

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

[Out]

1/2*b*arcsinh(c*x)^2/(c*sqrt(d)) + a*arcsinh(c*x)/(c*sqrt(d))

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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