\(\int (a+b \text {arcsinh}(c+d x))^2 \, dx\) [131]

Optimal result

Integrand size = 12, antiderivative size = 57 \[ \int (a+b \text {arcsinh}(c+d x))^2 \, dx=2 b^2 x-\frac {2 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{d}+\frac {(c+d x) (a+b \text {arcsinh}(c+d x))^2}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5858, 5772, 5798, 8} \[ \int (a+b \text {arcsinh}(c+d x))^2 \, dx=-\frac {2 b \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))}{d}+\frac {(c+d x) (a+b \text {arcsinh}(c+d x))^2}{d}+2 b^2 x \]

[In]

Int[(a + b*ArcSinh[c + d*x])^2,x]

[Out]

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

Rule 8

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

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 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 5858

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcSinh[x])^n, x
], x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.53 \[ \int (a+b \text {arcsinh}(c+d x))^2 \, dx=\frac {\left (a^2+2 b^2\right ) (c+d x)-2 a b \sqrt {1+(c+d x)^2}+2 b \left (a c+a d x-b \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)+b^2 (c+d x) \text {arcsinh}(c+d x)^2}{d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.54

method	result	size
parts	\(a^{2} x +\frac {b^{2} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{2}-2 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 d x +2 c \right )}{d}+\frac {2 a b \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )-\sqrt {1+\left (d x +c \right )^{2}}\right )}{d}\)	\(88\)
derivativedivides	\(\frac {\left (d x +c \right ) a^{2}+b^{2} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{2}-2 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 d x +2 c \right )+2 a b \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )-\sqrt {1+\left (d x +c \right )^{2}}\right )}{d}\)	\(90\)
default	\(\frac {\left (d x +c \right ) a^{2}+b^{2} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{2}-2 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 d x +2 c \right )+2 a b \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )-\sqrt {1+\left (d x +c \right )^{2}}\right )}{d}\)	\(90\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (55) = 110\).

Time = 0.27 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.47 \[ \int (a+b \text {arcsinh}(c+d x))^2 \, dx=\frac {{\left (a^{2} + 2 \, b^{2}\right )} d x + {\left (b^{2} d x + b^{2} c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} - 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} a b + 2 \, {\left (a b d x + a b c - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} b^{2}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )}{d} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (51) = 102\).

Time = 0.12 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.51 \[ \int (a+b \text {arcsinh}(c+d x))^2 \, dx=\begin {cases} a^{2} x + \frac {2 a b c \operatorname {asinh}{\left (c + d x \right )}}{d} + 2 a b x \operatorname {asinh}{\left (c + d x \right )} - \frac {2 a b \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{d} + \frac {b^{2} c \operatorname {asinh}^{2}{\left (c + d x \right )}}{d} + b^{2} x \operatorname {asinh}^{2}{\left (c + d x \right )} + 2 b^{2} x - \frac {2 b^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {asinh}{\left (c \right )}\right )^{2} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a**2*x + 2*a*b*c*asinh(c + d*x)/d + 2*a*b*x*asinh(c + d*x) - 2*a*b*sqrt(c**2 + 2*c*d*x + d**2*x**2
+ 1)/d + b**2*c*asinh(c + d*x)**2/d + b**2*x*asinh(c + d*x)**2 + 2*b**2*x - 2*b**2*sqrt(c**2 + 2*c*d*x + d**2*
x**2 + 1)*asinh(c + d*x)/d, Ne(d, 0)), (x*(a + b*asinh(c))**2, True))

Maxima [F]

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

[In]

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

[Out]

(x*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2 - integrate(2*(d^3*x^3 + 2*c*d^2*x^2 + (c^2*d + d)*x + s
qrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(d^2*x^2 + c*d*x))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))/(d^3*x^3
+ 3*c*d^2*x^2 + c^3 + (3*c^2*d + d)*x + (d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + c), x))*b^2 + a^2*x + 2*((d*x +
c)*arcsinh(d*x + c) - sqrt((d*x + c)^2 + 1))*a*b/d

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((a + b*asinh(c + d*x))^2,x)

[Out]

int((a + b*asinh(c + d*x))^2, x)