\(\int (c e+d e x) (a+b \text {arcsinh}(c+d x)) \, dx\) [118]

Optimal result

Integrand size = 19, antiderivative size = 68 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x)) \, dx=-\frac {b e (c+d x) \sqrt {1+(c+d x)^2}}{4 d}+\frac {b e \text {arcsinh}(c+d x)}{4 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))}{2 d} \]

[Out]

1/4*b*e*arcsinh(d*x+c)/d+1/2*e*(d*x+c)^2*(a+b*arcsinh(d*x+c))/d-1/4*b*e*(d*x+c)*(1+(d*x+c)^2)^(1/2)/d

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5859, 12, 5776, 327, 221} \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x)) \, dx=\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))}{2 d}+\frac {b e \text {arcsinh}(c+d x)}{4 d}-\frac {b e \sqrt {(c+d x)^2+1} (c+d x)}{4 d} \]

[In]

Int[(c*e + d*e*x)*(a + b*ArcSinh[c + d*x]),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 5776

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcS
inh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[
1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5859

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}(\int e x (a+b \text {arcsinh}(x)) \, dx,x,c+d x)}{d} \\ & = \frac {e \text {Subst}(\int x (a+b \text {arcsinh}(x)) \, dx,x,c+d x)}{d} \\ & = \frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))}{2 d}-\frac {(b e) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{2 d} \\ & = -\frac {b e (c+d x) \sqrt {1+(c+d x)^2}}{4 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))}{2 d}+\frac {(b e) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{4 d} \\ & = -\frac {b e (c+d x) \sqrt {1+(c+d x)^2}}{4 d}+\frac {b e \text {arcsinh}(c+d x)}{4 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))}{2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.84 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x)) \, dx=\frac {e \left (-b (c+d x) \sqrt {1+(c+d x)^2}+b \text {arcsinh}(c+d x)+2 (c+d x)^2 (a+b \text {arcsinh}(c+d x))\right )}{4 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.91

method	result	size
derivativedivides	\(\frac {\frac {e a \left (d x +c \right )^{2}}{2}+e b \left (\frac {\left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {\operatorname {arcsinh}\left (d x +c \right )}{4}\right )}{d}\)	\(62\)
default	\(\frac {\frac {e a \left (d x +c \right )^{2}}{2}+e b \left (\frac {\left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {\operatorname {arcsinh}\left (d x +c \right )}{4}\right )}{d}\)	\(62\)
parts	\(e a \left (\frac {1}{2} d \,x^{2}+c x \right )+\frac {e b \left (\frac {\left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {\operatorname {arcsinh}\left (d x +c \right )}{4}\right )}{d}\)	\(63\)

[In]

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

[Out]

1/d*(1/2*e*a*(d*x+c)^2+e*b*(1/2*(d*x+c)^2*arcsinh(d*x+c)-1/4*(d*x+c)*(1+(d*x+c)^2)^(1/2)+1/4*arcsinh(d*x+c)))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.59 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x)) \, dx=\frac {2 \, a d^{2} e x^{2} + 4 \, a c d e x + {\left (2 \, b d^{2} e x^{2} + 4 \, b c d e x + {\left (2 \, b c^{2} + b\right )} e\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - {\left (b d e x + b c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{4 \, d} \]

[In]

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

[Out]

1/4*(2*a*d^2*e*x^2 + 4*a*c*d*e*x + (2*b*d^2*e*x^2 + 4*b*c*d*e*x + (2*b*c^2 + b)*e)*log(d*x + c + sqrt(d^2*x^2
+ 2*c*d*x + c^2 + 1)) - (b*d*e*x + b*c*e)*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. 148 vs. \(2 (58) = 116\).

Time = 0.13 (sec) , antiderivative size = 148, normalized size of antiderivative = 2.18 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x)) \, dx=\begin {cases} a c e x + \frac {a d e x^{2}}{2} + \frac {b c^{2} e \operatorname {asinh}{\left (c + d x \right )}}{2 d} + b c e x \operatorname {asinh}{\left (c + d x \right )} - \frac {b c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{4 d} + \frac {b d e x^{2} \operatorname {asinh}{\left (c + d x \right )}}{2} - \frac {b e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{4} + \frac {b e \operatorname {asinh}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\c e x \left (a + b \operatorname {asinh}{\left (c \right )}\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a*c*e*x + a*d*e*x**2/2 + b*c**2*e*asinh(c + d*x)/(2*d) + b*c*e*x*asinh(c + d*x) - b*c*e*sqrt(c**2 +
2*c*d*x + d**2*x**2 + 1)/(4*d) + b*d*e*x**2*asinh(c + d*x)/2 - b*e*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/4 +
b*e*asinh(c + d*x)/(4*d), Ne(d, 0)), (c*e*x*(a + b*asinh(c)), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (60) = 120\).

Time = 0.19 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.96 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x)) \, dx=\frac {1}{2} \, a d e x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (d x + c\right ) - d {\left (\frac {3 \, c^{2} \operatorname {arsinh}\left (\frac {2 \, {\left (d^{2} x + c d\right )}}{\sqrt {-4 \, c^{2} d^{2} + 4 \, {\left (c^{2} + 1\right )} d^{2}}}\right )}{d^{3}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} x}{d^{2}} - \frac {{\left (c^{2} + 1\right )} \operatorname {arsinh}\left (\frac {2 \, {\left (d^{2} x + c d\right )}}{\sqrt {-4 \, c^{2} d^{2} + 4 \, {\left (c^{2} + 1\right )} d^{2}}}\right )}{d^{3}} - \frac {3 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} c}{d^{3}}\right )}\right )} b d e + a c e x + \frac {{\left ({\left (d x + c\right )} \operatorname {arsinh}\left (d x + c\right ) - \sqrt {{\left (d x + c\right )}^{2} + 1}\right )} b c e}{d} \]

[In]

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

[Out]

1/2*a*d*e*x^2 + 1/4*(2*x^2*arcsinh(d*x + c) - d*(3*c^2*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d
^2))/d^3 + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*x/d^2 - (c^2 + 1)*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^
2 + 1)*d^2))/d^3 - 3*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c/d^3))*b*d*e + a*c*e*x + ((d*x + c)*arcsinh(d*x + c) -
sqrt((d*x + c)^2 + 1))*b*c*e/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (60) = 120\).

Time = 0.49 (sec) , antiderivative size = 243, normalized size of antiderivative = 3.57 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x)) \, dx=\frac {1}{2} \, a d e x^{2} - {\left (d {\left (\frac {c \log \left (-c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} {\left | d \right |}\right )}{d {\left | d \right |}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{d^{2}}\right )} - x \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )\right )} b c e + \frac {1}{4} \, {\left (2 \, x^{2} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left (\frac {x}{d^{2}} - \frac {3 \, c}{d^{3}}\right )} - \frac {{\left (2 \, c^{2} - 1\right )} \log \left (-c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} {\left | d \right |}\right )}{d^{2} {\left | d \right |}}\right )} d\right )} b d e + a c e x \]

[In]

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

[Out]

1/2*a*d*e*x^2 - (d*(c*log(-c*d - (x*abs(d) - sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*abs(d))/(d*abs(d)) + sqrt(d^2*
x^2 + 2*c*d*x + c^2 + 1)/d^2) - x*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)))*b*c*e + 1/4*(2*x^2*log(d*x
+ c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) - (sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(x/d^2 - 3*c/d^3) - (2*c^2 - 1)
*log(-c*d - (x*abs(d) - sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*abs(d))/(d^2*abs(d)))*d)*b*d*e + a*c*e*x

Mupad [F(-1)]

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

[In]

int((c*e + d*e*x)*(a + b*asinh(c + d*x)),x)

[Out]

int((c*e + d*e*x)*(a + b*asinh(c + d*x)), x)