Integrand size = 14, antiderivative size = 97 \[ \int (d+e x) (a+b \text {arcsinh}(c x)) \, dx=-\frac {3 b d \sqrt {1+c^2 x^2}}{4 c}-\frac {b (d+e x) \sqrt {1+c^2 x^2}}{4 c}-\frac {b \left (2 d^2-\frac {e^2}{c^2}\right ) \text {arcsinh}(c x)}{4 e}+\frac {(d+e x)^2 (a+b \text {arcsinh}(c x))}{2 e} \]
-1/4*b*(2*d^2-e^2/c^2)*arcsinh(c*x)/e+1/2*(e*x+d)^2*(a+b*arcsinh(c*x))/e-3 /4*b*d*(c^2*x^2+1)^(1/2)/c-1/4*b*(e*x+d)*(c^2*x^2+1)^(1/2)/c
Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.94 \[ \int (d+e x) (a+b \text {arcsinh}(c x)) \, dx=a d x+\frac {1}{2} a e x^2-\frac {b d \sqrt {1+c^2 x^2}}{c}-\frac {b e x \sqrt {1+c^2 x^2}}{4 c}+\frac {b e \text {arcsinh}(c x)}{4 c^2}+b d x \text {arcsinh}(c x)+\frac {1}{2} b e x^2 \text {arcsinh}(c x) \]
a*d*x + (a*e*x^2)/2 - (b*d*Sqrt[1 + c^2*x^2])/c - (b*e*x*Sqrt[1 + c^2*x^2] )/(4*c) + (b*e*ArcSinh[c*x])/(4*c^2) + b*d*x*ArcSinh[c*x] + (b*e*x^2*ArcSi nh[c*x])/2
Time = 0.26 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6243, 497, 455, 222}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (d+e x) (a+b \text {arcsinh}(c x)) \, dx\) |
\(\Big \downarrow \) 6243 |
\(\displaystyle \frac {(d+e x)^2 (a+b \text {arcsinh}(c x))}{2 e}-\frac {b c \int \frac {(d+e x)^2}{\sqrt {c^2 x^2+1}}dx}{2 e}\) |
\(\Big \downarrow \) 497 |
\(\displaystyle \frac {(d+e x)^2 (a+b \text {arcsinh}(c x))}{2 e}-\frac {b c \left (\frac {\int \frac {2 d^2 c^2+3 d e x c^2-e^2}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {e \sqrt {c^2 x^2+1} (d+e x)}{2 c^2}\right )}{2 e}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {(d+e x)^2 (a+b \text {arcsinh}(c x))}{2 e}-\frac {b c \left (\frac {\left (2 c^2 d^2-e^2\right ) \int \frac {1}{\sqrt {c^2 x^2+1}}dx+3 d e \sqrt {c^2 x^2+1}}{2 c^2}+\frac {e \sqrt {c^2 x^2+1} (d+e x)}{2 c^2}\right )}{2 e}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {(d+e x)^2 (a+b \text {arcsinh}(c x))}{2 e}-\frac {b c \left (\frac {\frac {\text {arcsinh}(c x) \left (2 c^2 d^2-e^2\right )}{c}+3 d e \sqrt {c^2 x^2+1}}{2 c^2}+\frac {e \sqrt {c^2 x^2+1} (d+e x)}{2 c^2}\right )}{2 e}\) |
((d + e*x)^2*(a + b*ArcSinh[c*x]))/(2*e) - (b*c*((e*(d + e*x)*Sqrt[1 + c^2 *x^2])/(2*c^2) + (3*d*e*Sqrt[1 + c^2*x^2] + ((2*c^2*d^2 - e^2)*ArcSinh[c*x ])/c)/(2*c^2)))/(2*e)
3.1.6.3.1 Defintions of rubi rules used
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]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[1/(b *(n + 2*p + 1)) Int[(c + d*x)^(n - 2)*(a + b*x^2)^p*Simp[b*c^2*(n + 2*p + 1) - a*d^2*(n - 1) + 2*b*c*d*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, n , p}, x] && If[RationalQ[n], GtQ[n, 1], SumSimplerQ[n, -2]] && NeQ[n + 2*p + 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x _Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 1))), x] - Simp[b*c*(n/(e*(m + 1))) Int[(d + e*x)^(m + 1)*((a + b*ArcSinh[c*x])^( n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n , 0] && NeQ[m, -1]
Time = 0.02 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.87
method | result | size |
parts | \(a \left (\frac {1}{2} e \,x^{2}+d x \right )+\frac {b \left (\frac {c \,\operatorname {arcsinh}\left (c x \right ) x^{2} e}{2}+\operatorname {arcsinh}\left (c x \right ) d c x -\frac {e \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )+2 d c \sqrt {c^{2} x^{2}+1}}{2 c}\right )}{c}\) | \(84\) |
derivativedivides | \(\frac {\frac {a \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b \left (\operatorname {arcsinh}\left (c x \right ) d \,c^{2} x +\frac {\operatorname {arcsinh}\left (c x \right ) e \,c^{2} x^{2}}{2}-\frac {e \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )}{2}-d c \sqrt {c^{2} x^{2}+1}\right )}{c}}{c}\) | \(96\) |
default | \(\frac {\frac {a \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b \left (\operatorname {arcsinh}\left (c x \right ) d \,c^{2} x +\frac {\operatorname {arcsinh}\left (c x \right ) e \,c^{2} x^{2}}{2}-\frac {e \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )}{2}-d c \sqrt {c^{2} x^{2}+1}\right )}{c}}{c}\) | \(96\) |
a*(1/2*e*x^2+d*x)+b/c*(1/2*c*arcsinh(c*x)*x^2*e+arcsinh(c*x)*d*c*x-1/2/c*( e*(1/2*c*x*(c^2*x^2+1)^(1/2)-1/2*arcsinh(c*x))+2*d*c*(c^2*x^2+1)^(1/2)))
Time = 0.26 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.90 \[ \int (d+e x) (a+b \text {arcsinh}(c x)) \, dx=\frac {2 \, a c^{2} e x^{2} + 4 \, a c^{2} d x + {\left (2 \, b c^{2} e x^{2} + 4 \, b c^{2} d x + b e\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (b c e x + 4 \, b c d\right )} \sqrt {c^{2} x^{2} + 1}}{4 \, c^{2}} \]
1/4*(2*a*c^2*e*x^2 + 4*a*c^2*d*x + (2*b*c^2*e*x^2 + 4*b*c^2*d*x + b*e)*log (c*x + sqrt(c^2*x^2 + 1)) - (b*c*e*x + 4*b*c*d)*sqrt(c^2*x^2 + 1))/c^2
Time = 0.17 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.02 \[ \int (d+e x) (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} a d x + \frac {a e x^{2}}{2} + b d x \operatorname {asinh}{\left (c x \right )} + \frac {b e x^{2} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {b d \sqrt {c^{2} x^{2} + 1}}{c} - \frac {b e x \sqrt {c^{2} x^{2} + 1}}{4 c} + \frac {b e \operatorname {asinh}{\left (c x \right )}}{4 c^{2}} & \text {for}\: c \neq 0 \\a \left (d x + \frac {e x^{2}}{2}\right ) & \text {otherwise} \end {cases} \]
Piecewise((a*d*x + a*e*x**2/2 + b*d*x*asinh(c*x) + b*e*x**2*asinh(c*x)/2 - b*d*sqrt(c**2*x**2 + 1)/c - b*e*x*sqrt(c**2*x**2 + 1)/(4*c) + b*e*asinh(c *x)/(4*c**2), Ne(c, 0)), (a*(d*x + e*x**2/2), True))
Time = 0.18 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85 \[ \int (d+e x) (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{2} \, a e x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )}\right )} b e + a d x + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b d}{c} \]
1/2*a*e*x^2 + 1/4*(2*x^2*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x/c^2 - arcsi nh(c*x)/c^3))*b*e + a*d*x + (c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*b*d/c
Time = 0.34 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.26 \[ \int (d+e x) (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{2} \, a e x^{2} + {\left (x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - \frac {\sqrt {c^{2} x^{2} + 1}}{c}\right )} b d + \frac {1}{4} \, {\left (2 \, x^{2} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} + \frac {\log \left (-x {\left | c \right |} + \sqrt {c^{2} x^{2} + 1}\right )}{c^{2} {\left | c \right |}}\right )}\right )} b e + a d x \]
1/2*a*e*x^2 + (x*log(c*x + sqrt(c^2*x^2 + 1)) - sqrt(c^2*x^2 + 1)/c)*b*d + 1/4*(2*x^2*log(c*x + sqrt(c^2*x^2 + 1)) - c*(sqrt(c^2*x^2 + 1)*x/c^2 + lo g(-x*abs(c) + sqrt(c^2*x^2 + 1))/(c^2*abs(c))))*b*e + a*d*x
Time = 2.76 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.80 \[ \int (d+e x) (a+b \text {arcsinh}(c x)) \, dx=\frac {a\,x\,\left (2\,d+e\,x\right )}{2}-\frac {b\,d\,\left (\sqrt {c^2\,x^2+1}-c\,x\,\mathrm {asinh}\left (c\,x\right )\right )}{c}-\frac {b\,e\,x\,\sqrt {c^2\,x^2+1}}{4\,c}+b\,e\,x\,\mathrm {asinh}\left (c\,x\right )\,\left (\frac {x}{2}+\frac {1}{4\,c^2\,x}\right ) \]