Integrand size = 21, antiderivative size = 161 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^3 \, dx=-\frac {3 b^3 e (c+d x) \sqrt {1+(c+d x)^2}}{8 d}+\frac {3 b^3 e \text {arcsinh}(c+d x)}{8 d}+\frac {3 b^2 e (c+d x)^2 (a+b \text {arcsinh}(c+d x))}{4 d}-\frac {3 b e (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{4 d}+\frac {e (a+b \text {arcsinh}(c+d x))^3}{4 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^3}{2 d} \]
[Out]
Time = 0.15 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5859, 12, 5776, 5812, 5783, 327, 221} \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^3 \, dx=\frac {3 b^2 e (c+d x)^2 (a+b \text {arcsinh}(c+d x))}{4 d}-\frac {3 b e (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2}{4 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^3}{2 d}+\frac {e (a+b \text {arcsinh}(c+d x))^3}{4 d}+\frac {3 b^3 e \text {arcsinh}(c+d x)}{8 d}-\frac {3 b^3 e (c+d x) \sqrt {(c+d x)^2+1}}{8 d} \]
[In]
[Out]
Rule 12
Rule 221
Rule 327
Rule 5776
Rule 5783
Rule 5812
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e x (a+b \text {arcsinh}(x))^3 \, dx,x,c+d x\right )}{d} \\ & = \frac {e \text {Subst}\left (\int x (a+b \text {arcsinh}(x))^3 \, dx,x,c+d x\right )}{d} \\ & = \frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^3}{2 d}-\frac {(3 b e) \text {Subst}\left (\int \frac {x^2 (a+b \text {arcsinh}(x))^2}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{2 d} \\ & = -\frac {3 b e (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{4 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^3}{2 d}+\frac {(3 b e) \text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^2}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{4 d}+\frac {\left (3 b^2 e\right ) \text {Subst}(\int x (a+b \text {arcsinh}(x)) \, dx,x,c+d x)}{2 d} \\ & = \frac {3 b^2 e (c+d x)^2 (a+b \text {arcsinh}(c+d x))}{4 d}-\frac {3 b e (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{4 d}+\frac {e (a+b \text {arcsinh}(c+d x))^3}{4 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^3}{2 d}-\frac {\left (3 b^3 e\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{4 d} \\ & = -\frac {3 b^3 e (c+d x) \sqrt {1+(c+d x)^2}}{8 d}+\frac {3 b^2 e (c+d x)^2 (a+b \text {arcsinh}(c+d x))}{4 d}-\frac {3 b e (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{4 d}+\frac {e (a+b \text {arcsinh}(c+d x))^3}{4 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^3}{2 d}+\frac {\left (3 b^3 e\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{8 d} \\ & = -\frac {3 b^3 e (c+d x) \sqrt {1+(c+d x)^2}}{8 d}+\frac {3 b^3 e \text {arcsinh}(c+d x)}{8 d}+\frac {3 b^2 e (c+d x)^2 (a+b \text {arcsinh}(c+d x))}{4 d}-\frac {3 b e (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{4 d}+\frac {e (a+b \text {arcsinh}(c+d x))^3}{4 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^3}{2 d} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.24 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^3 \, dx=\frac {e \left (2 a \left (2 a^2+3 b^2\right ) (c+d x)^2-3 b \left (2 a^2+b^2\right ) (c+d x) \sqrt {1+(c+d x)^2}+3 b \left (2 a^2+b^2\right ) \text {arcsinh}(c+d x)-6 b (c+d x) \left (-2 a^2 (c+d x)-b^2 (c+d x)+2 a b \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)+6 b^2 \left (a+2 a (c+d x)^2-b (c+d x) \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)^2+2 b^3 \left (1+2 (c+d x)^2\right ) \text {arcsinh}(c+d x)^3\right )}{8 d} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.51
method | result | size |
derivativedivides | \(\frac {\frac {e \,a^{3} \left (d x +c \right )^{2}}{2}+e \,b^{3} \left (\frac {\operatorname {arcsinh}\left (d x +c \right )^{3} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{4}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3}}{4}+\frac {3 \left (1+\left (d x +c \right )^{2}\right ) \operatorname {arcsinh}\left (d x +c \right )}{4}-\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}-\frac {3 \,\operatorname {arcsinh}\left (d x +c \right )}{8}\right )+3 e a \,b^{2} \left (\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {\sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}+\frac {1}{4}\right )+3 e \,a^{2} b \left (\frac {\left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {\operatorname {arcsinh}\left (d x +c \right )}{4}\right )}{d}\) | \(243\) |
default | \(\frac {\frac {e \,a^{3} \left (d x +c \right )^{2}}{2}+e \,b^{3} \left (\frac {\operatorname {arcsinh}\left (d x +c \right )^{3} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{4}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3}}{4}+\frac {3 \left (1+\left (d x +c \right )^{2}\right ) \operatorname {arcsinh}\left (d x +c \right )}{4}-\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}-\frac {3 \,\operatorname {arcsinh}\left (d x +c \right )}{8}\right )+3 e a \,b^{2} \left (\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {\sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}+\frac {1}{4}\right )+3 e \,a^{2} b \left (\frac {\left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {\operatorname {arcsinh}\left (d x +c \right )}{4}\right )}{d}\) | \(243\) |
parts | \(e \,a^{3} \left (\frac {1}{2} d \,x^{2}+c x \right )+\frac {e \,b^{3} \left (\frac {\operatorname {arcsinh}\left (d x +c \right )^{3} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{4}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3}}{4}+\frac {3 \left (1+\left (d x +c \right )^{2}\right ) \operatorname {arcsinh}\left (d x +c \right )}{4}-\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}-\frac {3 \,\operatorname {arcsinh}\left (d x +c \right )}{8}\right )}{d}+\frac {3 e a \,b^{2} \left (\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {\sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}+\frac {1}{4}\right )}{d}+\frac {3 e \,a^{2} b \left (\frac {\left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {\operatorname {arcsinh}\left (d x +c \right )}{4}\right )}{d}\) | \(250\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 391 vs. \(2 (145) = 290\).
Time = 0.31 (sec) , antiderivative size = 391, normalized size of antiderivative = 2.43 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^3 \, dx=\frac {2 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} d^{2} e x^{2} + 4 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c d e x + 2 \, {\left (2 \, b^{3} d^{2} e x^{2} + 4 \, b^{3} c d e x + {\left (2 \, b^{3} c^{2} + b^{3}\right )} e\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{3} + 6 \, {\left (2 \, a b^{2} d^{2} e x^{2} + 4 \, a b^{2} c d e x + {\left (2 \, a b^{2} c^{2} + a b^{2}\right )} e - {\left (b^{3} d e x + b^{3} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + 3 \, {\left (2 \, {\left (2 \, a^{2} b + b^{3}\right )} d^{2} e x^{2} + 4 \, {\left (2 \, a^{2} b + b^{3}\right )} c d e x + {\left (2 \, a^{2} b + b^{3} + 2 \, {\left (2 \, a^{2} b + b^{3}\right )} c^{2}\right )} e - 4 \, {\left (a b^{2} d e x + a b^{2} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 3 \, {\left ({\left (2 \, a^{2} b + b^{3}\right )} d e x + {\left (2 \, a^{2} b + b^{3}\right )} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{8 \, d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 685 vs. \(2 (148) = 296\).
Time = 0.31 (sec) , antiderivative size = 685, normalized size of antiderivative = 4.25 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^3 \, dx=\begin {cases} a^{3} c e x + \frac {a^{3} d e x^{2}}{2} + \frac {3 a^{2} b c^{2} e \operatorname {asinh}{\left (c + d x \right )}}{2 d} + 3 a^{2} b c e x \operatorname {asinh}{\left (c + d x \right )} - \frac {3 a^{2} b c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{4 d} + \frac {3 a^{2} b d e x^{2} \operatorname {asinh}{\left (c + d x \right )}}{2} - \frac {3 a^{2} b e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{4} + \frac {3 a^{2} b e \operatorname {asinh}{\left (c + d x \right )}}{4 d} + \frac {3 a b^{2} c^{2} e \operatorname {asinh}^{2}{\left (c + d x \right )}}{2 d} + 3 a b^{2} c e x \operatorname {asinh}^{2}{\left (c + d x \right )} + \frac {3 a b^{2} c e x}{2} - \frac {3 a b^{2} c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{2 d} + \frac {3 a b^{2} d e x^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{2} + \frac {3 a b^{2} d e x^{2}}{4} - \frac {3 a b^{2} e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{2} + \frac {3 a b^{2} e \operatorname {asinh}^{2}{\left (c + d x \right )}}{4 d} + \frac {b^{3} c^{2} e \operatorname {asinh}^{3}{\left (c + d x \right )}}{2 d} + \frac {3 b^{3} c^{2} e \operatorname {asinh}{\left (c + d x \right )}}{4 d} + b^{3} c e x \operatorname {asinh}^{3}{\left (c + d x \right )} + \frac {3 b^{3} c e x \operatorname {asinh}{\left (c + d x \right )}}{2} - \frac {3 b^{3} c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c + d x \right )}}{4 d} - \frac {3 b^{3} c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{8 d} + \frac {b^{3} d e x^{2} \operatorname {asinh}^{3}{\left (c + d x \right )}}{2} + \frac {3 b^{3} d e x^{2} \operatorname {asinh}{\left (c + d x \right )}}{4} - \frac {3 b^{3} e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c + d x \right )}}{4} - \frac {3 b^{3} e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{8} + \frac {b^{3} e \operatorname {asinh}^{3}{\left (c + d x \right )}}{4 d} + \frac {3 b^{3} e \operatorname {asinh}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\c e x \left (a + b \operatorname {asinh}{\left (c \right )}\right )^{3} & \text {otherwise} \end {cases} \]
[In]
[Out]
\[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^3 \, dx=\int { {\left (d e x + c e\right )} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
[In]
[Out]
\[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^3 \, dx=\int { {\left (d e x + c e\right )} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
[In]
[Out]
Timed out. \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^3 \, dx=\int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3 \,d x \]
[In]
[Out]