Integrand size = 12, antiderivative size = 115 \[ \int (a+b \text {arcsinh}(c+d x))^4 \, dx=24 b^4 x-\frac {24 b^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{d}+\frac {12 b^2 (c+d x) (a+b \text {arcsinh}(c+d x))^2}{d}-\frac {4 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{d}+\frac {(c+d x) (a+b \text {arcsinh}(c+d x))^4}{d} \]
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Time = 0.11 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5858, 5772, 5798, 8} \[ \int (a+b \text {arcsinh}(c+d x))^4 \, dx=-\frac {24 b^3 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))}{d}+\frac {12 b^2 (c+d x) (a+b \text {arcsinh}(c+d x))^2}{d}-\frac {4 b \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{d}+\frac {(c+d x) (a+b \text {arcsinh}(c+d x))^4}{d}+24 b^4 x \]
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Rule 8
Rule 5772
Rule 5798
Rule 5858
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a+b \text {arcsinh}(x))^4 \, dx,x,c+d x\right )}{d} \\ & = \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^4}{d}-\frac {(4 b) \text {Subst}\left (\int \frac {x (a+b \text {arcsinh}(x))^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {4 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{d}+\frac {(c+d x) (a+b \text {arcsinh}(c+d x))^4}{d}+\frac {\left (12 b^2\right ) \text {Subst}\left (\int (a+b \text {arcsinh}(x))^2 \, dx,x,c+d x\right )}{d} \\ & = \frac {12 b^2 (c+d x) (a+b \text {arcsinh}(c+d x))^2}{d}-\frac {4 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{d}+\frac {(c+d x) (a+b \text {arcsinh}(c+d x))^4}{d}-\frac {\left (24 b^3\right ) \text {Subst}\left (\int \frac {x (a+b \text {arcsinh}(x))}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {24 b^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{d}+\frac {12 b^2 (c+d x) (a+b \text {arcsinh}(c+d x))^2}{d}-\frac {4 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{d}+\frac {(c+d x) (a+b \text {arcsinh}(c+d x))^4}{d}+\frac {\left (24 b^4\right ) \text {Subst}(\int 1 \, dx,x,c+d x)}{d} \\ & = 24 b^4 x-\frac {24 b^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{d}+\frac {12 b^2 (c+d x) (a+b \text {arcsinh}(c+d x))^2}{d}-\frac {4 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{d}+\frac {(c+d x) (a+b \text {arcsinh}(c+d x))^4}{d} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.97 \[ \int (a+b \text {arcsinh}(c+d x))^4 \, dx=\frac {\left (a^4+12 a^2 b^2+24 b^4\right ) (c+d x)-4 a b \left (a^2+6 b^2\right ) \sqrt {1+(c+d x)^2}-4 b \left (-a^3 (c+d x)-6 a b^2 (c+d x)+3 a^2 b \sqrt {1+(c+d x)^2}+6 b^3 \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)+6 b^2 \left (a^2 (c+d x)+2 b^2 (c+d x)-2 a b \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)^2-4 b^3 \left (-a (c+d x)+b \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)^3+b^4 (c+d x) \text {arcsinh}(c+d x)^4}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(244\) vs. \(2(111)=222\).
Time = 0.17 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.13
method | result | size |
derivativedivides | \(\frac {\left (d x +c \right ) a^{4}+b^{4} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{4}-4 \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+12 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{2}-24 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+24 d x +24 c \right )+4 a \,b^{3} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{3}-3 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+6 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )-6 \sqrt {1+\left (d x +c \right )^{2}}\right )+6 a^{2} b^{2} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{2}-2 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 d x +2 c \right )+4 b \,a^{3} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )-\sqrt {1+\left (d x +c \right )^{2}}\right )}{d}\) | \(245\) |
default | \(\frac {\left (d x +c \right ) a^{4}+b^{4} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{4}-4 \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+12 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{2}-24 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+24 d x +24 c \right )+4 a \,b^{3} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{3}-3 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+6 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )-6 \sqrt {1+\left (d x +c \right )^{2}}\right )+6 a^{2} b^{2} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{2}-2 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 d x +2 c \right )+4 b \,a^{3} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )-\sqrt {1+\left (d x +c \right )^{2}}\right )}{d}\) | \(245\) |
parts | \(a^{4} x +\frac {b^{4} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{4}-4 \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+12 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{2}-24 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+24 d x +24 c \right )}{d}+\frac {4 a \,b^{3} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{3}-3 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+6 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )-6 \sqrt {1+\left (d x +c \right )^{2}}\right )}{d}+\frac {6 a^{2} b^{2} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{2}-2 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 d x +2 c \right )}{d}+\frac {4 b \,a^{3} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )-\sqrt {1+\left (d x +c \right )^{2}}\right )}{d}\) | \(249\) |
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Leaf count of result is larger than twice the leaf count of optimal. 344 vs. \(2 (111) = 222\).
Time = 0.28 (sec) , antiderivative size = 344, normalized size of antiderivative = 2.99 \[ \int (a+b \text {arcsinh}(c+d x))^4 \, dx=\frac {{\left (b^{4} d x + b^{4} c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{4} + 4 \, {\left (a b^{3} d x + a b^{3} c - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} b^{4}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{3} + {\left (a^{4} + 12 \, a^{2} b^{2} + 24 \, b^{4}\right )} d x - 6 \, {\left (2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} a b^{3} - {\left (a^{2} b^{2} + 2 \, b^{4}\right )} d x - {\left (a^{2} b^{2} + 2 \, b^{4}\right )} c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + 4 \, {\left ({\left (a^{3} b + 6 \, a b^{3}\right )} d x + {\left (a^{3} b + 6 \, a b^{3}\right )} c - 3 \, {\left (a^{2} b^{2} + 2 \, b^{4}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 4 \, {\left (a^{3} b + 6 \, a b^{3}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (105) = 210\).
Time = 0.27 (sec) , antiderivative size = 444, normalized size of antiderivative = 3.86 \[ \int (a+b \text {arcsinh}(c+d x))^4 \, dx=\begin {cases} a^{4} x + \frac {4 a^{3} b c \operatorname {asinh}{\left (c + d x \right )}}{d} + 4 a^{3} b x \operatorname {asinh}{\left (c + d x \right )} - \frac {4 a^{3} b \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{d} + \frac {6 a^{2} b^{2} c \operatorname {asinh}^{2}{\left (c + d x \right )}}{d} + 6 a^{2} b^{2} x \operatorname {asinh}^{2}{\left (c + d x \right )} + 12 a^{2} b^{2} x - \frac {12 a^{2} b^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{d} + \frac {4 a b^{3} c \operatorname {asinh}^{3}{\left (c + d x \right )}}{d} + \frac {24 a b^{3} c \operatorname {asinh}{\left (c + d x \right )}}{d} + 4 a b^{3} x \operatorname {asinh}^{3}{\left (c + d x \right )} + 24 a b^{3} x \operatorname {asinh}{\left (c + d x \right )} - \frac {12 a b^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c + d x \right )}}{d} - \frac {24 a b^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{d} + \frac {b^{4} c \operatorname {asinh}^{4}{\left (c + d x \right )}}{d} + \frac {12 b^{4} c \operatorname {asinh}^{2}{\left (c + d x \right )}}{d} + b^{4} x \operatorname {asinh}^{4}{\left (c + d x \right )} + 12 b^{4} x \operatorname {asinh}^{2}{\left (c + d x \right )} + 24 b^{4} x - \frac {4 b^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (c + d x \right )}}{d} - \frac {24 b^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {asinh}{\left (c \right )}\right )^{4} & \text {otherwise} \end {cases} \]
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\[ \int (a+b \text {arcsinh}(c+d x))^4 \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4} \,d x } \]
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\[ \int (a+b \text {arcsinh}(c+d x))^4 \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4} \,d x } \]
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Timed out. \[ \int (a+b \text {arcsinh}(c+d x))^4 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4 \,d x \]
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