Integrand size = 21, antiderivative size = 59 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^3} \, dx=-\frac {b \sqrt {1+(c+d x)^2}}{2 d e^3 (c+d x)}-\frac {a+b \text {arcsinh}(c+d x)}{2 d e^3 (c+d x)^2} \]
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Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5859, 12, 5776, 270} \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^3} \, dx=-\frac {a+b \text {arcsinh}(c+d x)}{2 d e^3 (c+d x)^2}-\frac {b \sqrt {(c+d x)^2+1}}{2 d e^3 (c+d x)} \]
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Rule 12
Rule 270
Rule 5776
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a+b \text {arcsinh}(x)}{e^3 x^3} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {a+b \text {arcsinh}(x)}{x^3} \, dx,x,c+d x\right )}{d e^3} \\ & = -\frac {a+b \text {arcsinh}(c+d x)}{2 d e^3 (c+d x)^2}+\frac {b \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x^2}} \, dx,x,c+d x\right )}{2 d e^3} \\ & = -\frac {b \sqrt {1+(c+d x)^2}}{2 d e^3 (c+d x)}-\frac {a+b \text {arcsinh}(c+d x)}{2 d e^3 (c+d x)^2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.97 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^3} \, dx=\frac {-\frac {b \sqrt {1+(c+d x)^2}}{2 (c+d x)}+\frac {-a-b \text {arcsinh}(c+d x)}{2 (c+d x)^2}}{d e^3} \]
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Time = 0.37 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.02
method | result | size |
derivativedivides | \(\frac {-\frac {a}{2 e^{3} \left (d x +c \right )^{2}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{2 \left (d x +c \right )}\right )}{e^{3}}}{d}\) | \(60\) |
default | \(\frac {-\frac {a}{2 e^{3} \left (d x +c \right )^{2}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{2 \left (d x +c \right )}\right )}{e^{3}}}{d}\) | \(60\) |
parts | \(-\frac {a}{2 e^{3} \left (d x +c \right )^{2} d}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{2 \left (d x +c \right )}\right )}{e^{3} d}\) | \(62\) |
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Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (53) = 106\).
Time = 0.28 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.00 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^3} \, dx=\frac {a d^{2} x^{2} + 2 \, a c d x - b c^{2} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - {\left (b c^{2} d x + b c^{3}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{2 \, {\left (c^{2} d^{3} e^{3} x^{2} + 2 \, c^{3} d^{2} e^{3} x + c^{4} d e^{3}\right )}} \]
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\[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^3} \, dx=\frac {\int \frac {a}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {b \operatorname {asinh}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx}{e^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (53) = 106\).
Time = 0.21 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.98 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^3} \, dx=-\frac {1}{2} \, b {\left (\frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} d}{d^{3} e^{3} x + c d^{2} e^{3}} + \frac {\operatorname {arsinh}\left (d x + c\right )}{d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}}\right )} - \frac {a}{2 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} \]
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\[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^3} \, dx=\int { \frac {b \operatorname {arsinh}\left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^3} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^3} \,d x \]
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