Integrand size = 21, antiderivative size = 84 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^4} \, dx=-\frac {b \sqrt {1+(c+d x)^2}}{6 d e^4 (c+d x)^2}-\frac {a+b \text {arcsinh}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \text {arctanh}\left (\sqrt {1+(c+d x)^2}\right )}{6 d e^4} \]
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Time = 0.05 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5859, 12, 5776, 272, 44, 65, 213} \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^4} \, dx=-\frac {a+b \text {arcsinh}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \text {arctanh}\left (\sqrt {(c+d x)^2+1}\right )}{6 d e^4}-\frac {b \sqrt {(c+d x)^2+1}}{6 d e^4 (c+d x)^2} \]
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Rule 12
Rule 44
Rule 65
Rule 213
Rule 272
Rule 5776
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a+b \text {arcsinh}(x)}{e^4 x^4} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {a+b \text {arcsinh}(x)}{x^4} \, dx,x,c+d x\right )}{d e^4} \\ & = -\frac {a+b \text {arcsinh}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \text {Subst}\left (\int \frac {1}{x^3 \sqrt {1+x^2}} \, dx,x,c+d x\right )}{3 d e^4} \\ & = -\frac {a+b \text {arcsinh}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,(c+d x)^2\right )}{6 d e^4} \\ & = -\frac {b \sqrt {1+(c+d x)^2}}{6 d e^4 (c+d x)^2}-\frac {a+b \text {arcsinh}(c+d x)}{3 d e^4 (c+d x)^3}-\frac {b \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,(c+d x)^2\right )}{12 d e^4} \\ & = -\frac {b \sqrt {1+(c+d x)^2}}{6 d e^4 (c+d x)^2}-\frac {a+b \text {arcsinh}(c+d x)}{3 d e^4 (c+d x)^3}-\frac {b \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+(c+d x)^2}\right )}{6 d e^4} \\ & = -\frac {b \sqrt {1+(c+d x)^2}}{6 d e^4 (c+d x)^2}-\frac {a+b \text {arcsinh}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \text {arctanh}\left (\sqrt {1+(c+d x)^2}\right )}{6 d e^4} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.82 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^4} \, dx=\frac {-\frac {2 (a+b \text {arcsinh}(c+d x))}{(c+d x)^3}+b \left (-\frac {\sqrt {1+(c+d x)^2}}{(c+d x)^2}+\text {arctanh}\left (\sqrt {1+(c+d x)^2}\right )\right )}{6 d e^4} \]
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Time = 0.45 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {-\frac {a}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{6 \left (d x +c \right )^{2}}+\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )}{6}\right )}{e^{4}}}{d}\) | \(74\) |
default | \(\frac {-\frac {a}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{6 \left (d x +c \right )^{2}}+\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )}{6}\right )}{e^{4}}}{d}\) | \(74\) |
parts | \(-\frac {a}{3 e^{4} \left (d x +c \right )^{3} d}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{6 \left (d x +c \right )^{2}}+\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )}{6}\right )}{e^{4} d}\) | \(76\) |
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Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (74) = 148\).
Time = 0.29 (sec) , antiderivative size = 343, normalized size of antiderivative = 4.08 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^4} \, dx=-\frac {2 \, a c^{3} - 2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - {\left (b c^{3} d^{3} x^{3} + 3 \, b c^{4} d^{2} x^{2} + 3 \, b c^{5} d x + b c^{6}\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} + 1\right ) - 2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) + {\left (b c^{3} d^{3} x^{3} + 3 \, b c^{4} d^{2} x^{2} + 3 \, b c^{5} d x + b c^{6}\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} - 1\right ) + {\left (b c^{3} d x + b c^{4}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{6 \, {\left (c^{3} d^{4} e^{4} x^{3} + 3 \, c^{4} d^{3} e^{4} x^{2} + 3 \, c^{5} d^{2} e^{4} x + c^{6} d e^{4}\right )}} \]
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\[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^4} \, dx=\frac {\int \frac {a}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {b \operatorname {asinh}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{e^{4}} \]
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\[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^4} \, dx=\int { \frac {b \operatorname {arsinh}\left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{4}} \,d x } \]
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\[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^4} \, dx=\int { \frac {b \operatorname {arsinh}\left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{4}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^4} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^4} \,d x \]
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