Integrand size = 21, antiderivative size = 115 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^6} \, dx=-\frac {b \sqrt {1+(c+d x)^2}}{20 d e^6 (c+d x)^4}+\frac {3 b \sqrt {1+(c+d x)^2}}{40 d e^6 (c+d x)^2}-\frac {a+b \text {arcsinh}(c+d x)}{5 d e^6 (c+d x)^5}-\frac {3 b \text {arctanh}\left (\sqrt {1+(c+d x)^2}\right )}{40 d e^6} \]
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Time = 0.06 (sec) , antiderivative size = 115, 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, 272, 44, 65, 213} \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^6} \, dx=-\frac {a+b \text {arcsinh}(c+d x)}{5 d e^6 (c+d x)^5}-\frac {3 b \text {arctanh}\left (\sqrt {(c+d x)^2+1}\right )}{40 d e^6}+\frac {3 b \sqrt {(c+d x)^2+1}}{40 d e^6 (c+d x)^2}-\frac {b \sqrt {(c+d x)^2+1}}{20 d e^6 (c+d x)^4} \]
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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^6 x^6} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {a+b \text {arcsinh}(x)}{x^6} \, dx,x,c+d x\right )}{d e^6} \\ & = -\frac {a+b \text {arcsinh}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {b \text {Subst}\left (\int \frac {1}{x^5 \sqrt {1+x^2}} \, dx,x,c+d x\right )}{5 d e^6} \\ & = -\frac {a+b \text {arcsinh}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {b \text {Subst}\left (\int \frac {1}{x^3 \sqrt {1+x}} \, dx,x,(c+d x)^2\right )}{10 d e^6} \\ & = -\frac {b \sqrt {1+(c+d x)^2}}{20 d e^6 (c+d x)^4}-\frac {a+b \text {arcsinh}(c+d x)}{5 d e^6 (c+d x)^5}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,(c+d x)^2\right )}{40 d e^6} \\ & = -\frac {b \sqrt {1+(c+d x)^2}}{20 d e^6 (c+d x)^4}+\frac {3 b \sqrt {1+(c+d x)^2}}{40 d e^6 (c+d x)^2}-\frac {a+b \text {arcsinh}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,(c+d x)^2\right )}{80 d e^6} \\ & = -\frac {b \sqrt {1+(c+d x)^2}}{20 d e^6 (c+d x)^4}+\frac {3 b \sqrt {1+(c+d x)^2}}{40 d e^6 (c+d x)^2}-\frac {a+b \text {arcsinh}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+(c+d x)^2}\right )}{40 d e^6} \\ & = -\frac {b \sqrt {1+(c+d x)^2}}{20 d e^6 (c+d x)^4}+\frac {3 b \sqrt {1+(c+d x)^2}}{40 d e^6 (c+d x)^2}-\frac {a+b \text {arcsinh}(c+d x)}{5 d e^6 (c+d x)^5}-\frac {3 b \text {arctanh}\left (\sqrt {1+(c+d x)^2}\right )}{40 d e^6} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.53 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^6} \, dx=-\frac {\frac {a+b \text {arcsinh}(c+d x)}{(c+d x)^5}+b \sqrt {1+(c+d x)^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},3,\frac {3}{2},1+(c+d x)^2\right )}{5 d e^6} \]
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Time = 0.40 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {-\frac {a}{5 e^{6} \left (d x +c \right )^{5}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{5 \left (d x +c \right )^{5}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{20 \left (d x +c \right )^{4}}+\frac {3 \sqrt {1+\left (d x +c \right )^{2}}}{40 \left (d x +c \right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )}{40}\right )}{e^{6}}}{d}\) | \(94\) |
default | \(\frac {-\frac {a}{5 e^{6} \left (d x +c \right )^{5}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{5 \left (d x +c \right )^{5}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{20 \left (d x +c \right )^{4}}+\frac {3 \sqrt {1+\left (d x +c \right )^{2}}}{40 \left (d x +c \right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )}{40}\right )}{e^{6}}}{d}\) | \(94\) |
parts | \(-\frac {a}{5 e^{6} \left (d x +c \right )^{5} d}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{5 \left (d x +c \right )^{5}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{20 \left (d x +c \right )^{4}}+\frac {3 \sqrt {1+\left (d x +c \right )^{2}}}{40 \left (d x +c \right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )}{40}\right )}{e^{6} d}\) | \(96\) |
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Leaf count of result is larger than twice the leaf count of optimal. 509 vs. \(2 (101) = 202\).
Time = 0.31 (sec) , antiderivative size = 509, normalized size of antiderivative = 4.43 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^6} \, dx=-\frac {8 \, a c^{5} - 8 \, {\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) + 3 \, {\left (b c^{5} d^{5} x^{5} + 5 \, b c^{6} d^{4} x^{4} + 10 \, b c^{7} d^{3} x^{3} + 10 \, b c^{8} d^{2} x^{2} + 5 \, b c^{9} d x + b c^{10}\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} + 1\right ) - 8 \, {\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x + b c^{5}\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 3 \, {\left (b c^{5} d^{5} x^{5} + 5 \, b c^{6} d^{4} x^{4} + 10 \, b c^{7} d^{3} x^{3} + 10 \, b c^{8} d^{2} x^{2} + 5 \, b c^{9} d x + b c^{10}\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} - 1\right ) - {\left (3 \, b c^{5} d^{3} x^{3} + 9 \, b c^{6} d^{2} x^{2} + 3 \, b c^{8} - 2 \, b c^{6} + {\left (9 \, b c^{7} - 2 \, b c^{5}\right )} d x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{40 \, {\left (c^{5} d^{6} e^{6} x^{5} + 5 \, c^{6} d^{5} e^{6} x^{4} + 10 \, c^{7} d^{4} e^{6} x^{3} + 10 \, c^{8} d^{3} e^{6} x^{2} + 5 \, c^{9} d^{2} e^{6} x + c^{10} d e^{6}\right )}} \]
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\[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^6} \, dx=\frac {\int \frac {a}{c^{6} + 6 c^{5} d x + 15 c^{4} d^{2} x^{2} + 20 c^{3} d^{3} x^{3} + 15 c^{2} d^{4} x^{4} + 6 c d^{5} x^{5} + d^{6} x^{6}}\, dx + \int \frac {b \operatorname {asinh}{\left (c + d x \right )}}{c^{6} + 6 c^{5} d x + 15 c^{4} d^{2} x^{2} + 20 c^{3} d^{3} x^{3} + 15 c^{2} d^{4} x^{4} + 6 c d^{5} x^{5} + d^{6} x^{6}}\, dx}{e^{6}} \]
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\[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^6} \, dx=\int { \frac {b \operatorname {arsinh}\left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{6}} \,d x } \]
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\[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^6} \, dx=\int { \frac {b \operatorname {arsinh}\left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{6}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^6} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^6} \,d x \]
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