Integrand size = 26, antiderivative size = 287 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^2} \, dx=-\frac {b c (a+b \text {arcsinh}(c x))}{d^2 \sqrt {1+c^2 x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{d^2 x \left (1+c^2 x^2\right )}-\frac {3 c^2 x (a+b \text {arcsinh}(c x))^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac {3 c (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {b^2 c \arctan (c x)}{d^2}-\frac {4 b c (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {2 b^2 c \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {3 i b c (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {3 i b c (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {2 b^2 c \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {3 i b^2 c \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {3 i b^2 c \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{d^2} \]
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Time = 0.40 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {5809, 5788, 5789, 4265, 2611, 2320, 6724, 5798, 209, 5811, 5816, 4267, 2317, 2438} \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^2} \, dx=-\frac {3 c \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{d^2}-\frac {4 b c \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d^2}-\frac {b c (a+b \text {arcsinh}(c x))}{d^2 \sqrt {c^2 x^2+1}}-\frac {3 c^2 x (a+b \text {arcsinh}(c x))^2}{2 d^2 \left (c^2 x^2+1\right )}-\frac {(a+b \text {arcsinh}(c x))^2}{d^2 x \left (c^2 x^2+1\right )}+\frac {3 i b c \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d^2}-\frac {3 i b c \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d^2}-\frac {2 b^2 c \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {2 b^2 c \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {3 i b^2 c \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {3 i b^2 c \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {b^2 c \arctan (c x)}{d^2} \]
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Rule 209
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 4265
Rule 4267
Rule 5788
Rule 5789
Rule 5798
Rule 5809
Rule 5811
Rule 5816
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b \text {arcsinh}(c x))^2}{d^2 x \left (1+c^2 x^2\right )}-\left (3 c^2\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^2} \, dx+\frac {(2 b c) \int \frac {a+b \text {arcsinh}(c x)}{x \left (1+c^2 x^2\right )^{3/2}} \, dx}{d^2} \\ & = \frac {2 b c (a+b \text {arcsinh}(c x))}{d^2 \sqrt {1+c^2 x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{d^2 x \left (1+c^2 x^2\right )}-\frac {3 c^2 x (a+b \text {arcsinh}(c x))^2}{2 d^2 \left (1+c^2 x^2\right )}+\frac {(2 b c) \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {1+c^2 x^2}} \, dx}{d^2}-\frac {\left (2 b^2 c^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{d^2}+\frac {\left (3 b c^3\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{d^2}-\frac {\left (3 c^2\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx}{2 d} \\ & = -\frac {b c (a+b \text {arcsinh}(c x))}{d^2 \sqrt {1+c^2 x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{d^2 x \left (1+c^2 x^2\right )}-\frac {3 c^2 x (a+b \text {arcsinh}(c x))^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac {2 b^2 c \arctan (c x)}{d^2}-\frac {(3 c) \text {Subst}\left (\int (a+b x)^2 \text {sech}(x) \, dx,x,\text {arcsinh}(c x)\right )}{2 d^2}+\frac {(2 b c) \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arcsinh}(c x))}{d^2}+\frac {\left (3 b^2 c^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{d^2} \\ & = -\frac {b c (a+b \text {arcsinh}(c x))}{d^2 \sqrt {1+c^2 x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{d^2 x \left (1+c^2 x^2\right )}-\frac {3 c^2 x (a+b \text {arcsinh}(c x))^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac {3 c (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {b^2 c \arctan (c x)}{d^2}-\frac {4 b c (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {(3 i b c) \text {Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d^2}-\frac {(3 i b c) \text {Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d^2}-\frac {\left (2 b^2 c\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d^2}+\frac {\left (2 b^2 c\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d^2} \\ & = -\frac {b c (a+b \text {arcsinh}(c x))}{d^2 \sqrt {1+c^2 x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{d^2 x \left (1+c^2 x^2\right )}-\frac {3 c^2 x (a+b \text {arcsinh}(c x))^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac {3 c (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {b^2 c \arctan (c x)}{d^2}-\frac {4 b c (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {3 i b c (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {3 i b c (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {\left (3 i b^2 c\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d^2}+\frac {\left (3 i b^2 c\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{d^2}-\frac {\left (2 b^2 c\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {\left (2 b^2 c\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{d^2} \\ & = -\frac {b c (a+b \text {arcsinh}(c x))}{d^2 \sqrt {1+c^2 x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{d^2 x \left (1+c^2 x^2\right )}-\frac {3 c^2 x (a+b \text {arcsinh}(c x))^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac {3 c (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {b^2 c \arctan (c x)}{d^2}-\frac {4 b c (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {2 b^2 c \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {3 i b c (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {3 i b c (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {2 b^2 c \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {\left (3 i b^2 c\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {\left (3 i b^2 c\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{d^2} \\ & = -\frac {b c (a+b \text {arcsinh}(c x))}{d^2 \sqrt {1+c^2 x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{d^2 x \left (1+c^2 x^2\right )}-\frac {3 c^2 x (a+b \text {arcsinh}(c x))^2}{2 d^2 \left (1+c^2 x^2\right )}-\frac {3 c (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {b^2 c \arctan (c x)}{d^2}-\frac {4 b c (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {2 b^2 c \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {3 i b c (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {3 i b c (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {2 b^2 c \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {3 i b^2 c \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {3 i b^2 c \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{d^2} \\ \end{align*}
Time = 7.06 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.91 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^2} \, dx=-\frac {a^2}{d^2 x}-\frac {a^2 c^2 x}{2 d^2 \left (1+c^2 x^2\right )}-\frac {3 a^2 c \arctan (c x)}{2 d^2}+\frac {2 a b c \left (\frac {\sqrt {1+c^2 x^2}+i \text {arcsinh}(c x)}{4 (-1-i c x)}-\frac {\text {arcsinh}(c x)}{c x}-\frac {i \sqrt {1+c^2 x^2}+\text {arcsinh}(c x)}{4 (i+c x)}-\text {arctanh}\left (\sqrt {1+c^2 x^2}\right )+\frac {3}{4} i \left (-\frac {1}{2} \text {arcsinh}(c x)^2+2 \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )+2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )\right )-\frac {3}{4} i \left (-\frac {1}{2} \text {arcsinh}(c x)^2+2 \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )+2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )\right )}{d^2}+\frac {b^2 c \left (-\frac {2 \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}-\frac {c x \text {arcsinh}(c x)^2}{1+c^2 x^2}+4 \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-\text {arcsinh}(c x)^2 \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )+4 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )+3 i \text {arcsinh}(c x)^2 \log \left (1-i e^{-\text {arcsinh}(c x)}\right )-3 i \text {arcsinh}(c x)^2 \log \left (1+i e^{-\text {arcsinh}(c x)}\right )-4 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+4 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )+6 i \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )-6 i \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )-4 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )+6 i \operatorname {PolyLog}\left (3,-i e^{-\text {arcsinh}(c x)}\right )-6 i \operatorname {PolyLog}\left (3,i e^{-\text {arcsinh}(c x)}\right )+\text {arcsinh}(c x)^2 \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{2 d^2} \]
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\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}{x^{2} \left (c^{2} d \,x^{2}+d \right )^{2}}d x\]
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\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{2} x^{2}} \,d x } \]
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\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a^{2}}{c^{4} x^{6} + 2 c^{2} x^{4} + x^{2}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{4} x^{6} + 2 c^{2} x^{4} + x^{2}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{6} + 2 c^{2} x^{4} + x^{2}}\, dx}{d^{2}} \]
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\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{2} x^{2}} \,d x } \]
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\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{2} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^2\,{\left (d\,c^2\,x^2+d\right )}^2} \,d x \]
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