Integrand size = 26, antiderivative size = 204 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )} \, dx=-\frac {(a+b \text {arcsinh}(c x))^2}{d x}-\frac {2 c (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d}-\frac {4 b c (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d}-\frac {2 b^2 c \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d}+\frac {2 i b c (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{d}-\frac {2 i b c (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{d}+\frac {2 b^2 c \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d}-\frac {2 i b^2 c \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{d}+\frac {2 i b^2 c \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{d} \]
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Time = 0.24 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5809, 5789, 4265, 2611, 2320, 6724, 5816, 4267, 2317, 2438} \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )} \, dx=-\frac {2 c \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{d}-\frac {4 b c \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d}+\frac {2 i b c \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d}-\frac {2 i b c \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{d x}-\frac {2 b^2 c \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d}+\frac {2 b^2 c \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d}-\frac {2 i b^2 c \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{d}+\frac {2 i b^2 c \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{d} \]
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Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 4265
Rule 4267
Rule 5789
Rule 5809
Rule 5816
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b \text {arcsinh}(c x))^2}{d x}-c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx+\frac {(2 b c) \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {1+c^2 x^2}} \, dx}{d} \\ & = -\frac {(a+b \text {arcsinh}(c x))^2}{d x}-\frac {c \text {Subst}\left (\int (a+b x)^2 \text {sech}(x) \, dx,x,\text {arcsinh}(c x)\right )}{d}+\frac {(2 b c) \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arcsinh}(c x))}{d} \\ & = -\frac {(a+b \text {arcsinh}(c x))^2}{d x}-\frac {2 c (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d}-\frac {4 b c (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d}+\frac {(2 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}-\frac {(2 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}-\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}+\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} \\ & = -\frac {(a+b \text {arcsinh}(c x))^2}{d x}-\frac {2 c (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d}-\frac {4 b c (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d}+\frac {2 i b c (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{d}-\frac {2 i b c (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{d}-\frac {\left (2 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}+\frac {\left (2 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}-\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}+\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} \\ & = -\frac {(a+b \text {arcsinh}(c x))^2}{d x}-\frac {2 c (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d}-\frac {4 b c (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d}-\frac {2 b^2 c \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d}+\frac {2 i b c (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{d}-\frac {2 i b c (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{d}+\frac {2 b^2 c \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d}-\frac {\left (2 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}+\frac {\left (2 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} \\ & = -\frac {(a+b \text {arcsinh}(c x))^2}{d x}-\frac {2 c (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d}-\frac {4 b c (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d}-\frac {2 b^2 c \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d}+\frac {2 i b c (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{d}-\frac {2 i b c (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{d}+\frac {2 b^2 c \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d}-\frac {2 i b^2 c \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{d}+\frac {2 i b^2 c \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{d} \\ \end{align*}
Time = 0.80 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.78 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )} \, dx=-\frac {\frac {a^2}{x}+\frac {2 a b \text {arcsinh}(c x)}{x}+a^2 c \arctan (c x)+2 a b c \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )+\frac {1}{2} i a b c \left (\text {arcsinh}(c x) \left (\text {arcsinh}(c x)-4 \log \left (1+i e^{\text {arcsinh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )\right )-\frac {1}{2} i a b c \left (\text {arcsinh}(c x) \left (\text {arcsinh}(c x)-4 \log \left (1-i e^{\text {arcsinh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )-b^2 c \left (-\frac {\text {arcsinh}(c x)^2}{c x}+2 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )+i \text {arcsinh}(c x)^2 \log \left (1-i e^{-\text {arcsinh}(c x)}\right )-i \text {arcsinh}(c x)^2 \log \left (1+i e^{-\text {arcsinh}(c x)}\right )-2 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+2 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )+2 i \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )-2 i \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )-2 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )+2 i \operatorname {PolyLog}\left (3,-i e^{-\text {arcsinh}(c x)}\right )-2 i \operatorname {PolyLog}\left (3,i e^{-\text {arcsinh}(c x)}\right )\right )}{d} \]
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\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}{x^{2} \left (c^{2} d \,x^{2}+d \right )}d x\]
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\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )} x^{2}} \,d x } \]
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\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )} \, dx=\frac {\int \frac {a^{2}}{c^{2} x^{4} + x^{2}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{2} x^{4} + x^{2}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{4} + x^{2}}\, dx}{d} \]
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\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )} x^{2}} \,d x } \]
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\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^2\,\left (d\,c^2\,x^2+d\right )} \,d x \]
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