Integrand size = 37, antiderivative size = 464 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} \sqrt {f-i c f x}} \, dx=\frac {i f \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {f x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {f \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {4 i b f \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {2 b f \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {2 b^2 f \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 b^2 f \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b^2 f \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]
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Time = 0.54 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.297, Rules used = {5796, 5838, 5787, 5797, 3799, 2221, 2317, 2438, 5798, 5789, 4265} \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} \sqrt {f-i c f x}} \, dx=-\frac {4 i b f \left (c^2 x^2+1\right )^{3/2} \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {f \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {i f \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {f x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {2 b f \left (c^2 x^2+1\right )^{3/2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {2 b^2 f \left (c^2 x^2+1\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 b^2 f \left (c^2 x^2+1\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b^2 f \left (c^2 x^2+1\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]
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Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 4265
Rule 5787
Rule 5789
Rule 5796
Rule 5797
Rule 5798
Rule 5838
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+c^2 x^2\right )^{3/2} \int \frac {(f-i c f x) (a+b \text {arcsinh}(c x))^2}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = \frac {\left (1+c^2 x^2\right )^{3/2} \int \left (\frac {f (a+b \text {arcsinh}(c x))^2}{\left (1+c^2 x^2\right )^{3/2}}-\frac {i c f x (a+b \text {arcsinh}(c x))^2}{\left (1+c^2 x^2\right )^{3/2}}\right ) \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = \frac {\left (f \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (i c f \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))^2}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = \frac {i f \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {f x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (2 i b f \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{1+c^2 x^2} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (2 b c f \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{1+c^2 x^2} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = \frac {i f \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {f x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (2 i b f \left (1+c^2 x^2\right )^{3/2}\right ) \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (2 b f \left (1+c^2 x^2\right )^{3/2}\right ) \text {Subst}(\int (a+b x) \tanh (x) \, dx,x,\text {arcsinh}(c x))}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = \frac {i f \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {f x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {f \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {4 i b f \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (4 b f \left (1+c^2 x^2\right )^{3/2}\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\text {arcsinh}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (2 b^2 f \left (1+c^2 x^2\right )^{3/2}\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (2 b^2 f \left (1+c^2 x^2\right )^{3/2}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = \frac {i f \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {f x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {f \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {4 i b f \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {2 b f \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (2 b^2 f \left (1+c^2 x^2\right )^{3/2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (2 b^2 f \left (1+c^2 x^2\right )^{3/2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (2 b^2 f \left (1+c^2 x^2\right )^{3/2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = \frac {i f \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {f x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {f \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {4 i b f \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {2 b f \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {2 b^2 f \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 b^2 f \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (b^2 f \left (1+c^2 x^2\right )^{3/2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = \frac {i f \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {f x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {f \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {4 i b f \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {2 b f \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {2 b^2 f \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 b^2 f \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b^2 f \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ \end{align*}
Time = 3.26 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.09 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} \sqrt {f-i c f x}} \, dx=\frac {\sqrt {d+i c d x} \sqrt {f-i c f x} \left ((-1+i) b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)^2 \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-\sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+\left (i a^2+a^2 c x-4 i a b \sqrt {1+c^2 x^2} \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+2 i b^2 \pi \sqrt {1+c^2 x^2} \log \left (1-i e^{-\text {arcsinh}(c x)}\right )+4 i b^2 \pi \sqrt {1+c^2 x^2} \log \left (1+e^{\text {arcsinh}(c x)}\right )-a b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )-4 i b^2 \pi \sqrt {1+c^2 x^2} \log \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-2 i b^2 \pi \sqrt {1+c^2 x^2} \log \left (\sin \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )\right ) \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+4 b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right ) \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+b \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \left (i \cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right ) \left (2 a-b \pi +4 i b \log \left (1-i e^{-\text {arcsinh}(c x)}\right )\right )+\left (2 a+b \pi -4 i b \log \left (1-i e^{-\text {arcsinh}(c x)}\right )\right ) \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )\right )}{c d^2 f (-i+c x) (i+c x) \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )} \]
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\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}{\left (i c d x +d \right )^{\frac {3}{2}} \sqrt {-i c f x +f}}d x\]
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\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} \sqrt {f-i c f x}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (i \, c d x + d\right )}^{\frac {3}{2}} \sqrt {-i \, c f x + f}} \,d x } \]
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\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} \sqrt {f-i c f x}} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (i d \left (c x - i\right )\right )^{\frac {3}{2}} \sqrt {- i f \left (c x + i\right )}}\, dx \]
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\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} \sqrt {f-i c f x}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (i \, c d x + d\right )}^{\frac {3}{2}} \sqrt {-i \, c f x + f}} \,d x } \]
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\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} \sqrt {f-i c f x}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (i \, c d x + d\right )}^{\frac {3}{2}} \sqrt {-i \, c f x + f}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} \sqrt {f-i c f x}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{3/2}\,\sqrt {f-c\,f\,x\,1{}\mathrm {i}}} \,d x \]
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