Integrand size = 37, antiderivative size = 743 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{3/2}} \, dx=-\frac {i b^2 f \left (1+c^2 x^2\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b^2 f x \left (1+c^2 x^2\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b f \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {i b f x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {i f \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {f x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 f x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 f \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i b f \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {4 b f \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b^2 f \left (1+c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b^2 f \left (1+c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b^2 f \left (1+c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]
[Out]
Time = 0.64 (sec) , antiderivative size = 743, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.378, Rules used = {5796, 5838, 5788, 5787, 5797, 3799, 2221, 2317, 2438, 5798, 197, 5789, 4265, 267} \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{3/2}} \, dx=-\frac {2 i b f \left (c^2 x^2+1\right )^{5/2} \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 f \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 f x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b f \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {i b f x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {i f \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {f x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {4 b f \left (c^2 x^2+1\right )^{5/2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b^2 f \left (c^2 x^2+1\right )^{5/2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b^2 f \left (c^2 x^2+1\right )^{5/2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b^2 f \left (c^2 x^2+1\right )^{5/2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {i b^2 f \left (c^2 x^2+1\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b^2 f x \left (c^2 x^2+1\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]
[In]
[Out]
Rule 197
Rule 267
Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 4265
Rule 5787
Rule 5788
Rule 5789
Rule 5796
Rule 5797
Rule 5798
Rule 5838
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+c^2 x^2\right )^{5/2} \int \frac {(f-i c f x) (a+b \text {arcsinh}(c x))^2}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {\left (1+c^2 x^2\right )^{5/2} \int \left (\frac {f (a+b \text {arcsinh}(c x))^2}{\left (1+c^2 x^2\right )^{5/2}}-\frac {i c f x (a+b \text {arcsinh}(c x))^2}{\left (1+c^2 x^2\right )^{5/2}}\right ) \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {\left (f \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (i c f \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))^2}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {i f \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {f x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 f \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (2 i b f \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{\left (1+c^2 x^2\right )^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (2 b c f \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{\left (1+c^2 x^2\right )^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = \frac {b f \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {i b f x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {i f \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {f x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 f x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (i b f \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{1+c^2 x^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (b^2 f \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {1}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (4 b c f \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{1+c^2 x^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (i b^2 c f \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {x}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = -\frac {i b^2 f \left (1+c^2 x^2\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b^2 f x \left (1+c^2 x^2\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b f \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {i b f x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {i f \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {f x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 f x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (i b f \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (4 b f \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}(\int (a+b x) \tanh (x) \, dx,x,\text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = -\frac {i b^2 f \left (1+c^2 x^2\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b^2 f x \left (1+c^2 x^2\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b f \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {i b f x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {i f \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {f x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 f x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 f \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i b f \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (8 b f \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (b^2 f \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (b^2 f \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = -\frac {i b^2 f \left (1+c^2 x^2\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b^2 f x \left (1+c^2 x^2\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b f \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {i b f x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {i f \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {f x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 f x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 f \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i b f \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {4 b f \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (b^2 f \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (b^2 f \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (4 b^2 f \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = -\frac {i b^2 f \left (1+c^2 x^2\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b^2 f x \left (1+c^2 x^2\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b f \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {i b f x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {i f \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {f x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 f x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 f \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i b f \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {4 b f \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b^2 f \left (1+c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b^2 f \left (1+c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 b^2 f \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ & = -\frac {i b^2 f \left (1+c^2 x^2\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b^2 f x \left (1+c^2 x^2\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b f \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {i b f x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {i f \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {f x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 f x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 f \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i b f \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {4 b f \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b^2 f \left (1+c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b^2 f \left (1+c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b^2 f \left (1+c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \\ \end{align*}
Time = 9.91 (sec) , antiderivative size = 754, normalized size of antiderivative = 1.01 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{3/2}} \, dx=\frac {\sqrt {i d (-i+c x)} \sqrt {-i f (i+c x)} \left (-\frac {i a^2}{6 d^3 f^2 (-i+c x)^2}+\frac {5 a^2}{12 d^3 f^2 (-i+c x)}+\frac {a^2}{4 d^3 f^2 (i+c x)}\right )}{c}+\frac {i a b \sqrt {i (-i d+c d x)} \sqrt {-i (i f+c f x)} \left (4 c x \text {arcsinh}(c x)+2 i \text {arcsinh}(c x) \cosh (2 \text {arcsinh}(c x))+\sqrt {1+c^2 x^2} \left (1-2 i \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+2 c x \left (\arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-2 i \log \left (\sqrt {1+c^2 x^2}\right )\right )-4 \log \left (\sqrt {1+c^2 x^2}\right )\right )\right )}{3 c d^2 f (-i+c x) \sqrt {-((-i d+c d x) (i f+c f x))} \sqrt {-d f \left (1+c^2 x^2\right )}}+\frac {i b^2 \sqrt {i (-i d+c d x)} \sqrt {-i (i f+c f x)} \sqrt {1+c^2 x^2} \left (7 \pi \text {arcsinh}(c x)+\frac {(2+i \text {arcsinh}(c x)) \text {arcsinh}(c x)}{-i+c x}-(1+4 i) \text {arcsinh}(c x)^2-5 (\pi +2 i \text {arcsinh}(c x)) \log \left (1-i e^{-\text {arcsinh}(c x)}\right )+3 (\pi -2 i \text {arcsinh}(c x)) \log \left (1+i e^{-\text {arcsinh}(c x)}\right )-16 \pi \log \left (1+e^{\text {arcsinh}(c x)}\right )-3 \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )+16 \pi \log \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+5 \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )+6 i \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )+10 i \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )+\frac {3 i \text {arcsinh}(c x)^2 \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )}{\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )}+\frac {2 i \text {arcsinh}(c x)^2 \sinh \left (\frac {1}{2} \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 )^3}+\frac {\left (-4+5 \text {arcsinh}(c x)^2\right ) \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )}{-i \cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+\sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )}\right )}{6 c d^2 f \sqrt {-((-i d+c d x) (i f+c f x))} \sqrt {-d f \left (1+c^2 x^2\right )}} \]
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\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}{\left (i c d x +d \right )^{\frac {5}{2}} \left (-i c f x +f \right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (i \, c d x + d\right )}^{\frac {5}{2}} {\left (-i \, c f x + f\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{3/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{5/2}\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]
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