Integrand size = 37, antiderivative size = 972 \[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=\frac {8 i a b d^4 x \left (1+c^2 x^2\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {8 i b^2 d^4 \left (1+c^2 x^2\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {b^2 d^4 x \left (1+c^2 x^2\right )^2}{4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b^2 d^4 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {8 i b^2 d^4 x \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b c d^4 x^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {8 i d^4 \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 {8 d^4 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 {8 d^4 \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 d^4 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {d^4 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {5 d^4 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^3}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {32 i b d^4 \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 {16 b d^4 \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 {16 b^2 d^4 \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 {16 b^2 d^4 \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 {8 b^2 d^4 \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}} \]
[Out]
Time = 0.93 (sec) , antiderivative size = 972, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.514, Rules used = {5796, 5844, 5838, 5787, 5797, 3799, 2221, 2317, 2438, 5798, 5789, 4265, 5783, 5772, 267, 5812, 5776, 327, 221} \[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=-\frac {5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^3 d^4}{2 b c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac {b^2 x \left (c^2 x^2+1\right )^2 d^4}{4 (i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac {8 i b^2 \left (c^2 x^2+1\right )^2 d^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac {x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2 d^4}{2 (i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac {4 i \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2 d^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac {8 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2 d^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac {8 x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2 d^4}{(i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac {8 i \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2 d^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac {8 i a b x \left (c^2 x^2+1\right )^{3/2} d^4}{(i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac {8 i b^2 x \left (c^2 x^2+1\right )^{3/2} \text {arcsinh}(c x) d^4}{(i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac {b^2 \left (c^2 x^2+1\right )^{3/2} \text {arcsinh}(c x) d^4}{4 c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac {b c x^2 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x)) d^4}{2 (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac {32 i b \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right ) d^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac {16 b \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right ) d^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac {16 b^2 \left (c^2 x^2+1\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) d^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac {16 b^2 \left (c^2 x^2+1\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) d^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac {8 b^2 \left (c^2 x^2+1\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right ) d^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}} \]
[In]
[Out]
Rule 221
Rule 267
Rule 327
Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 4265
Rule 5772
Rule 5776
Rule 5783
Rule 5787
Rule 5789
Rule 5796
Rule 5797
Rule 5798
Rule 5812
Rule 5838
Rule 5844
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+c^2 x^2\right )^{3/2} \int \frac {(d+i c d x)^4 (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 {8 i \left (i d^4-c d^4 x\right ) (a+b \text {arcsinh}(c x))^2}{\left (1+c^2 x^2\right )^{3/2}}-\frac {7 d^4 (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}}-\frac {4 i c d^4 x (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}}+\frac {c^2 d^4 x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}}\right ) \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = -\frac {\left (8 i \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {\left (i d^4-c d^4 x\right ) (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 (7 d^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (4 i c d^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (c^2 d^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = -\frac {4 i d^4 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {d^4 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {7 d^4 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^3}{3 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (8 i \left (1+c^2 x^2\right )^{3/2}\right ) \int \left (\frac {i d^4 (a+b \text {arcsinh}(c x))^2}{\left (1+c^2 x^2\right )^{3/2}}-\frac {c d^4 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 (d^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}} \, dx}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (8 i b d^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int (a+b \text {arcsinh}(c x)) \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (b c d^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int x (a+b \text {arcsinh}(c x)) \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = \frac {8 i a b d^4 x \left (1+c^2 x^2\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b c d^4 x^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {4 i d^4 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {d^4 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {5 d^4 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^3}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (8 d^4 \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 (8 i b^2 d^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \text {arcsinh}(c x) \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (8 i c d^4 \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 {\left (b^2 c^2 d^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = \frac {8 i a b d^4 x \left (1+c^2 x^2\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {b^2 d^4 x \left (1+c^2 x^2\right )^2}{4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {8 i b^2 d^4 x \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b c d^4 x^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {8 i d^4 \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 {8 d^4 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 {4 i d^4 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {d^4 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {5 d^4 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^3}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (16 i b d^4 \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 (b^2 d^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (16 b c d^4 \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 {\left (8 i b^2 c d^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = \frac {8 i a b d^4 x \left (1+c^2 x^2\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {8 i b^2 d^4 \left (1+c^2 x^2\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {b^2 d^4 x \left (1+c^2 x^2\right )^2}{4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b^2 d^4 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {8 i b^2 d^4 x \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b c d^4 x^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {8 i d^4 \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 {8 d^4 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 {4 i d^4 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {d^4 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {5 d^4 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^3}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (16 i b d^4 \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 (16 b d^4 \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 {8 i a b d^4 x \left (1+c^2 x^2\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {8 i b^2 d^4 \left (1+c^2 x^2\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {b^2 d^4 x \left (1+c^2 x^2\right )^2}{4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b^2 d^4 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {8 i b^2 d^4 x \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b c d^4 x^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {8 i d^4 \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 {8 d^4 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 {8 d^4 \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 d^4 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {d^4 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {5 d^4 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^3}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {32 i b d^4 \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 (32 b d^4 \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 (16 b^2 d^4 \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 (16 b^2 d^4 \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 {8 i a b d^4 x \left (1+c^2 x^2\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {8 i b^2 d^4 \left (1+c^2 x^2\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {b^2 d^4 x \left (1+c^2 x^2\right )^2}{4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b^2 d^4 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {8 i b^2 d^4 x \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b c d^4 x^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {8 i d^4 \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 {8 d^4 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 {8 d^4 \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 d^4 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {d^4 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {5 d^4 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^3}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {32 i b d^4 \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 {16 b d^4 \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 (16 b^2 d^4 \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 (16 b^2 d^4 \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 (16 b^2 d^4 \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 {8 i a b d^4 x \left (1+c^2 x^2\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {8 i b^2 d^4 \left (1+c^2 x^2\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {b^2 d^4 x \left (1+c^2 x^2\right )^2}{4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b^2 d^4 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {8 i b^2 d^4 x \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b c d^4 x^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {8 i d^4 \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 {8 d^4 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 {8 d^4 \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 d^4 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {d^4 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {5 d^4 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^3}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {32 i b d^4 \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 {16 b d^4 \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 {16 b^2 d^4 \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 {16 b^2 d^4 \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 (8 b^2 d^4 \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 {8 i a b d^4 x \left (1+c^2 x^2\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {8 i b^2 d^4 \left (1+c^2 x^2\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {b^2 d^4 x \left (1+c^2 x^2\right )^2}{4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b^2 d^4 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {8 i b^2 d^4 x \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b c d^4 x^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {8 i d^4 \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 {8 d^4 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 {8 d^4 \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 d^4 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {d^4 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {5 d^4 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^3}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {32 i b d^4 \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 {16 b d^4 \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 {16 b^2 d^4 \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 {16 b^2 d^4 \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 {8 b^2 d^4 \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*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2143\) vs. \(2(972)=1944\).
Time = 25.04 (sec) , antiderivative size = 2143, normalized size of antiderivative = 2.20 \[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=\text {Result too large to show} \]
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\[\int \frac {\left (i c d x +d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}{\left (-i c f x +f \right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (-i \, c f x + f\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (-i \, c f x + f\right )}^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{5/2}}{{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]
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