Optimal. Leaf size=942 \[ \frac {2 i b^2 d^2 \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 {2 b^2 d^2 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^2 d^2 \left (1+c^2 x^2\right )^{5/2} \sinh ^{-1}(c x)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b d^2 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b c d^2 x^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {c^2 d^2 x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 d^2 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {4 i b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \text {ArcTan}\left (e^{\sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 b^2 d^2 \left (1+c^2 x^2\right )^{5/2} \text {PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b^2 d^2 \left (1+c^2 x^2\right )^{5/2} \text {PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b^2 d^2 \left (1+c^2 x^2\right )^{5/2} \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.86, antiderivative size = 942, normalized size of antiderivative = 1.00, number of steps
used = 30, number of rules used = 18, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.486, Rules used = {5796, 5838,
5788, 5787, 5797, 3799, 2221, 2317, 2438, 5798, 197, 5789, 4265, 267, 5800, 5810, 294, 221}
\begin {gather*} -\frac {c^2 d^2 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 x^3}{3 (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac {b c d^2 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) x^2}{3 (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b^2 d^2 \left (c^2 x^2+1\right )^2 x}{3 (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac {2 d^2 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 x}{3 (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 x}{3 (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b d^2 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) x}{3 (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b^2 d^2 \left (c^2 x^2+1\right )^2}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i d^2 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac {b^2 d^2 \left (c^2 x^2+1\right )^{5/2} \sinh ^{-1}(c x)}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac {b d^2 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac {4 i b d^2 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \text {ArcTan}\left (e^{\sinh ^{-1}(c x)}\right )}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b d^2 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac {2 b^2 d^2 \left (c^2 x^2+1\right )^{5/2} \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b^2 d^2 \left (c^2 x^2+1\right )^{5/2} \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac {b^2 d^2 \left (c^2 x^2+1\right )^{5/2} \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 197
Rule 221
Rule 267
Rule 294
Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 4265
Rule 5787
Rule 5788
Rule 5789
Rule 5796
Rule 5797
Rule 5798
Rule 5800
Rule 5810
Rule 5838
Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+i c d x} (f-i c f x)^{5/2}} \, dx &=\frac {\left (1+c^2 x^2\right )^{5/2} \int \frac {(d+i c d x)^2 \left (a+b \sinh ^{-1}(c x)\right )^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 {d^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (1+c^2 x^2\right )^{5/2}}+\frac {2 i c d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (1+c^2 x^2\right )^{5/2}}-\frac {c^2 d^2 x^2 \left (a+b \sinh ^{-1}(c x)\right )^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 (d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^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 (2 i c d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )^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 (c^2 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )^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 {2 i d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {c^2 d^2 x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^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 (4 i b d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {a+b \sinh ^{-1}(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 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\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^3 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\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 d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b d^2 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b c d^2 x^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {c^2 d^2 x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 d^2 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 i b d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {a+b \sinh ^{-1}(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 d^2 \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 (2 b c d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (4 b c d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (2 i b^2 c d^2 \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 {\left (b^2 c^2 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {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 {2 i b^2 d^2 \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 {2 b^2 d^2 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 d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b d^2 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b c d^2 x^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {c^2 d^2 x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 d^2 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (b^2 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 i b d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int (a+b x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 b d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (4 b d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=\frac {2 i b^2 d^2 \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 {2 b^2 d^2 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^2 d^2 \left (1+c^2 x^2\right )^{5/2} \sinh ^{-1}(c x)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b d^2 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b c d^2 x^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {c^2 d^2 x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 d^2 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {4 i b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (4 b d^2 \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,\sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (8 b d^2 \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,\sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 b^2 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (2 b^2 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=\frac {2 i b^2 d^2 \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 {2 b^2 d^2 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^2 d^2 \left (1+c^2 x^2\right )^{5/2} \sinh ^{-1}(c x)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b d^2 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b c d^2 x^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {c^2 d^2 x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 d^2 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {4 i b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (2 b^2 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 b^2 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (2 b^2 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (4 b^2 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=\frac {2 i b^2 d^2 \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 {2 b^2 d^2 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^2 d^2 \left (1+c^2 x^2\right )^{5/2} \sinh ^{-1}(c x)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b d^2 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b c d^2 x^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {c^2 d^2 x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 d^2 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {4 i b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 b^2 d^2 \left (1+c^2 x^2\right )^{5/2} \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b^2 d^2 \left (1+c^2 x^2\right )^{5/2} \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (b^2 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 b^2 d^2 \left (1+c^2 x^2\right )^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=\frac {2 i b^2 d^2 \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 {2 b^2 d^2 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^2 d^2 \left (1+c^2 x^2\right )^{5/2} \sinh ^{-1}(c x)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {b d^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i b d^2 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b c d^2 x^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {c^2 d^2 x^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 d^2 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {4 i b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 b^2 d^2 \left (1+c^2 x^2\right )^{5/2} \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 b^2 d^2 \left (1+c^2 x^2\right )^{5/2} \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b^2 d^2 \left (1+c^2 x^2\right )^{5/2} \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 4.67, size = 528, normalized size = 0.56 \begin {gather*} \frac {\sqrt {d+i c d x} \sqrt {f-i c f x} \left (\frac {a^2 (2 i+c x)}{(i+c x)^2}-\frac {a b \left (i \cosh \left (\frac {3}{2} \sinh ^{-1}(c x)\right ) \left (\sinh ^{-1}(c x)-2 \text {ArcTan}\left (\coth \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )+\frac {1}{2} i \log \left (1+c^2 x^2\right )\right )+\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right ) \left (-2+3 i \sinh ^{-1}(c x)+6 i \text {ArcTan}\left (\coth \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )+\frac {3}{2} \log \left (1+c^2 x^2\right )\right )+2 \left (i+\left (-1+\sqrt {1+c^2 x^2}\right ) \sinh ^{-1}(c x)+2 \left (2+\sqrt {1+c^2 x^2}\right ) \text {ArcTan}\left (\coth \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )-\frac {1}{2} i \left (2+\sqrt {1+c^2 x^2}\right ) \log \left (1+c^2 x^2\right )\right ) \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )}{\sqrt {1+c^2 x^2} \left (\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )-i \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )^3}-\frac {b^2 \left ((1+i) \sinh ^{-1}(c x)^2-\frac {\sinh ^{-1}(c x) \left (2 i+\sinh ^{-1}(c x)\right )}{i+c x}+2 \left (i \pi +2 \sinh ^{-1}(c x)\right ) \log \left (1+i e^{-\sinh ^{-1}(c x)}\right )+i \pi \left (3 \sinh ^{-1}(c x)-4 \log \left (1+e^{\sinh ^{-1}(c x)}\right )-2 \log \left (-\cos \left (\frac {1}{4} \left (\pi +2 i \sinh ^{-1}(c x)\right )\right )\right )+4 \log \left (\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )\right )-4 \text {PolyLog}\left (2,-i e^{-\sinh ^{-1}(c x)}\right )-\frac {2 \sinh ^{-1}(c x)^2 \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )}{\left (\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )-i \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )^3}-\frac {2 \left (-2+\sinh ^{-1}(c x)^2\right ) \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )}{\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )-i \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )}\right )}{\sqrt {1+c^2 x^2}}\right )}{3 c d f^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arcsinh \left (c x \right )\right )^{2}}{\left (-i c f x +f \right )^{\frac {5}{2}} \sqrt {i c d x +d}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\sqrt {i d \left (c x - i\right )} \left (- i f \left (c x + i\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{\sqrt {d+c\,d\,x\,1{}\mathrm {i}}\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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