Optimal. Leaf size=518 \[ \frac{4 b^2 f^3 \left (c^2 x^2+1\right )^{5/2} \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{f^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{4 b f^3 \left (c^2 x^2+1\right )^{5/2} \log \left (1+i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{i f^3 \left (c^2 x^2+1\right )^{5/2} \cot \left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 b f^3 \left (c^2 x^2+1\right )^{5/2} \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{i f^3 \left (c^2 x^2+1\right )^{5/2} \cot \left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{4 i b^2 f^3 \left (c^2 x^2+1\right )^{5/2} \cot \left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]
[Out]
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Rubi [A] time = 1.14856, antiderivative size = 518, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 12, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.324, Rules used = {5712, 5833, 5831, 3318, 4186, 3767, 8, 4184, 3716, 2190, 2279, 2391} \[ \frac{4 b^2 f^3 \left (c^2 x^2+1\right )^{5/2} \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{f^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{4 b f^3 \left (c^2 x^2+1\right )^{5/2} \log \left (1+i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{i f^3 \left (c^2 x^2+1\right )^{5/2} \cot \left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 b f^3 \left (c^2 x^2+1\right )^{5/2} \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{i f^3 \left (c^2 x^2+1\right )^{5/2} \cot \left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{4 i b^2 f^3 \left (c^2 x^2+1\right )^{5/2} \cot \left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 5712
Rule 5833
Rule 5831
Rule 3318
Rule 4186
Rule 3767
Rule 8
Rule 4184
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\sqrt{f-i c f x} \left (a+b \sinh ^{-1}(c x)\right )^2}{(d+i c d x)^{5/2}} \, dx &=\frac{\left (1+c^2 x^2\right )^{5/2} \int \frac{(f-i c f x)^3 \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{2 f^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{(-i+c x)^2 \sqrt{1+c^2 x^2}}+\frac{i f^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{(-i+c x) \sqrt{1+c^2 x^2}}\right ) \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=\frac{\left (i f^3 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{(-i+c x) \sqrt{1+c^2 x^2}} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{\left (2 f^3 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{(-i+c x)^2 \sqrt{1+c^2 x^2}} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=\frac{\left (i f^3 \left (1+c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^2}{-i c+c \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{\left (2 c f^3 \left (1+c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^2}{(-i c+c \sinh (x))^2} \, dx,x,\sinh ^{-1}(c x)\right )}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=-\frac{\left (f^3 \left (1+c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \csc ^2\left (\frac{\pi }{4}+\frac{i x}{2}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{\left (f^3 \left (1+c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \csc ^4\left (\frac{\pi }{4}+\frac{i x}{2}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=-\frac{i f^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 b f^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{i f^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{\left (f^3 \left (1+c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \csc ^2\left (\frac{\pi }{4}+\frac{i x}{2}\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 i b f^3 \left (1+c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int (a+b x) \cot \left (\frac{\pi }{4}+\frac{i x}{2}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{\left (2 b^2 f^3 \left (1+c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \csc ^2\left (\frac{\pi }{4}+\frac{i x}{2}\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{f^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{i f^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 b f^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{i f^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{\left (4 i b f^3 \left (1+c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int (a+b x) \cot \left (\frac{\pi }{4}+\frac{i x}{2}\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 (4 i b f^3 \left (1+c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{1+i e^x} \, dx,x,\sinh ^{-1}(c x)\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{\left (4 i b^2 f^3 \left (1+c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int 1 \, dx,x,\cot \left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right )\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=-\frac{f^3 \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^2 f^3 \left (1+c^2 x^2\right )^{5/2} \cot \left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{i f^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 b f^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{i f^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{4 b f^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+i e^{\sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{\left (8 i b f^3 \left (1+c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{1+i e^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^2 f^3 \left (1+c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=-\frac{f^3 \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^2 f^3 \left (1+c^2 x^2\right )^{5/2} \cot \left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{i f^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 b f^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{i f^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{4 b f^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+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 (8 b^2 f^3 \left (1+c^2 x^2\right )^{5/2}\right ) \operatorname{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 (4 b^2 f^3 \left (1+c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=-\frac{f^3 \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^2 f^3 \left (1+c^2 x^2\right )^{5/2} \cot \left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{i f^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 b f^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{i f^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{4 b f^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+i e^{\sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{4 b^2 f^3 \left (1+c^2 x^2\right )^{5/2} \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{\left (8 b^2 f^3 \left (1+c^2 x^2\right )^{5/2}\right ) \operatorname{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{f^3 \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^2 f^3 \left (1+c^2 x^2\right )^{5/2} \cot \left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{i f^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 b f^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{i f^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{4 b f^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+i e^{\sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{4 b^2 f^3 \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}}\\ \end{align*}
Mathematica [A] time = 8.06206, size = 783, normalized size = 1.51 \[ \frac{i b^2 (c x+i) \sqrt{i (c d x-i d)} \sqrt{-i (c f x+i f)} \sqrt{-d f \left (c^2 x^2+1\right )} \left (4 \text{PolyLog}\left (2,i e^{-\sinh ^{-1}(c x)}\right )-(1-i) \sinh ^{-1}(c x)^2-\frac{2 \left (\sinh ^{-1}(c x)-2 i\right ) \sinh ^{-1}(c x)}{c x-i}+2 i \left (\pi +2 i \sinh ^{-1}(c x)\right ) \log \left (1-i e^{-\sinh ^{-1}(c x)}\right )-\frac{4 \sinh \left (\frac{1}{2} \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c 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{2 \left (\sinh ^{-1}(c x)^2+4\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 )}-i \pi \left (\sinh ^{-1}(c x)-4 \log \left (e^{\sinh ^{-1}(c x)}+1\right )+2 \log \left (\sin \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 )\right )}{3 c d^3 \sqrt{c^2 x^2+1} \sqrt{-(c d x-i d) (c f x+i f)} \left (\cosh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-i \sinh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )^2}+\frac{\sqrt{i d (c x-i)} \sqrt{-i f (c x+i)} \left (-\frac{a^2}{3 d^3 (c x-i)}-\frac{2 i a^2}{3 d^3 (c x-i)^2}\right )}{c}+\frac{i a b \sqrt{i (c d x-i d)} \sqrt{-i (c f x+i f)} \sqrt{-d f \left (c^2 x^2+1\right )} \left (\cosh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-i \sinh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right ) \left (2 \sinh \left (\frac{1}{2} \sinh ^{-1}(c x)\right ) \left (\sqrt{c^2 x^2+1} \left (i \log \left (\sqrt{c^2 x^2+1}\right )+\sinh ^{-1}(c x)+2 \tan ^{-1}\left (\coth \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )\right )+2 \left (i \log \left (\sqrt{c^2 x^2+1}\right )+\sinh ^{-1}(c x)+2 \tan ^{-1}\left (\coth \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )+i\right )\right )-i \cosh \left (\frac{3}{2} \sinh ^{-1}(c x)\right ) \left (-i \log \left (\sqrt{c^2 x^2+1}\right )+\sinh ^{-1}(c x)-2 \tan ^{-1}\left (\coth \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )\right )+\cosh \left (\frac{1}{2} \sinh ^{-1}(c x)\right ) \left (3 \log \left (\sqrt{c^2 x^2+1}\right )+3 i \sinh ^{-1}(c x)-6 i \tan ^{-1}\left (\coth \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )+4\right )\right )}{3 c d^3 (c x+i) \sqrt{-(c d x-i d) (c f x+i f)} \left (\cosh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )+i \sinh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )^4} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.323, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}\sqrt{f-icfx} \left ( d+icdx \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (b^{2} c x + i \, b^{2}\right )} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} -{\left (3 \, c^{3} d^{3} x^{2} - 6 i \, c^{2} d^{3} x - 3 \, c d^{3}\right )}{\rm integral}\left (\frac{3 i \, \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} a^{2} + 2 \,{\left (\sqrt{c^{2} x^{2} + 1} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} b^{2} + 3 i \, \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} a b\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{3 \, c^{3} d^{3} x^{3} - 9 i \, c^{2} d^{3} x^{2} - 9 \, c d^{3} x + 3 i \, d^{3}}, x\right )}{3 \, c^{3} d^{3} x^{2} - 6 i \, c^{2} d^{3} x - 3 \, c d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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