Optimal. Leaf size=580 \[ \frac {f^4 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {8 f^4 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {32 b f^4 \left (c^2 x^2+1\right )^{5/2} \log \left (1+i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {8 i f^4 \left (c^2 x^2+1\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} i \sinh ^{-1}(c x)\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {4 b f^4 \left (c^2 x^2+1\right )^{5/2} \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \sinh ^{-1}(c x)\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i f^4 \left (c^2 x^2+1\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} i \sinh ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \sinh ^{-1}(c x)\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {32 b^2 f^4 \left (c^2 x^2+1\right )^{5/2} \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {8 i b^2 f^4 \left (c^2 x^2+1\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} i \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]
[Out]
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Rubi [A] time = 1.20, antiderivative size = 580, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 13, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.351, Rules used = {5712, 5833, 5675, 5831, 3318, 4186, 3767, 8, 4184, 3716, 2190, 2279, 2391} \[ \frac {32 b^2 f^4 \left (c^2 x^2+1\right )^{5/2} \text {PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {f^4 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {8 f^4 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {32 b f^4 \left (c^2 x^2+1\right )^{5/2} \log \left (1+i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {8 i f^4 \left (c^2 x^2+1\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} i \sinh ^{-1}(c x)\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {4 b f^4 \left (c^2 x^2+1\right )^{5/2} \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \sinh ^{-1}(c x)\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i f^4 \left (c^2 x^2+1\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} i \sinh ^{-1}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \sinh ^{-1}(c x)\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {8 i b^2 f^4 \left (c^2 x^2+1\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} i \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2190
Rule 2279
Rule 2391
Rule 3318
Rule 3716
Rule 3767
Rule 4184
Rule 4186
Rule 5675
Rule 5712
Rule 5831
Rule 5833
Rubi steps
\begin {align*} \int \frac {(f-i c f x)^{3/2} \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)^4 \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 {f^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}}-\frac {4 f^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{(-i+c x)^2 \sqrt {1+c^2 x^2}}+\frac {4 i f^4 \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 (4 i f^4 \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 (f^4 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (4 f^4 \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 {f^4 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (4 i f^4 \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 (4 c f^4 \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 {f^4 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (f^4 \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 )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (2 f^4 \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 )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=\frac {f^4 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {4 i f^4 \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 {4 b f^4 \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 {2 i f^4 \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 (2 f^4 \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 (8 i b f^4 \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 (4 b^2 f^4 \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 {4 f^4 \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 {f^4 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {8 i f^4 \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 {4 b f^4 \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 {2 i f^4 \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 (8 i b f^4 \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 (16 i b f^4 \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 (8 i b^2 f^4 \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 {8 f^4 \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 {f^4 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {8 i b^2 f^4 \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 {8 i f^4 \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 {4 b f^4 \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 {2 i f^4 \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 {16 b f^4 \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 (16 i b f^4 \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 (16 b^2 f^4 \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 {8 f^4 \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 {f^4 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {8 i b^2 f^4 \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 {8 i f^4 \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 {4 b f^4 \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 {2 i f^4 \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 {32 b f^4 \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 (16 b^2 f^4 \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 (16 b^2 f^4 \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 {8 f^4 \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 {f^4 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {8 i b^2 f^4 \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 {8 i f^4 \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 {4 b f^4 \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 {2 i f^4 \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 {32 b f^4 \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 {16 b^2 f^4 \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 (16 b^2 f^4 \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 {8 f^4 \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 {f^4 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {8 i b^2 f^4 \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 {8 i f^4 \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 {4 b f^4 \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 {2 i f^4 \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 {32 b f^4 \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 {32 b^2 f^4 \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*}
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Mathematica [B] time = 10.24, size = 1609, normalized size = 2.77 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} c f x + i \, b^{2} f\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 2 \, {\left (a b c f x + i \, a b f\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (a^{2} c f x + i \, a^{2} f\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f}}{c^{3} d^{3} x^{3} - 3 i \, c^{2} d^{3} x^{2} - 3 \, c d^{3} x + i \, d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {\left (-i c f x +f \right )^{\frac {3}{2}} \left (a +b \arcsinh \left (c x \right )\right )^{2}}{\left (i c d x +d \right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{3/2}}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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