Optimal. Leaf size=224 \[ \frac {\left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {2 b \left (c^2 x^2+1\right )^{3/2} \log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b^2 \left (c^2 x^2+1\right )^{3/2} \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]
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Rubi [A] time = 0.42, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {5712, 5687, 5714, 3718, 2190, 2279, 2391} \[ -\frac {b^2 \left (c^2 x^2+1\right )^{3/2} \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {2 b \left (c^2 x^2+1\right )^{3/2} \log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 5687
Rule 5712
Rule 5714
Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \, dx &=\frac {\left (1+c^2 x^2\right )^{3/2} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^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 {x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (2 b c \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=\frac {x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (2 b \left (1+c^2 x^2\right )^{3/2}\right ) \operatorname {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=\frac {x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (4 b \left (1+c^2 x^2\right )^{3/2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=\frac {x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {2 b \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (2 b^2 \left (1+c^2 x^2\right )^{3/2}\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=\frac {x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {2 b \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (b^2 \left (1+c^2 x^2\right )^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=\frac {x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {2 b \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b^2 \left (1+c^2 x^2\right )^{3/2} \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ \end {align*}
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Mathematica [B] time = 1.36, size = 488, normalized size = 2.18 \[ \frac {a^2 c x-a b \sqrt {c^2 x^2+1} \log \left (c^2 x^2+1\right )+2 a b c x \sinh ^{-1}(c x)+2 b^2 \sqrt {c^2 x^2+1} \text {Li}_2\left (-i e^{-\sinh ^{-1}(c x)}\right )+2 b^2 \sqrt {c^2 x^2+1} \text {Li}_2\left (i e^{-\sinh ^{-1}(c x)}\right )-b^2 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)^2-2 i \pi b^2 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)-2 b^2 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x) \log \left (1-i e^{-\sinh ^{-1}(c x)}\right )+i \pi b^2 \sqrt {c^2 x^2+1} \log \left (1-i e^{-\sinh ^{-1}(c x)}\right )-2 b^2 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x) \log \left (1+i e^{-\sinh ^{-1}(c x)}\right )-i \pi b^2 \sqrt {c^2 x^2+1} \log \left (1+i e^{-\sinh ^{-1}(c x)}\right )+4 i \pi b^2 \sqrt {c^2 x^2+1} \log \left (e^{\sinh ^{-1}(c x)}+1\right )-i \pi b^2 \sqrt {c^2 x^2+1} \log \left (\sin \left (\frac {1}{4} \left (\pi +2 i \sinh ^{-1}(c x)\right )\right )\right )+i \pi b^2 \sqrt {c^2 x^2+1} \log \left (-\cos \left (\frac {1}{4} \left (\pi +2 i \sinh ^{-1}(c x)\right )\right )\right )-4 i \pi b^2 \sqrt {c^2 x^2+1} \log \left (\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )+b^2 c x \sinh ^{-1}(c x)^2}{c d f \sqrt {d+i c d x} \sqrt {f-i c f x}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ \frac {\sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} b^{2} x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + {\left (c^{2} d^{2} f^{2} x^{2} + d^{2} f^{2}\right )} {\rm integral}\left (\frac {\sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} a^{2} - 2 \, {\left (\sqrt {c^{2} x^{2} + 1} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} b^{2} c x - \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} a b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{c^{4} d^{2} f^{2} x^{4} + 2 \, c^{2} d^{2} f^{2} x^{2} + d^{2} f^{2}}, x\right )}{c^{2} d^{2} f^{2} x^{2} + d^{2} f^{2}} \]
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 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (i \, c d x + d\right )}^{\frac {3}{2}} {\left (-i \, c f x + f\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arcsinh \left (c x \right )\right )^{2}}{\left (i c d x +d \right )^{\frac {3}{2}} \left (-i c f x +f \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ b^{2} \int \frac {\log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{{\left (i \, c d x + d\right )}^{\frac {3}{2}} {\left (-i \, c f x + f\right )}^{\frac {3}{2}}}\,{d x} + \frac {2 \, a b x \operatorname {arsinh}\left (c x\right )}{\sqrt {c^{2} d f x^{2} + d f} d f} + \frac {a^{2} x}{\sqrt {c^{2} d f x^{2} + d f} d f} - \frac {a b \sqrt {\frac {1}{d f}} \log \left (x^{2} + \frac {1}{c^{2}}\right )}{c d f} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{3/2}\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (i d \left (c x - i\right )\right )^{\frac {3}{2}} \left (- i f \left (c x + i\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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