Optimal. Leaf size=103 \[ \frac{x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{b \left (c^2 x^2+1\right )^{3/2} \log \left (c^2 x^2+1\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]
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Rubi [A] time = 0.205475, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {5712, 5687, 260} \[ \frac{x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{b \left (c^2 x^2+1\right )^{3/2} \log \left (c^2 x^2+1\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5712
Rule 5687
Rule 260
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c x)}{(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{a+b \sinh ^{-1}(c x)}{\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 )}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{\left (b c \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac{x}{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 )}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{b \left (1+c^2 x^2\right )^{3/2} \log \left (1+c^2 x^2\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.465136, size = 118, normalized size = 1.15 \[ \frac{i \sqrt{f-i c f x} \left (2 a c x-b \sqrt{c^2 x^2+1} \log (d (-1+i c x))-b \sqrt{c^2 x^2+1} \log (d+i c d x)+2 b c x \sinh ^{-1}(c x)\right )}{2 c d f^2 (c x+i) \sqrt{d+i c d x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.243, size = 0, normalized size = 0. \begin{align*} \int{(a+b{\it Arcsinh} \left ( cx \right ) ) \left ( d+icdx \right ) ^{-{\frac{3}{2}}} \left ( f-icfx \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.22233, size = 112, normalized size = 1.09 \begin{align*} -\frac{b c \sqrt{\frac{1}{c^{4} d f}} \log \left (x^{2} + \frac{1}{c^{2}}\right )}{2 \, d f} + \frac{b x \operatorname{arsinh}\left (c x\right )}{\sqrt{c^{2} d f x^{2} + d f} d f} + \frac{a x}{\sqrt{c^{2} d f x^{2} + d f} d f} \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{4 \, \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} b x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + 4 \, \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} a x +{\left (c^{2} d^{2} f^{2} x^{2} + d^{2} f^{2}\right )} \sqrt{\frac{b^{2}}{c^{2} d^{3} f^{3}}} \log \left (\frac{b c^{2} x^{4} + \sqrt{c^{2} x^{2} + 1} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} c d f x^{2} \sqrt{\frac{b^{2}}{c^{2} d^{3} f^{3}}} + b x^{2}}{b c^{4} x^{4} + 2 \, b c^{2} x^{2} + b}\right ) -{\left (c^{2} d^{2} f^{2} x^{2} + d^{2} f^{2}\right )} \sqrt{\frac{b^{2}}{c^{2} d^{3} f^{3}}} \log \left (\frac{b c^{2} x^{4} - \sqrt{c^{2} x^{2} + 1} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} c d f x^{2} \sqrt{\frac{b^{2}}{c^{2} d^{3} f^{3}}} + b x^{2}}{b c^{4} x^{4} + 2 \, b c^{2} x^{2} + b}\right ) - 2 \,{\left (c^{2} d^{2} f^{2} x^{2} + d^{2} f^{2}\right )} \sqrt{\frac{b^{2}}{c^{2} d^{3} f^{3}}} \log \left (\frac{b c^{2} x^{3} + \sqrt{c^{2} x^{2} + 1} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} c d f x \sqrt{\frac{b^{2}}{c^{2} d^{3} f^{3}}} + b x}{b c^{2} x^{2} + b}\right ) + 2 \,{\left (c^{2} d^{2} f^{2} x^{2} + d^{2} f^{2}\right )} \sqrt{\frac{b^{2}}{c^{2} d^{3} f^{3}}} \log \left (\frac{b c^{2} x^{3} - \sqrt{c^{2} x^{2} + 1} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} c d f x \sqrt{\frac{b^{2}}{c^{2} d^{3} f^{3}}} + b x}{b c^{2} x^{2} + b}\right ) + 4 \,{\left (c^{2} d^{2} f^{2} x^{2} + d^{2} f^{2}\right )}{\rm integral}\left (-\frac{\sqrt{c^{2} x^{2} + 1} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} b c x}{c^{4} d^{2} f^{2} x^{4} + 2 \, c^{2} d^{2} f^{2} x^{2} + d^{2} f^{2}}, x\right )}{4 \,{\left (c^{2} d^{2} f^{2} x^{2} + d^{2} f^{2}\right )}} \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: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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