Optimal. Leaf size=158 \[ \frac{d \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{i d \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{i b d x \sqrt{c^2 x^2+1}}{\sqrt{d+i c d x} \sqrt{f-i c f x}} \]
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Rubi [A] time = 0.290386, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5712, 5821, 5675, 5717, 8} \[ \frac{d \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{i d \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{i b d x \sqrt{c^2 x^2+1}}{\sqrt{d+i c d x} \sqrt{f-i c f x}} \]
Antiderivative was successfully verified.
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Rule 5712
Rule 5821
Rule 5675
Rule 5717
Rule 8
Rubi steps
\begin{align*} \int \frac{\sqrt{d+i c d x} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{f-i c f x}} \, dx &=\frac{\sqrt{1+c^2 x^2} \int \frac{(d+i c d x) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{\sqrt{d+i c d x} \sqrt{f-i c f x}}\\ &=\frac{\sqrt{1+c^2 x^2} \int \left (\frac{d \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}+\frac{i c d x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}\right ) \, dx}{\sqrt{d+i c d x} \sqrt{f-i c f x}}\\ &=\frac{\left (d \sqrt{1+c^2 x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx}{\sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{\left (i c d \sqrt{1+c^2 x^2}\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{\sqrt{d+i c d x} \sqrt{f-i c f x}}\\ &=\frac{i d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{d \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{\left (i b d \sqrt{1+c^2 x^2}\right ) \int 1 \, dx}{\sqrt{d+i c d x} \sqrt{f-i c f x}}\\ &=-\frac{i b d x \sqrt{1+c^2 x^2}}{\sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{i d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{d \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}\\ \end{align*}
Mathematica [A] time = 0.583145, size = 227, normalized size = 1.44 \[ \frac{-2 i \sqrt{d+i c d x} \sqrt{f-i c f x} \left (b c x-a \sqrt{c^2 x^2+1}\right )+2 a \sqrt{d} \sqrt{f} \sqrt{c^2 x^2+1} \log \left (c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right )+2 i b \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)+b \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2}{2 c f \sqrt{c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.298, size = 0, normalized size = 0. \begin{align*} \int{(a+b{\it Arcsinh} \left ( cx \right ) )\sqrt{d+icdx}{\frac{1}{\sqrt{f-icfx}}}}\, 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*}{\rm integral}\left (\frac{i \, \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} b \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + i \, \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} a}{c f x + i \, f}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d \left (i c x + 1\right )} \left (a + b \operatorname{asinh}{\left (c x \right )}\right )}{\sqrt{- f \left (i c x - 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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