Optimal. Leaf size=59 \[ \frac {\sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c \sqrt {d+i c d x} \sqrt {f-i c f x}} \]
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Rubi [A] time = 0.30, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {5712, 5675} \[ \frac {\sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c \sqrt {d+i c d x} \sqrt {f-i c f x}} \]
Antiderivative was successfully verified.
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Rule 5675
Rule 5712
Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+i c d x} \sqrt {f-i c f x}} \, dx &=\frac {\sqrt {1+c^2 x^2} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\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} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}\\ \end {align*}
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Mathematica [B] time = 0.85, size = 168, normalized size = 2.85 \[ \frac {a^2 \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )}{c \sqrt {d} \sqrt {f}}+\frac {a b \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)^2}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b^2 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)^3}{3 c \sqrt {d+i c d x} \sqrt {f-i c f x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} b^{2} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 2 \, \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} a b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} a^{2}}{c^{2} d f x^{2} + d f}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{\sqrt {i \, c d x + d} \sqrt {-i \, c f x + f}}\,{d x} \]
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 \[ \int \frac {\left (a +b \arcsinh \left (c x \right )\right )^{2}}{\sqrt {i c d x +d}\, \sqrt {-i c f x +f}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 53, normalized size = 0.90 \[ \frac {b^{2} \operatorname {arsinh}\left (c x\right )^{3}}{3 \, \sqrt {d f} c} + \frac {a b \operatorname {arsinh}\left (c x\right )^{2}}{\sqrt {d f} c} + \frac {a^{2} \operatorname {arsinh}\left (c x\right )}{\sqrt {d f} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{\sqrt {d+c\,d\,x\,1{}\mathrm {i}}\,\sqrt {f-c\,f\,x\,1{}\mathrm {i}}} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\sqrt {i d \left (c x - i\right )} \sqrt {- i f \left (c x + i\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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