Optimal. Leaf size=111 \[ \frac{f (c x+i) \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{b f \left (c^2 x^2+1\right )^{3/2} \log (-c x+i)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]
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Rubi [A] time = 0.241307, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {5712, 637, 5819, 12, 627, 31} \[ \frac{f (c x+i) \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{b f \left (c^2 x^2+1\right )^{3/2} \log (-c x+i)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5712
Rule 637
Rule 5819
Rule 12
Rule 627
Rule 31
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c x)}{(d+i c d x)^{3/2} \sqrt{f-i c f x}} \, dx &=\frac{\left (1+c^2 x^2\right )^{3/2} \int \frac{(f-i c f x) \left (a+b \sinh ^{-1}(c x)\right )}{\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{f (i+c x) \left (1+c^2 x^2\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{\left (b c \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac{f (i+c x)}{c \left (1+c^2 x^2\right )} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=\frac{f (i+c x) \left (1+c^2 x^2\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{\left (b f \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac{i+c x}{1+c^2 x^2} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=\frac{f (i+c x) \left (1+c^2 x^2\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{\left (b f \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac{1}{-i+c x} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=\frac{f (i+c x) \left (1+c^2 x^2\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 f \left (1+c^2 x^2\right )^{3/2} \log (i-c x)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.397476, size = 113, normalized size = 1.02 \[ \frac{\sqrt{d+i c d x} \sqrt{f-i c f x} \left (a \sqrt{c^2 x^2+1}+b \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+b (-c x+i) \log (d+i c d x)\right )}{c d^2 f (c x-i) \sqrt{c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.255, size = 0, normalized size = 0. \begin{align*} \int{(a+b{\it Arcsinh} \left ( cx \right ) ) \left ( d+icdx \right ) ^{-{\frac{3}{2}}}{\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 [B] time = 2.31198, size = 1011, normalized size = 9.11 \begin{align*} \frac{2 \, \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} b \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) +{\left (c^{2} d^{2} f x - i \, c d^{2} f\right )} \sqrt{\frac{b^{2}}{c^{2} d^{3} f}} \log \left (\frac{{\left (-2 i \, b c^{6} x^{2} - 4 \, b c^{5} x + 4 i \, b c^{4}\right )} \sqrt{c^{2} x^{2} + 1} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} + 2 \,{\left (i \, c^{9} d^{2} f x^{4} + 2 \, c^{8} d^{2} f x^{3} + i \, c^{7} d^{2} f x^{2} + 2 \, c^{6} d^{2} f x\right )} \sqrt{\frac{b^{2}}{c^{2} d^{3} f}}}{16 \, b c^{3} x^{3} - 16 i \, b c^{2} x^{2} + 16 \, b c x - 16 i \, b}\right ) -{\left (c^{2} d^{2} f x - i \, c d^{2} f\right )} \sqrt{\frac{b^{2}}{c^{2} d^{3} f}} \log \left (\frac{{\left (-2 i \, b c^{6} x^{2} - 4 \, b c^{5} x + 4 i \, b c^{4}\right )} \sqrt{c^{2} x^{2} + 1} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} + 2 \,{\left (-i \, c^{9} d^{2} f x^{4} - 2 \, c^{8} d^{2} f x^{3} - i \, c^{7} d^{2} f x^{2} - 2 \, c^{6} d^{2} f x\right )} \sqrt{\frac{b^{2}}{c^{2} d^{3} f}}}{16 \, b c^{3} x^{3} - 16 i \, b c^{2} x^{2} + 16 \, b c x - 16 i \, b}\right ) + 2 \, \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} a}{2 \,{\left (c^{2} d^{2} f x - i \, c d^{2} f\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{a + b \operatorname{asinh}{\left (c x \right )}}{\left (d \left (i c x + 1\right )\right )^{\frac{3}{2}} \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(-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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