Optimal. Leaf size=283 \[ -\frac{3 d^3 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{i d^3 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{4 i d^3 (1+i c x) \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{i b d^3 x \left (c^2 x^2+1\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{4 b d^3 \left (c^2 x^2+1\right )^{3/2} \log (c x+i)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]
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Rubi [A] time = 0.477803, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5712, 5833, 637, 5819, 12, 627, 31, 5675, 5717, 8} \[ -\frac{3 d^3 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{i d^3 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{4 i d^3 (1+i c x) \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{i b d^3 x \left (c^2 x^2+1\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{4 b d^3 \left (c^2 x^2+1\right )^{3/2} \log (c x+i)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5712
Rule 5833
Rule 637
Rule 5819
Rule 12
Rule 627
Rule 31
Rule 5675
Rule 5717
Rule 8
Rubi steps
\begin{align*} \int \frac{(d+i c d x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{(f-i c f x)^{3/2}} \, dx &=\frac{\left (1+c^2 x^2\right )^{3/2} \int \frac{(d+i c d x)^3 \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{\left (1+c^2 x^2\right )^{3/2} \int \left (-\frac{4 i \left (i d^3-c d^3 x\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^{3/2}}-\frac{3 d^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}-\frac{i c d^3 x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}\right ) \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=-\frac{\left (4 i \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac{\left (i d^3-c d^3 x\right ) \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{\left (3 d^3 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{\left (i c d^3 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=-\frac{4 i d^3 (1+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{i d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{3 d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{\left (4 i b c \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac{d^3 (1+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{\left (i b d^3 \left (1+c^2 x^2\right )^{3/2}\right ) \int 1 \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=\frac{i b d^3 x \left (1+c^2 x^2\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{4 i d^3 (1+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{i d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{3 d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{\left (4 i b d^3 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac{1+i c x}{1+c^2 x^2} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=\frac{i b d^3 x \left (1+c^2 x^2\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{4 i d^3 (1+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{i d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{3 d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{\left (4 i b d^3 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac{1}{1-i c x} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=\frac{i b d^3 x \left (1+c^2 x^2\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{4 i d^3 (1+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{i d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{3 d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{4 b d^3 \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 = 2.35144, size = 515, normalized size = 1.82 \[ \frac{-\frac{6 a d^{3/2} \log \left (c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right )}{f^{3/2}}+\frac{2 a d (5-i c x) \sqrt{d+i c d x} \sqrt{f-i c f x}}{f^2 (c x+i)}+\frac{b d \sqrt{d+i c d x} \sqrt{f-i c f x} \left (2 \left (\sinh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )+i \cosh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right ) \left (4 \tan ^{-1}\left (\tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )+i \log \left (c^2 x^2+1\right )\right )+\sinh ^{-1}(c x)^2 \left (-\left (\cosh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-i \sinh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )\right )+4 \sinh ^{-1}(c x) \left (\sinh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-i \cosh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )\right )}{f^2 \sqrt{c^2 x^2+1} \left (\cosh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-i \sinh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )}+\frac{2 b d \sqrt{d+i c d x} \sqrt{f-i c f x} \left (\sinh ^{-1}(c x) \left (-\left (\sqrt{c^2 x^2+1}-2\right ) \sinh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-i \left (\sqrt{c^2 x^2+1}+2\right ) \cosh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )+\left (\sinh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )+i \cosh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right ) \left (i \log \left (c^2 x^2+1\right )+c x-4 \tan ^{-1}\left (\coth \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )\right )+\sinh ^{-1}(c x)^2 \left (-\left (\cosh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-i \sinh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )\right )\right )}{f^2 \sqrt{c^2 x^2+1} \left (\cosh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-i \sinh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )}}{2 c} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.3, 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 [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{{\left (-i \, b c d x - b d\right )} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) +{\left (-i \, a c d x - a d\right )} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f}}{c^{2} f^{2} x^{2} + 2 i \, c f^{2} x - f^{2}}, x\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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