Optimal. Leaf size=436 \[ \frac{f^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{2 i f^2 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{f^2 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{b c f^2 x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{4 i b f^2 x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{4 i b^2 f^2 \left (c^2 x^2+1\right )}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{b^2 f^2 x \left (c^2 x^2+1\right )}{4 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{b^2 f^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{4 c \sqrt{d+i c d x} \sqrt{f-i c f x}} \]
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Rubi [A] time = 0.632811, antiderivative size = 436, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.243, Rules used = {5712, 5831, 3317, 3296, 2638, 3311, 32, 2635, 8} \[ \frac{f^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{2 i f^2 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{f^2 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{b c f^2 x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{4 i b f^2 x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{4 i b^2 f^2 \left (c^2 x^2+1\right )}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{b^2 f^2 x \left (c^2 x^2+1\right )}{4 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{b^2 f^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{4 c \sqrt{d+i c d x} \sqrt{f-i c f x}} \]
Antiderivative was successfully verified.
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Rule 5712
Rule 5831
Rule 3317
Rule 3296
Rule 2638
Rule 3311
Rule 32
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{(f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{d+i c d x}} \, dx &=\frac{\sqrt{1+c^2 x^2} \int \frac{(f-i c f x)^2 \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} \operatorname{Subst}\left (\int (a+b x)^2 (c f-i c f \sinh (x))^2 \, dx,x,\sinh ^{-1}(c x)\right )}{c^3 \sqrt{d+i c d x} \sqrt{f-i c f x}}\\ &=\frac{\sqrt{1+c^2 x^2} \operatorname{Subst}\left (\int \left (c^2 f^2 (a+b x)^2-2 i c^2 f^2 (a+b x)^2 \sinh (x)-c^2 f^2 (a+b x)^2 \sinh ^2(x)\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^3 \sqrt{d+i c d x} \sqrt{f-i c f x}}\\ &=\frac{f^2 \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}}-\frac{\left (2 i f^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \sinh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{\left (f^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \sinh ^2(x) \, dx,x,\sinh ^{-1}(c x)\right )}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}\\ &=\frac{b c f^2 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{2 i f^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{f^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{f^2 \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}}+\frac{\left (f^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \, dx,x,\sinh ^{-1}(c x)\right )}{2 c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{\left (4 i b f^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \cosh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{\left (b^2 f^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \sinh ^2(x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 c \sqrt{d+i c d x} \sqrt{f-i c f x}}\\ &=-\frac{b^2 f^2 x \left (1+c^2 x^2\right )}{4 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{4 i b f^2 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{b c f^2 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{2 i f^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{f^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{f^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{\left (4 i b^2 f^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \sinh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{\left (b^2 f^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int 1 \, dx,x,\sinh ^{-1}(c x)\right )}{4 c \sqrt{d+i c d x} \sqrt{f-i c f x}}\\ &=-\frac{4 i b^2 f^2 \left (1+c^2 x^2\right )}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{b^2 f^2 x \left (1+c^2 x^2\right )}{4 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{b^2 f^2 \sqrt{1+c^2 x^2} \sinh ^{-1}(c x)}{4 c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{4 i b f^2 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{b c f^2 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{2 i f^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{f^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{f^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}\\ \end{align*}
Mathematica [A] time = 2.18331, size = 532, normalized size = 1.22 \[ \frac{12 a^2 \sqrt{d} f^{3/2} \sqrt{c^2 x^2+1} \log \left (c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right )-16 i a^2 f \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}-4 a^2 c f x \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+2 b f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2 \left (6 a-8 i b \sqrt{c^2 x^2+1}-b \sinh \left (2 \sinh ^{-1}(c x)\right )\right )+2 b f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left (b \cosh \left (2 \sinh ^{-1}(c x)\right )+4 i \left (4 b c x+a (-4+i c x) \sqrt{c^2 x^2+1}\right )\right )+32 i a b c f x \sqrt{d+i c d x} \sqrt{f-i c f x}+2 a b f \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left (2 \sinh ^{-1}(c x)\right )-32 i b^2 f \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+4 b^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^3-b^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left (2 \sinh ^{-1}(c x)\right )}{8 c d \sqrt{c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.316, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2} \left ( f-icfx \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{d+icdx}}}}\, 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 (b^{2} c f x + i \, b^{2} f\right )} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 2 \,{\left (a b c f x + i \, a b f\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 (a^{2} c f x + i \, a^{2} f\right )} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f}}{c d x - i \, d}, 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(-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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