Optimal. Leaf size=396 \[ \frac{(d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \left (c^2 x^2+1\right )^{3/2}}+\frac{3 x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{8 \left (c^2 x^2+1\right )}-\frac{b \sqrt{c^2 x^2+1} (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}-\frac{3 b c x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \left (c^2 x^2+1\right )^{3/2}}+\frac{1}{4} x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{15 b^2 x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{64 \left (c^2 x^2+1\right )}-\frac{9 b^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \sinh ^{-1}(c x)}{64 c \left (c^2 x^2+1\right )^{3/2}}+\frac{1}{32} b^2 x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \]
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Rubi [A] time = 0.481928, antiderivative size = 396, 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, 5684, 5682, 5675, 5661, 321, 215, 5717, 195} \[ \frac{(d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \left (c^2 x^2+1\right )^{3/2}}+\frac{3 x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{8 \left (c^2 x^2+1\right )}-\frac{b \sqrt{c^2 x^2+1} (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}-\frac{3 b c x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \left (c^2 x^2+1\right )^{3/2}}+\frac{1}{4} x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{15 b^2 x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{64 \left (c^2 x^2+1\right )}-\frac{9 b^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \sinh ^{-1}(c x)}{64 c \left (c^2 x^2+1\right )^{3/2}}+\frac{1}{32} b^2 x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \]
Antiderivative was successfully verified.
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Rule 5712
Rule 5684
Rule 5682
Rule 5675
Rule 5661
Rule 321
Rule 215
Rule 5717
Rule 195
Rubi steps
\begin{align*} \int (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac{\left ((d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx}{\left (1+c^2 x^2\right )^{3/2}}\\ &=\frac{1}{4} x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (3 (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx}{4 \left (1+c^2 x^2\right )^{3/2}}-\frac{\left (b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{2 \left (1+c^2 x^2\right )^{3/2}}\\ &=-\frac{b (d+i c d x)^{3/2} (f-i c f x)^{3/2} \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac{1}{4} x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3 x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{8 \left (1+c^2 x^2\right )}+\frac{\left (3 (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}} \, dx}{8 \left (1+c^2 x^2\right )^{3/2}}+\frac{\left (b^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx}{8 \left (1+c^2 x^2\right )^{3/2}}-\frac{\left (3 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{4 \left (1+c^2 x^2\right )^{3/2}}\\ &=\frac{1}{32} b^2 x (d+i c d x)^{3/2} (f-i c f x)^{3/2}-\frac{3 b c x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \left (1+c^2 x^2\right )^{3/2}}-\frac{b (d+i c d x)^{3/2} (f-i c f x)^{3/2} \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac{1}{4} x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3 x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{8 \left (1+c^2 x^2\right )}+\frac{(d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \left (1+c^2 x^2\right )^{3/2}}+\frac{\left (3 b^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \sqrt{1+c^2 x^2} \, dx}{32 \left (1+c^2 x^2\right )^{3/2}}+\frac{\left (3 b^2 c^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \frac{x^2}{\sqrt{1+c^2 x^2}} \, dx}{8 \left (1+c^2 x^2\right )^{3/2}}\\ &=\frac{1}{32} b^2 x (d+i c d x)^{3/2} (f-i c f x)^{3/2}+\frac{15 b^2 x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{64 \left (1+c^2 x^2\right )}-\frac{3 b c x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \left (1+c^2 x^2\right )^{3/2}}-\frac{b (d+i c d x)^{3/2} (f-i c f x)^{3/2} \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac{1}{4} x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3 x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{8 \left (1+c^2 x^2\right )}+\frac{(d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \left (1+c^2 x^2\right )^{3/2}}+\frac{\left (3 b^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{64 \left (1+c^2 x^2\right )^{3/2}}-\frac{\left (3 b^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{16 \left (1+c^2 x^2\right )^{3/2}}\\ &=\frac{1}{32} b^2 x (d+i c d x)^{3/2} (f-i c f x)^{3/2}+\frac{15 b^2 x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{64 \left (1+c^2 x^2\right )}-\frac{9 b^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \sinh ^{-1}(c x)}{64 c \left (1+c^2 x^2\right )^{3/2}}-\frac{3 b c x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \left (1+c^2 x^2\right )^{3/2}}-\frac{b (d+i c d x)^{3/2} (f-i c f x)^{3/2} \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac{1}{4} x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3 x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{8 \left (1+c^2 x^2\right )}+\frac{(d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \left (1+c^2 x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 1.76358, size = 524, normalized size = 1.32 \[ \frac{96 a^2 d^{3/2} f^{3/2} \sqrt{c^2 x^2+1} \log \left (c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right )+64 a^2 c^3 d f x^3 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+160 a^2 c d f x \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+8 b d f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2 \left (12 a+8 b \sinh \left (2 \sinh ^{-1}(c x)\right )+b \sinh \left (4 \sinh ^{-1}(c x)\right )\right )-64 a b d f \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left (2 \sinh ^{-1}(c x)\right )-4 a b d f \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left (4 \sinh ^{-1}(c x)\right )-4 b d f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left (-4 a \left (8 \sinh \left (2 \sinh ^{-1}(c x)\right )+\sinh \left (4 \sinh ^{-1}(c x)\right )\right )+16 b \cosh \left (2 \sinh ^{-1}(c x)\right )+b \cosh \left (4 \sinh ^{-1}(c x)\right )\right )+32 b^2 d f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^3+32 b^2 d f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left (2 \sinh ^{-1}(c x)\right )+b^2 d f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left (4 \sinh ^{-1}(c x)\right )}{256 c \sqrt{c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.263, size = 0, normalized size = 0. \begin{align*} \int \left ( d+icdx \right ) ^{{\frac{3}{2}}} \left ( f-icfx \right ) ^{{\frac{3}{2}}} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{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 ({\left (b^{2} c^{2} d f x^{2} + b^{2} d 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^{2} d f x^{2} + a b d 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^{2} d f x^{2} + a^{2} d f\right )} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f}, 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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