Optimal. Leaf size=141 \[ \frac{2 c^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x}-\frac{\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^3}-\frac{b c \sqrt{c^2 x^2+1}}{6 x^2 \sqrt{c^2 d x^2+d}}-\frac{2 b c^3 \sqrt{c^2 x^2+1} \log (x)}{3 \sqrt{c^2 d x^2+d}} \]
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Rubi [A] time = 0.18419, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {5747, 5723, 29, 30} \[ \frac{2 c^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x}-\frac{\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^3}-\frac{b c \sqrt{c^2 x^2+1}}{6 x^2 \sqrt{c^2 d x^2+d}}-\frac{2 b c^3 \sqrt{c^2 x^2+1} \log (x)}{3 \sqrt{c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 5747
Rule 5723
Rule 29
Rule 30
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c x)}{x^4 \sqrt{d+c^2 d x^2}} \, dx &=-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^3}-\frac{1}{3} \left (2 c^2\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x^2 \sqrt{d+c^2 d x^2}} \, dx+\frac{\left (b c \sqrt{1+c^2 x^2}\right ) \int \frac{1}{x^3} \, dx}{3 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{6 x^2 \sqrt{d+c^2 d x^2}}-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^3}+\frac{2 c^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x}-\frac{\left (2 b c^3 \sqrt{1+c^2 x^2}\right ) \int \frac{1}{x} \, dx}{3 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{6 x^2 \sqrt{d+c^2 d x^2}}-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^3}+\frac{2 c^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x}-\frac{2 b c^3 \sqrt{1+c^2 x^2} \log (x)}{3 \sqrt{d+c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.209244, size = 135, normalized size = 0.96 \[ \frac{2 a \left (2 c^4 x^4+c^2 x^2-1\right )+b c x \sqrt{c^2 x^2+1} \left (6 c^2 x^2-1\right )+2 b \left (2 c^4 x^4+c^2 x^2-1\right ) \sinh ^{-1}(c x)}{6 x^3 \sqrt{c^2 d x^2+d}}-\frac{2 b c^3 \log (x) \sqrt{d \left (c^2 x^2+1\right )}}{3 d \sqrt{c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.17, size = 791, normalized size = 5.6 \begin{align*} \text{result too large to display} \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 [A] time = 3.18487, size = 477, normalized size = 3.38 \begin{align*} \frac{2 \,{\left (2 \, b c^{4} x^{4} + b c^{2} x^{2} - b\right )} \sqrt{c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + 2 \,{\left (b c^{5} x^{5} + b c^{3} x^{3}\right )} \sqrt{d} \log \left (\frac{c^{2} d x^{6} + c^{2} d x^{2} + d x^{4} - \sqrt{c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} + 1}{\left (x^{4} - 1\right )} \sqrt{d} + d}{c^{2} x^{4} + x^{2}}\right ) +{\left (4 \, a c^{4} x^{4} + 2 \, a c^{2} x^{2} +{\left (b c x^{3} - b c x\right )} \sqrt{c^{2} x^{2} + 1} - 2 \, a\right )} \sqrt{c^{2} d x^{2} + d}}{6 \,{\left (c^{2} d x^{5} + d x^{3}\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 )}}{x^{4} \sqrt{d \left (c^{2} x^{2} + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arsinh}\left (c x\right ) + a}{\sqrt{c^{2} d x^{2} + d} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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