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 = 0.18, antiderivative size = 141, normalized size of antiderivative = 1.00, 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 29
Rule 30
Rule 5723
Rule 5747
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*}
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Mathematica [A] time = 0.20, 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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fricas [A] time = 0.71, size = 222, normalized size = 1.57 \[ \frac {2 \, {\left (2 \, b c^{4} x^{4} + b c^{2} x^{2} - b\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, {\left (b c^{5} x^{5} + b c^{3} x^{3}\right )} \sqrt {d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} + d x^{4} - \sqrt {c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} + 1} {\left (x^{4} - 1\right )} \sqrt {d} + d}{c^{2} x^{4} + x^{2}}\right ) + {\left (4 \, a c^{4} x^{4} + 2 \, a c^{2} x^{2} + {\left (b c x^{3} - b c x\right )} \sqrt {c^{2} x^{2} + 1} - 2 \, a\right )} \sqrt {c^{2} d x^{2} + d}}{6 \, {\left (c^{2} d x^{5} + d x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arsinh}\left (c x\right ) + a}{\sqrt {c^{2} d x^{2} + d} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.28, size = 791, normalized size = 5.61 \[ -\frac {a \sqrt {c^{2} d \,x^{2}+d}}{3 d \,x^{3}}+\frac {2 a \,c^{2} \sqrt {c^{2} d \,x^{2}+d}}{3 d x}+\frac {4 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) c^{3}}{3 \sqrt {c^{2} x^{2}+1}\, d}+\frac {2 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{5} c^{8}}{3 \left (3 c^{4} x^{4}+2 c^{2} x^{2}-1\right ) d}-\frac {2 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{3} \left (c^{2} x^{2}+1\right ) c^{6}}{3 \left (3 c^{4} x^{4}+2 c^{2} x^{2}-1\right ) d}+\frac {2 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{3} \arcsinh \left (c x \right ) c^{6}}{\left (3 c^{4} x^{4}+2 c^{2} x^{2}-1\right ) d}-\frac {2 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{5}}{\left (3 c^{4} x^{4}+2 c^{2} x^{2}-1\right ) d}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{3} c^{6}}{3 \left (3 c^{4} x^{4}+2 c^{2} x^{2}-1\right ) d}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x \left (c^{2} x^{2}+1\right ) c^{4}}{3 \left (3 c^{4} x^{4}+2 c^{2} x^{2}-1\right ) d}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x \arcsinh \left (c x \right ) c^{4}}{3 \left (3 c^{4} x^{4}+2 c^{2} x^{2}-1\right ) d}+\frac {2 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3}}{3 \left (3 c^{4} x^{4}+2 c^{2} x^{2}-1\right ) d}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x \,c^{4}}{3 \left (3 c^{4} x^{4}+2 c^{2} x^{2}-1\right ) d}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{3} \sqrt {c^{2} x^{2}+1}}{2 \left (3 c^{4} x^{4}+2 c^{2} x^{2}-1\right ) d}-\frac {4 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) c^{2}}{3 \left (3 c^{4} x^{4}+2 c^{2} x^{2}-1\right ) d x}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c \sqrt {c^{2} x^{2}+1}}{6 \left (3 c^{4} x^{4}+2 c^{2} x^{2}-1\right ) d \,x^{2}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )}{3 \left (3 c^{4} x^{4}+2 c^{2} x^{2}-1\right ) d \,x^{3}}-\frac {2 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) c^{3}}{3 \sqrt {c^{2} x^{2}+1}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 121, normalized size = 0.86 \[ -\frac {1}{6} \, {\left (\frac {4 \, c^{2} \log \relax (x)}{\sqrt {d}} + \frac {1}{\sqrt {d} x^{2}}\right )} b c + \frac {1}{3} \, b {\left (\frac {2 \, \sqrt {c^{2} d x^{2} + d} c^{2}}{d x} - \frac {\sqrt {c^{2} d x^{2} + d}}{d x^{3}}\right )} \operatorname {arsinh}\left (c x\right ) + \frac {1}{3} \, a {\left (\frac {2 \, \sqrt {c^{2} d x^{2} + d} c^{2}}{d x} - \frac {\sqrt {c^{2} d x^{2} + d}}{d x^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^4\,\sqrt {d\,c^2\,x^2+d}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{x^{4} \sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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