Optimal. Leaf size=80 \[ -\frac {b c d \sqrt {1+c^2 x^2}}{6 x^2}-\frac {d \left (a+b \sinh ^{-1}(c x)\right )}{3 x^3}-\frac {c^2 d \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac {5}{6} b c^3 d \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right ) \]
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Rubi [A]
time = 0.06, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {14, 5803, 12,
457, 79, 65, 214} \begin {gather*} -\frac {c^2 d \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac {d \left (a+b \sinh ^{-1}(c x)\right )}{3 x^3}-\frac {b c d \sqrt {c^2 x^2+1}}{6 x^2}-\frac {5}{6} b c^3 d \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 65
Rule 79
Rule 214
Rule 457
Rule 5803
Rubi steps
\begin {align*} \int \frac {\left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{x^4} \, dx &=-\frac {d \left (a+b \sinh ^{-1}(c x)\right )}{3 x^3}-\frac {c^2 d \left (a+b \sinh ^{-1}(c x)\right )}{x}-(b c) \int \frac {d \left (-1-3 c^2 x^2\right )}{3 x^3 \sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {d \left (a+b \sinh ^{-1}(c x)\right )}{3 x^3}-\frac {c^2 d \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac {1}{3} (b c d) \int \frac {-1-3 c^2 x^2}{x^3 \sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {d \left (a+b \sinh ^{-1}(c x)\right )}{3 x^3}-\frac {c^2 d \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac {1}{6} (b c d) \text {Subst}\left (\int \frac {-1-3 c^2 x}{x^2 \sqrt {1+c^2 x}} \, dx,x,x^2\right )\\ &=-\frac {b c d \sqrt {1+c^2 x^2}}{6 x^2}-\frac {d \left (a+b \sinh ^{-1}(c x)\right )}{3 x^3}-\frac {c^2 d \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac {1}{12} \left (5 b c^3 d\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )\\ &=-\frac {b c d \sqrt {1+c^2 x^2}}{6 x^2}-\frac {d \left (a+b \sinh ^{-1}(c x)\right )}{3 x^3}-\frac {c^2 d \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac {1}{6} (5 b c d) \text {Subst}\left (\int \frac {1}{-\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {1+c^2 x^2}\right )\\ &=-\frac {b c d \sqrt {1+c^2 x^2}}{6 x^2}-\frac {d \left (a+b \sinh ^{-1}(c x)\right )}{3 x^3}-\frac {c^2 d \left (a+b \sinh ^{-1}(c x)\right )}{x}-\frac {5}{6} b c^3 d \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 93, normalized size = 1.16 \begin {gather*} -\frac {a d}{3 x^3}-\frac {a c^2 d}{x}-\frac {b c d \sqrt {1+c^2 x^2}}{6 x^2}-\frac {b d \sinh ^{-1}(c x)}{3 x^3}-\frac {b c^2 d \sinh ^{-1}(c x)}{x}-\frac {5}{6} b c^3 d \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.49, size = 87, normalized size = 1.09
method | result | size |
derivativedivides | \(c^{3} \left (a d \left (-\frac {1}{c x}-\frac {1}{3 c^{3} x^{3}}\right )+b d \left (-\frac {\arcsinh \left (c x \right )}{c x}-\frac {\arcsinh \left (c x \right )}{3 c^{3} x^{3}}-\frac {5 \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}-\frac {\sqrt {c^{2} x^{2}+1}}{6 c^{2} x^{2}}\right )\right )\) | \(87\) |
default | \(c^{3} \left (a d \left (-\frac {1}{c x}-\frac {1}{3 c^{3} x^{3}}\right )+b d \left (-\frac {\arcsinh \left (c x \right )}{c x}-\frac {\arcsinh \left (c x \right )}{3 c^{3} x^{3}}-\frac {5 \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}-\frac {\sqrt {c^{2} x^{2}+1}}{6 c^{2} x^{2}}\right )\right )\) | \(87\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 91, normalized size = 1.14 \begin {gather*} -{\left (c \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arsinh}\left (c x\right )}{x}\right )} b c^{2} d + \frac {1}{6} \, {\left ({\left (c^{2} \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) - \frac {\sqrt {c^{2} x^{2} + 1}}{x^{2}}\right )} c - \frac {2 \, \operatorname {arsinh}\left (c x\right )}{x^{3}}\right )} b d - \frac {a c^{2} d}{x} - \frac {a d}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 169 vs.
\(2 (70) = 140\).
time = 0.38, size = 169, normalized size = 2.11 \begin {gather*} -\frac {5 \, b c^{3} d x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} + 1} + 1\right ) - 5 \, b c^{3} d x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} + 1} - 1\right ) + 6 \, a c^{2} d x^{2} - 2 \, {\left (3 \, b c^{2} + b\right )} d x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} + 1}\right ) + \sqrt {c^{2} x^{2} + 1} b c d x + 2 \, a d + 2 \, {\left (3 \, b c^{2} d x^{2} - {\left (3 \, b c^{2} + b\right )} d x^{3} + b d\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{6 \, x^{3}} \end {gather*}
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 \begin {gather*} d \left (\int \frac {a}{x^{4}}\, dx + \int \frac {a c^{2}}{x^{2}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {b c^{2} \operatorname {asinh}{\left (c x \right )}}{x^{2}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\left (d\,c^2\,x^2+d\right )}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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