Optimal. Leaf size=94 \[ \frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{6 x^2}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac {1}{6} b c \left (c^2 d+6 e\right ) \text {ArcTan}\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \]
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Rubi [A]
time = 0.07, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {5956, 465, 94,
211} \begin {gather*} -\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac {1}{6} b c \text {ArcTan}\left (\sqrt {c x-1} \sqrt {c x+1}\right ) \left (c^2 d+6 e\right )+\frac {b c d \sqrt {c x-1} \sqrt {c x+1}}{6 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 94
Rule 211
Rule 465
Rule 5956
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{x^4} \, dx &=-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac {1}{3} (b c) \int \frac {-d-3 e x^2}{x^3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{6 x^2}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac {1}{6} \left (b c \left (c^2 d+6 e\right )\right ) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{6 x^2}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac {1}{6} \left (b c^2 \left (c^2 d+6 e\right )\right ) \text {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )\\ &=\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{6 x^2}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac {1}{6} b c \left (c^2 d+6 e\right ) \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 128, normalized size = 1.36 \begin {gather*} \frac {-2 b \left (d+3 e x^2\right ) \cosh ^{-1}(c x)+\frac {b c d x \left (-1+c^2 x^2\right )-2 a \sqrt {-1+c x} \sqrt {1+c x} \left (d+3 e x^2\right )+b c \left (c^2 d+6 e\right ) x^3 \sqrt {-1+c^2 x^2} \text {ArcTan}\left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}}{6 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(166\) vs.
\(2(80)=160\).
time = 1.86, size = 167, normalized size = 1.78
method | result | size |
derivativedivides | \(c^{3} \left (\frac {a \left (-\frac {d}{3 c \,x^{3}}-\frac {e}{c x}\right )}{c^{2}}-\frac {b \,\mathrm {arccosh}\left (c x \right ) d}{3 c^{3} x^{3}}-\frac {b \,\mathrm {arccosh}\left (c x \right ) e}{c^{3} x}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, d \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{6 \sqrt {c^{2} x^{2}-1}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) e}{c^{2} \sqrt {c^{2} x^{2}-1}}+\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, d}{6 c^{2} x^{2}}\right )\) | \(167\) |
default | \(c^{3} \left (\frac {a \left (-\frac {d}{3 c \,x^{3}}-\frac {e}{c x}\right )}{c^{2}}-\frac {b \,\mathrm {arccosh}\left (c x \right ) d}{3 c^{3} x^{3}}-\frac {b \,\mathrm {arccosh}\left (c x \right ) e}{c^{3} x}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, d \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{6 \sqrt {c^{2} x^{2}-1}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) e}{c^{2} \sqrt {c^{2} x^{2}-1}}+\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, d}{6 c^{2} x^{2}}\right )\) | \(167\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 87, normalized size = 0.93 \begin {gather*} -\frac {1}{6} \, {\left ({\left (c^{2} \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) - \frac {\sqrt {c^{2} x^{2} - 1}}{x^{2}}\right )} c + \frac {2 \, \operatorname {arcosh}\left (c x\right )}{x^{3}}\right )} b d - {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b e - \frac {a e}{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. 192 vs.
\(2 (82) = 164\).
time = 0.42, size = 192, normalized size = 2.04 \begin {gather*} \frac {\sqrt {c^{2} x^{2} - 1} b c d x - 6 \, a x^{2} \cosh \left (1\right ) - 6 \, a x^{2} \sinh \left (1\right ) - 2 \, a d + 2 \, {\left (b c^{3} d x^{3} + 6 \, b c x^{3} \cosh \left (1\right ) + 6 \, b c x^{3} \sinh \left (1\right )\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, {\left (b d x^{3} - b d + 3 \, {\left (b x^{3} - b x^{2}\right )} \cosh \left (1\right ) + 3 \, {\left (b x^{3} - b x^{2}\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, {\left (b d x^{3} + 3 \, b x^{3} \cosh \left (1\right ) + 3 \, b x^{3} \sinh \left (1\right )\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*} \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{4}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 {acosh}\left (c\,x\right )\right )\,\left (e\,x^2+d\right )}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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