Optimal. Leaf size=184 \[ \frac {b e^2 \left (1-c^2 x^2\right )}{c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \cosh ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {b c d \left (c^2 d+12 e\right ) \sqrt {-1+c^2 x^2} \text {ArcTan}\left (\sqrt {-1+c^2 x^2}\right )}{6 \sqrt {-1+c x} \sqrt {1+c x}} \]
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Rubi [A]
time = 0.20, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {276, 5958, 534,
1265, 911, 1171, 396, 211} \begin {gather*} -\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \cosh ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {b c d \sqrt {c^2 x^2-1} \text {ArcTan}\left (\sqrt {c^2 x^2-1}\right ) \left (c^2 d+12 e\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^2 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e^2 \left (1-c^2 x^2\right )}{c \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 276
Rule 396
Rule 534
Rule 911
Rule 1171
Rule 1265
Rule 5958
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{x^4} \, dx &=-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \cosh ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {-\frac {d^2}{3}-2 d e x^2+e^2 x^4}{x^3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \cosh ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {-\frac {d^2}{3}-2 d e x^2+e^2 x^4}{x^3 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \cosh ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {-\frac {d^2}{3}-2 d e x+e^2 x^2}{x^2 \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \cosh ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\frac {-\frac {1}{3} c^4 d^2-2 c^2 d e+e^2}{c^4}-\frac {\left (2 c^2 d e-2 e^2\right ) x^2}{c^4}+\frac {e^2 x^4}{c^4}}{\left (\frac {1}{c^2}+\frac {x^2}{c^2}\right )^2} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^2 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \cosh ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\frac {1}{3} \left (d^2+\frac {12 d e}{c^2}-\frac {6 e^2}{c^4}\right )-\frac {2 e^2 x^2}{c^4}}{\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b e^2 \left (1-c^2 x^2\right )}{c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \cosh ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {\left (b c d \left (d+\frac {12 e}{c^2}\right ) \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{6 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b e^2 \left (1-c^2 x^2\right )}{c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \cosh ^{-1}(c x)\right )}{x}+e^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {b c d \left (c^2 d+12 e\right ) \sqrt {-1+c^2 x^2} \tan ^{-1}\left (\sqrt {-1+c^2 x^2}\right )}{6 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 133, normalized size = 0.72 \begin {gather*} -\frac {a d^2}{3 x^3}-\frac {2 a d e}{x}+a e^2 x+b \left (-\frac {e^2}{c}+\frac {c d^2}{6 x^2}\right ) \sqrt {-1+c x} \sqrt {1+c x}-\frac {b \left (d^2+6 d e x^2-3 e^2 x^4\right ) \cosh ^{-1}(c x)}{3 x^3}-\frac {1}{6} b c d \left (c^2 d+12 e\right ) \text {ArcTan}\left (\frac {1}{\sqrt {-1+c x} \sqrt {1+c x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.90, size = 216, normalized size = 1.17
method | result | size |
derivativedivides | \(c^{3} \left (\frac {a \left (e^{2} c x -\frac {c \,d^{2}}{3 x^{3}}-\frac {2 c d e}{x}\right )}{c^{4}}+\frac {b \,\mathrm {arccosh}\left (c x \right ) e^{2} x}{c^{3}}-\frac {b \,\mathrm {arccosh}\left (c x \right ) d^{2}}{3 c^{3} x^{3}}-\frac {2 b \,\mathrm {arccosh}\left (c x \right ) d e}{c^{3} x}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, d^{2} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{6 \sqrt {c^{2} x^{2}-1}}-\frac {2 b \sqrt {c x -1}\, \sqrt {c x +1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) d e}{c^{2} \sqrt {c^{2} x^{2}-1}}+\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, d^{2}}{6 c^{2} x^{2}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{2}}{c^{4}}\right )\) | \(216\) |
default | \(c^{3} \left (\frac {a \left (e^{2} c x -\frac {c \,d^{2}}{3 x^{3}}-\frac {2 c d e}{x}\right )}{c^{4}}+\frac {b \,\mathrm {arccosh}\left (c x \right ) e^{2} x}{c^{3}}-\frac {b \,\mathrm {arccosh}\left (c x \right ) d^{2}}{3 c^{3} x^{3}}-\frac {2 b \,\mathrm {arccosh}\left (c x \right ) d e}{c^{3} x}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, d^{2} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{6 \sqrt {c^{2} x^{2}-1}}-\frac {2 b \sqrt {c x -1}\, \sqrt {c x +1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) d e}{c^{2} \sqrt {c^{2} x^{2}-1}}+\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, d^{2}}{6 c^{2} x^{2}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{2}}{c^{4}}\right )\) | \(216\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 126, normalized size = 0.68 \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^{2} - 2 \, {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b d e + a x e^{2} + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b e^{2}}{c} - \frac {2 \, a d e}{x} - \frac {a d^{2}}{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. 390 vs.
\(2 (161) = 322\).
time = 0.43, size = 390, normalized size = 2.12 \begin {gather*} \frac {6 \, a c x^{4} \cosh \left (1\right )^{2} + 6 \, a c x^{4} \sinh \left (1\right )^{2} - 12 \, a c d x^{2} \cosh \left (1\right ) - 2 \, a c d^{2} + 2 \, {\left (b c^{4} d^{2} x^{3} + 12 \, b c^{2} d x^{3} \cosh \left (1\right ) + 12 \, b c^{2} d x^{3} \sinh \left (1\right )\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, {\left (b c d^{2} x^{3} - b c d^{2} + 3 \, {\left (b c x^{4} - b c x^{3}\right )} \cosh \left (1\right )^{2} + 3 \, {\left (b c x^{4} - b c x^{3}\right )} \sinh \left (1\right )^{2} + 6 \, {\left (b c d x^{3} - b c d x^{2}\right )} \cosh \left (1\right ) + 6 \, {\left (b c d x^{3} - b c d x^{2} + {\left (b c x^{4} - b c x^{3}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, {\left (b c d^{2} x^{3} + 6 \, b c d x^{3} \cosh \left (1\right ) - 3 \, b c x^{3} \cosh \left (1\right )^{2} - 3 \, b c x^{3} \sinh \left (1\right )^{2} + 6 \, {\left (b c d x^{3} - b c x^{3} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 12 \, {\left (a c x^{4} \cosh \left (1\right ) - a c d x^{2}\right )} \sinh \left (1\right ) + {\left (b c^{2} d^{2} x - 6 \, b x^{3} \cosh \left (1\right )^{2} - 12 \, b x^{3} \cosh \left (1\right ) \sinh \left (1\right ) - 6 \, b x^{3} \sinh \left (1\right )^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{6 \, c 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 )^{2}}{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 )}^2}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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