Optimal. Leaf size=260 \[ -\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {3 d^2 e \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 d e^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} e^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {b c d^3 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c d^2 \sqrt {c^2 x^2-1} \left (c^2 d+18 e\right ) \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e^2 \left (1-c^2 x^2\right ) \left (9 c^2 d+e\right )}{3 c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b e^3 \left (1-c^2 x^2\right )^2}{9 c^3 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 0.46, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {270, 5790, 12, 1610, 1799, 1621, 897, 1153, 205} \[ -\frac {3 d^2 e \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+3 d e^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} e^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {b c d^2 \sqrt {c^2 x^2-1} \left (c^2 d+18 e\right ) \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^3 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e^2 \left (1-c^2 x^2\right ) \left (9 c^2 d+e\right )}{3 c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b e^3 \left (1-c^2 x^2\right )^2}{9 c^3 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 270
Rule 897
Rule 1153
Rule 1610
Rule 1621
Rule 1799
Rule 5790
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{x^4} \, dx &=-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {3 d^2 e \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 d e^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} e^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {-d^3-9 d^2 e x^2+9 d e^2 x^4+e^3 x^6}{3 x^3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {3 d^2 e \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 d e^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} e^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{3} (b c) \int \frac {-d^3-9 d^2 e x^2+9 d e^2 x^4+e^3 x^6}{x^3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {3 d^2 e \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 d e^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} e^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {-d^3-9 d^2 e x^2+9 d e^2 x^4+e^3 x^6}{x^3 \sqrt {-1+c^2 x^2}} \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {3 d^2 e \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 d e^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} e^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {-d^3-9 d^2 e x+9 d e^2 x^2+e^3 x^3}{x^2 \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{6 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^3 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {3 d^2 e \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 d e^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} e^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {-\frac {1}{2} d^2 \left (c^2 d+18 e\right )+9 d e^2 x+e^3 x^2}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{6 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^3 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {3 d^2 e \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 d e^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} e^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\frac {9 c^2 d e^2+e^3-\frac {1}{2} c^4 d^2 \left (c^2 d+18 e\right )}{c^4}-\frac {\left (-9 c^2 d e^2-2 e^3\right ) x^2}{c^4}+\frac {e^3 x^4}{c^4}}{\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{3 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^3 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {3 d^2 e \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 d e^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} e^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (e^2 \left (9 d+\frac {e}{c^2}\right )+\frac {e^3 x^2}{c^2}+\frac {-c^2 d^3-18 d^2 e}{2 \left (\frac {1}{c^2}+\frac {x^2}{c^2}\right )}\right ) \, dx,x,\sqrt {-1+c^2 x^2}\right )}{3 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b e^2 \left (9 c^2 d+e\right ) \left (1-c^2 x^2\right )}{3 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^3 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e^3 \left (1-c^2 x^2\right )^2}{9 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {3 d^2 e \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 d e^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} e^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {\left (b d^2 \left (c^2 d+18 e\right ) \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{6 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b e^2 \left (9 c^2 d+e\right ) \left (1-c^2 x^2\right )}{3 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^3 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e^3 \left (1-c^2 x^2\right )^2}{9 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {3 d^2 e \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 d e^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} e^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {b c d^2 \left (c^2 d+18 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 = 0.39, size = 184, normalized size = 0.71 \[ \frac {1}{6} \left (-\frac {2 a d^3}{x^3}-\frac {18 a d^2 e}{x}+18 a d e^2 x+2 a e^3 x^3-b c d^2 \left (c^2 d+18 e\right ) \tan ^{-1}\left (\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (-3 c^4 d^3+2 c^2 e^2 x^2 \left (27 d+e x^2\right )+4 e^3 x^2\right )}{3 c^3 x^2}+\frac {2 b \cosh ^{-1}(c x) \left (-d^3-9 d^2 e x^2+9 d e^2 x^4+e^3 x^6\right )}{x^3}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 322, normalized size = 1.24 \[ \frac {6 \, a c^{3} e^{3} x^{6} + 54 \, a c^{3} d e^{2} x^{4} - 54 \, a c^{3} d^{2} e x^{2} - 6 \, a c^{3} d^{3} + 6 \, {\left (b c^{6} d^{3} + 18 \, b c^{4} d^{2} e\right )} x^{3} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 6 \, {\left (b c^{3} d^{3} + 9 \, b c^{3} d^{2} e - 9 \, b c^{3} d e^{2} - b c^{3} e^{3}\right )} x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 6 \, {\left (b c^{3} e^{3} x^{6} + 9 \, b c^{3} d e^{2} x^{4} - 9 \, b c^{3} d^{2} e x^{2} - b c^{3} d^{3} + {\left (b c^{3} d^{3} + 9 \, b c^{3} d^{2} e - 9 \, b c^{3} d e^{2} - b c^{3} e^{3}\right )} x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (2 \, b c^{2} e^{3} x^{5} - 3 \, b c^{4} d^{3} x + 2 \, {\left (27 \, b c^{2} d e^{2} + 2 \, b e^{3}\right )} x^{3}\right )} \sqrt {c^{2} x^{2} - 1}}{18 \, c^{3} x^{3}} \]
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 \[ \int \frac {{\left (e x^{2} + d\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 278, normalized size = 1.07 \[ \frac {a \,x^{3} e^{3}}{3}+3 a x d \,e^{2}-\frac {3 a \,d^{2} e}{x}-\frac {a \,d^{3}}{3 x^{3}}+\frac {b \,\mathrm {arccosh}\left (c x \right ) x^{3} e^{3}}{3}+3 b \,\mathrm {arccosh}\left (c x \right ) x d \,e^{2}-\frac {3 b \,\mathrm {arccosh}\left (c x \right ) d^{2} e}{x}-\frac {b \,\mathrm {arccosh}\left (c x \right ) d^{3}}{3 x^{3}}-\frac {c^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, d^{3} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{6 \sqrt {c^{2} x^{2}-1}}+\frac {b c \,d^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{6 x^{2}}-\frac {3 c b \sqrt {c x -1}\, \sqrt {c x +1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) d^{2} e}{\sqrt {c^{2} x^{2}-1}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, x^{2} e^{3}}{9 c}-\frac {3 b \sqrt {c x -1}\, \sqrt {c x +1}\, d \,e^{2}}{c}-\frac {2 b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{3}}{9 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 197, normalized size = 0.76 \[ \frac {1}{3} \, a e^{3} x^{3} - \frac {1}{6} \, {\left ({\left (c^{2} \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) - \frac {\sqrt {c^{2} x^{2} - 1}}{x^{2}}\right )} c + \frac {2 \, \operatorname {arcosh}\left (c x\right )}{x^{3}}\right )} b d^{3} - 3 \, {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b d^{2} e + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b e^{3} + 3 \, a d e^{2} x + \frac {3 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d e^{2}}{c} - \frac {3 \, a d^{2} e}{x} - \frac {a d^{3}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3}{x^4} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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