Optimal. Leaf size=160 \[ \frac {b e \left (6 c^2 d+e\right ) \left (1-c^2 x^2\right )}{3 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e^2 \left (1-c^2 x^2\right )^2}{9 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+b c d^2 \text {ArcTan}\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.18, antiderivative size = 185, normalized size of antiderivative = 1.16, number of steps
used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {276, 5958, 534,
1265, 911, 1167, 211} \begin {gather*} -\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {b c d^2 \sqrt {c^2 x^2-1} \text {ArcTan}\left (\sqrt {c^2 x^2-1}\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {b e \left (1-c^2 x^2\right ) \left (6 c^2 d+e\right )}{3 c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b e^2 \left (1-c^2 x^2\right )^2}{9 c^3 \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 276
Rule 534
Rule 911
Rule 1167
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^2} \, dx &=-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {-d^2+2 d e x^2+\frac {e^2 x^4}{3}}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {-d^2+2 d e x^2+\frac {e^2 x^4}{3}}{x \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 )}{x}+2 d e x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {-d^2+2 d e x+\frac {e^2 x^2}{3}}{x \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 )}{x}+2 d e x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\frac {-c^4 d^2+2 c^2 d e+\frac {e^2}{3}}{c^4}-\frac {\left (-2 c^2 d e-\frac {2 e^2}{3}\right ) x^2}{c^4}+\frac {e^2 x^4}{3 c^4}}{\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{3} e \left (6 d+\frac {e}{c^2}\right )+\frac {e^2 x^2}{3 c^2}-\frac {d^2}{\frac {1}{c^2}+\frac {x^2}{c^2}}\right ) \, dx,x,\sqrt {-1+c^2 x^2}\right )}{c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b e \left (6 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 e^2 \left (1-c^2 x^2\right )^2}{9 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {\left (b d^2 \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 )}{c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b e \left (6 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 e^2 \left (1-c^2 x^2\right )^2}{9 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {b c d^2 \sqrt {-1+c^2 x^2} \tan ^{-1}\left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.15, size = 128, normalized size = 0.80 \begin {gather*} \frac {1}{3} \left (-\frac {3 a d^2}{x}+6 a d e x+a e^2 x^3-\frac {b e \sqrt {-1+c x} \sqrt {1+c x} \left (2 e+c^2 \left (18 d+e x^2\right )\right )}{3 c^3}+\frac {b \left (-3 d^2+6 d e x^2+e^2 x^4\right ) \cosh ^{-1}(c x)}{x}-3 b c d^2 \text {ArcTan}\left (\frac {1}{\sqrt {-1+c x} \sqrt {1+c x}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.92, size = 199, normalized size = 1.24
method | result | size |
derivativedivides | \(c \left (\frac {a \left (2 c^{3} d e x +\frac {e^{2} c^{3} x^{3}}{3}-\frac {c^{3} d^{2}}{x}\right )}{c^{4}}+\frac {2 b \,\mathrm {arccosh}\left (c x \right ) d e x}{c}+\frac {b \,\mathrm {arccosh}\left (c x \right ) e^{2} x^{3}}{3 c}-\frac {b \,\mathrm {arccosh}\left (c x \right ) d^{2}}{c x}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, d^{2} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{\sqrt {c^{2} x^{2}-1}}-\frac {2 b \sqrt {c x -1}\, \sqrt {c x +1}\, d e}{c^{2}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{2} x^{2}}{9 c^{2}}-\frac {2 b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{2}}{9 c^{4}}\right )\) | \(199\) |
default | \(c \left (\frac {a \left (2 c^{3} d e x +\frac {e^{2} c^{3} x^{3}}{3}-\frac {c^{3} d^{2}}{x}\right )}{c^{4}}+\frac {2 b \,\mathrm {arccosh}\left (c x \right ) d e x}{c}+\frac {b \,\mathrm {arccosh}\left (c x \right ) e^{2} x^{3}}{3 c}-\frac {b \,\mathrm {arccosh}\left (c x \right ) d^{2}}{c x}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, d^{2} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{\sqrt {c^{2} x^{2}-1}}-\frac {2 b \sqrt {c x -1}\, \sqrt {c x +1}\, d e}{c^{2}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{2} x^{2}}{9 c^{2}}-\frac {2 b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{2}}{9 c^{4}}\right )\) | \(199\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.48, size = 134, normalized size = 0.84 \begin {gather*} \frac {1}{3} \, a x^{3} e^{2} - {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b d^{2} + 2 \, a d x 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^{2} + \frac {2 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d e}{c} - \frac {a d^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 424 vs.
\(2 (141) = 282\).
time = 0.37, size = 424, normalized size = 2.65 \begin {gather*} \frac {3 \, a c^{3} x^{4} \cosh \left (1\right )^{2} + 3 \, a c^{3} x^{4} \sinh \left (1\right )^{2} + 18 \, b c^{4} d^{2} x \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 18 \, a c^{3} d x^{2} \cosh \left (1\right ) - 9 \, a c^{3} d^{2} + 3 \, {\left (3 \, b c^{3} d^{2} x - 3 \, b c^{3} d^{2} + {\left (b c^{3} x^{4} - b c^{3} x\right )} \cosh \left (1\right )^{2} + {\left (b c^{3} x^{4} - b c^{3} x\right )} \sinh \left (1\right )^{2} + 6 \, {\left (b c^{3} d x^{2} - b c^{3} d x\right )} \cosh \left (1\right ) + 2 \, {\left (3 \, b c^{3} d x^{2} - 3 \, b c^{3} d x + {\left (b c^{3} x^{4} - b c^{3} x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 3 \, {\left (3 \, b c^{3} d^{2} x - 6 \, b c^{3} d x \cosh \left (1\right ) - b c^{3} x \cosh \left (1\right )^{2} - b c^{3} x \sinh \left (1\right )^{2} - 2 \, {\left (3 \, b c^{3} d x + b c^{3} x \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 6 \, {\left (a c^{3} x^{4} \cosh \left (1\right ) + 3 \, a c^{3} d x^{2}\right )} \sinh \left (1\right ) - {\left (18 \, b c^{2} d x \cosh \left (1\right ) + {\left (b c^{2} x^{3} + 2 \, b x\right )} \cosh \left (1\right )^{2} + {\left (b c^{2} x^{3} + 2 \, b x\right )} \sinh \left (1\right )^{2} + 2 \, {\left (9 \, b c^{2} d x + {\left (b c^{2} x^{3} + 2 \, b x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} - 1}}{9 \, c^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________