Optimal. Leaf size=413 \[ -\frac {x \cosh ^{-1}(a x)}{c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \cosh ^{-1}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {PolyLog}\left (2,e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {PolyLog}\left (3,e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.35, antiderivative size = 413, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {5901, 5899,
5913, 3797, 2221, 2611, 2320, 6724, 5912, 5914, 5900, 266} \begin {gather*} -\frac {2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x) \text {Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {Li}_3\left (e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {a x-1} \sqrt {a x+1} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}}+\frac {2 x \cosh ^{-1}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {x \cosh ^{-1}(a x)}{c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 2221
Rule 2320
Rule 2611
Rule 3797
Rule 5899
Rule 5900
Rule 5901
Rule 5912
Rule 5913
Rule 5914
Rule 6724
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)^3}{(-1+a x)^{5/2} (1+a x)^{5/2}} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {x \cosh ^{-1}(a x)^3}{3 c^2 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}-\frac {\left (2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)^3}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {\left (a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x \cosh ^{-1}(a x)^2}{\left (-1+a^2 x^2\right )^2} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {2 x \cosh ^{-1}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)^3}{3 c^2 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}+\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{c^2 \sqrt {c-a^2 c x^2}}+\frac {\left (2 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x \cosh ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=-\frac {x \cosh ^{-1}(a x)}{c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {2 x \cosh ^{-1}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)^3}{3 c^2 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}-\frac {\left (2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int x^2 \coth (x) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2 \sqrt {c-a^2 c x^2}}-\frac {\left (a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x}{1-a^2 x^2} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=-\frac {x \cosh ^{-1}(a x)}{c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {2 x \cosh ^{-1}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)^3}{3 c^2 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}}+\frac {\left (4 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \frac {e^{2 x} x^2}{1-e^{2 x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2 \sqrt {c-a^2 c x^2}}\\ &=-\frac {x \cosh ^{-1}(a x)}{c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {2 x \cosh ^{-1}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)^3}{3 c^2 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}}+\frac {\left (4 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int x \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2 \sqrt {c-a^2 c x^2}}\\ &=-\frac {x \cosh ^{-1}(a x)}{c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {2 x \cosh ^{-1}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)^3}{3 c^2 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\left (2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \text {Li}_2\left (e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2 \sqrt {c-a^2 c x^2}}\\ &=-\frac {x \cosh ^{-1}(a x)}{c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {2 x \cosh ^{-1}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)^3}{3 c^2 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}\\ &=-\frac {x \cosh ^{-1}(a x)}{c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {2 x \cosh ^{-1}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)^3}{3 c^2 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {Li}_3\left (e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.71, size = 258, normalized size = 0.62 \begin {gather*} \frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \left (-i \pi ^3-\frac {12 a x \sqrt {\frac {-1+a x}{1+a x}} \cosh ^{-1}(a x)}{-1+a x}+\frac {6 \cosh ^{-1}(a x)^2}{1-a^2 x^2}+8 \cosh ^{-1}(a x)^3+\frac {8 a x \sqrt {\frac {-1+a x}{1+a x}} \cosh ^{-1}(a x)^3}{-1+a x}-\frac {4 a x \left (\frac {-1+a x}{1+a x}\right )^{3/2} \cosh ^{-1}(a x)^3}{(-1+a x)^3}-24 \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )+12 \log \left (\sqrt {\frac {-1+a x}{1+a x}} (1+a x)\right )-24 \cosh ^{-1}(a x) \text {PolyLog}\left (2,e^{2 \cosh ^{-1}(a x)}\right )+12 \text {PolyLog}\left (3,e^{2 \cosh ^{-1}(a x)}\right )\right )}{12 a c^2 \sqrt {c-a^2 c x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(954\) vs.
\(2(400)=800\).
time = 3.07, size = 955, normalized size = 2.31
method | result | size |
default | \(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 a^{3} x^{3}-3 a x -2 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}+2 \sqrt {a x -1}\, \sqrt {a x +1}\right ) \mathrm {arccosh}\left (a x \right ) \left (6 \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}\, a^{3} x^{3}+6 \,\mathrm {arccosh}\left (a x \right ) a^{4} x^{4}+6 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}+6 a^{4} x^{4}+6 \mathrm {arccosh}\left (a x \right )^{2} a^{2} x^{2}-9 a x \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}-12 \,\mathrm {arccosh}\left (a x \right ) a^{2} x^{2}-6 \sqrt {a x +1}\, \sqrt {a x -1}\, a x -18 a^{2} x^{2}-8 \mathrm {arccosh}\left (a x \right )^{2}+6 \,\mathrm {arccosh}\left (a x \right )+12\right )}{6 \left (3 a^{6} x^{6}-10 a^{4} x^{4}+11 a^{2} x^{2}-4\right ) a \,c^{3}}+\frac {2 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \ln \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \ln \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}-1\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}-\frac {4 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )^{3}}{3 c^{3} a \left (a^{2} x^{2}-1\right )}+\frac {2 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}+\frac {4 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right ) \polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}-\frac {4 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \polylog \left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}+\frac {2 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}+\frac {4 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right ) \polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}-\frac {4 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \polylog \left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}\) | \(955\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [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]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________