Optimal. Leaf size=355 \[ -\frac {b x^4 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt {d-c^2 d x^2}}-\frac {x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2 d}+\frac {\sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c^5 \sqrt {d-c^2 d x^2}}-\frac {3 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{8 c^4 d}-\frac {3 b x^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^3 \sqrt {d-c^2 d x^2}}-\frac {b^2 x^3 (1-c x) (c x+1)}{32 c^2 \sqrt {d-c^2 d x^2}}+\frac {15 b^2 \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{64 c^5 \sqrt {d-c^2 d x^2}}-\frac {15 b^2 x (1-c x) (c x+1)}{64 c^4 \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 1.02, antiderivative size = 371, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {5798, 5759, 5676, 5662, 90, 52, 100, 12} \[ -\frac {b x^4 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt {d-c^2 d x^2}}-\frac {x^3 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2 \sqrt {d-c^2 d x^2}}-\frac {3 b x^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^3 \sqrt {d-c^2 d x^2}}-\frac {3 x (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^2}{8 c^4 \sqrt {d-c^2 d x^2}}+\frac {\sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c^5 \sqrt {d-c^2 d x^2}}-\frac {b^2 x^3 (1-c x) (c x+1)}{32 c^2 \sqrt {d-c^2 d x^2}}-\frac {15 b^2 x (1-c x) (c x+1)}{64 c^4 \sqrt {d-c^2 d x^2}}+\frac {15 b^2 \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{64 c^5 \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 52
Rule 90
Rule 100
Rule 5662
Rule 5676
Rule 5759
Rule 5798
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=-\frac {x^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 c^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{2 c \sqrt {d-c^2 d x^2}}\\ &=-\frac {b x^4 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt {d-c^2 d x^2}}-\frac {3 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{8 c^4 \sqrt {d-c^2 d x^2}}-\frac {x^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^4}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 \sqrt {d-c^2 d x^2}}+\frac {\left (3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 c^4 \sqrt {d-c^2 d x^2}}-\frac {\left (3 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{4 c^3 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 x^3 (1-c x) (1+c x)}{32 c^2 \sqrt {d-c^2 d x^2}}-\frac {3 b x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^3 \sqrt {d-c^2 d x^2}}-\frac {b x^4 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt {d-c^2 d x^2}}-\frac {3 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{8 c^4 \sqrt {d-c^2 d x^2}}-\frac {x^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2 \sqrt {d-c^2 d x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c^5 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {3 x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{32 c^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 c^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {3 b^2 x (1-c x) (1+c x)}{16 c^4 \sqrt {d-c^2 d x^2}}-\frac {b^2 x^3 (1-c x) (1+c x)}{32 c^2 \sqrt {d-c^2 d x^2}}-\frac {3 b x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^3 \sqrt {d-c^2 d x^2}}-\frac {b x^4 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt {d-c^2 d x^2}}-\frac {3 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{8 c^4 \sqrt {d-c^2 d x^2}}-\frac {x^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2 \sqrt {d-c^2 d x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c^5 \sqrt {d-c^2 d x^2}}+\frac {\left (3 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{16 c^4 \sqrt {d-c^2 d x^2}}+\frac {\left (3 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{32 c^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {15 b^2 x (1-c x) (1+c x)}{64 c^4 \sqrt {d-c^2 d x^2}}-\frac {b^2 x^3 (1-c x) (1+c x)}{32 c^2 \sqrt {d-c^2 d x^2}}+\frac {3 b^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{16 c^5 \sqrt {d-c^2 d x^2}}-\frac {3 b x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^3 \sqrt {d-c^2 d x^2}}-\frac {b x^4 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt {d-c^2 d x^2}}-\frac {3 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{8 c^4 \sqrt {d-c^2 d x^2}}-\frac {x^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2 \sqrt {d-c^2 d x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c^5 \sqrt {d-c^2 d x^2}}+\frac {\left (3 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{64 c^4 \sqrt {d-c^2 d x^2}}\\ &=-\frac {15 b^2 x (1-c x) (1+c x)}{64 c^4 \sqrt {d-c^2 d x^2}}-\frac {b^2 x^3 (1-c x) (1+c x)}{32 c^2 \sqrt {d-c^2 d x^2}}+\frac {15 b^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{64 c^5 \sqrt {d-c^2 d x^2}}-\frac {3 b x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^3 \sqrt {d-c^2 d x^2}}-\frac {b x^4 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt {d-c^2 d x^2}}-\frac {3 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{8 c^4 \sqrt {d-c^2 d x^2}}-\frac {x^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2 \sqrt {d-c^2 d x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c^5 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 1.54, size = 295, normalized size = 0.83 \[ \frac {32 a^2 c \sqrt {d} x \left (c^2 x^2-1\right ) \left (2 c^2 x^2+3\right )-96 a^2 \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )-4 a b \sqrt {d} \sqrt {\frac {c x-1}{c x+1}} (c x+1) \left (16 \cosh \left (2 \cosh ^{-1}(c x)\right )+\cosh \left (4 \cosh ^{-1}(c x)\right )-4 \cosh ^{-1}(c x) \left (6 \cosh ^{-1}(c x)+8 \sinh \left (2 \cosh ^{-1}(c x)\right )+\sinh \left (4 \cosh ^{-1}(c x)\right )\right )\right )+b^2 \sqrt {d} \sqrt {\frac {c x-1}{c x+1}} (c x+1) \left (32 \cosh ^{-1}(c x)^3-4 \left (16 \cosh \left (2 \cosh ^{-1}(c x)\right )+\cosh \left (4 \cosh ^{-1}(c x)\right )\right ) \cosh ^{-1}(c x)+8 \cosh ^{-1}(c x)^2 \left (8 \sinh \left (2 \cosh ^{-1}(c x)\right )+\sinh \left (4 \cosh ^{-1}(c x)\right )\right )+32 \sinh \left (2 \cosh ^{-1}(c x)\right )+\sinh \left (4 \cosh ^{-1}(c x)\right )\right )}{256 c^5 \sqrt {d} \sqrt {d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (b^{2} x^{4} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b x^{4} \operatorname {arcosh}\left (c x\right ) + a^{2} x^{4}\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{2} d x^{2} - d}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{4}}{\sqrt {-c^{2} d x^{2} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.95, size = 887, normalized size = 2.50 \[ -\frac {a^{2} x^{3} \sqrt {-c^{2} d \,x^{2}+d}}{4 c^{2} d}-\frac {3 a^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{8 c^{4} d}+\frac {3 a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 c^{4} \sqrt {c^{2} d}}+\frac {3 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, x^{2}}{8 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{5}}{32 d \left (c^{2} x^{2}-1\right )}-\frac {13 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3}}{64 d \,c^{2} \left (c^{2} x^{2}-1\right )}+\frac {15 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x}{64 d \,c^{4} \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right )^{3}}{8 d \,c^{5} \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, x^{4}}{8 d c \left (c^{2} x^{2}-1\right )}-\frac {15 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{64 d \,c^{5} \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2} x^{5}}{4 d \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2} x^{3}}{8 d \,c^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2} x}{8 d \,c^{4} \left (c^{2} x^{2}-1\right )}-\frac {3 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right )^{2}}{8 d \,c^{5} \left (c^{2} x^{2}-1\right )}+\frac {3 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2}}{8 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x^{5}}{2 d \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x^{3}}{4 d \,c^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x}{4 d \,c^{4} \left (c^{2} x^{2}-1\right )}-\frac {15 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}}{64 d \,c^{5} \left (c^{2} x^{2}-1\right )}+\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x +1}\, \sqrt {c x -1}\, x^{4}}{8 d c \left (c^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{8} \, a^{2} {\left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} x^{3}}{c^{2} d} + \frac {3 \, \sqrt {-c^{2} d x^{2} + d} x}{c^{4} d} - \frac {3 \, \arcsin \left (c x\right )}{c^{5} \sqrt {d}}\right )} + \int \frac {b^{2} x^{4} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2}}{\sqrt {-c^{2} d x^{2} + d}} + \frac {2 \, a b x^{4} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{\sqrt {-c^{2} d x^{2} + d}}\,{d x} \]
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 \[ \int \frac {x^4\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{\sqrt {d-c^2\,d\,x^2}} \,d x \]
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 \[ \int \frac {x^{4} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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