Optimal. Leaf size=204 \[ -\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {b c x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} b^2 x \sqrt {d-c^2 d x^2}+\frac {b^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{4 c \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 0.35, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5713, 5683, 5676, 5662, 90, 52} \[ -\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {b c x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} b^2 x \sqrt {d-c^2 d x^2}+\frac {b^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{4 c \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 52
Rule 90
Rule 5662
Rule 5676
Rule 5683
Rule 5713
Rubi steps
\begin {align*} \int \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=\frac {\sqrt {d-c^2 d x^2} \int \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{2} x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\sqrt {d-c^2 d x^2} \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 c^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{4} b^2 x \sqrt {d-c^2 d x^2}-\frac {b c x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{4} b^2 x \sqrt {d-c^2 d x^2}+\frac {b^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{4 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b c \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 1.09, size = 235, normalized size = 1.15 \[ \frac {1}{24} \left (12 a^2 x \sqrt {d-c^2 d x^2}-\frac {12 a^2 \sqrt {d} \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )}{c}-\frac {6 a b \sqrt {d-c^2 d x^2} \left (2 \cosh ^{-1}(c x)^2+\cosh \left (2 \cosh ^{-1}(c x)\right )-2 \cosh ^{-1}(c x) \sinh \left (2 \cosh ^{-1}(c x)\right )\right )}{c \sqrt {\frac {c x-1}{c x+1}} (c x+1)}+\frac {b^2 \sqrt {d-c^2 d x^2} \left (-4 \cosh ^{-1}(c x)^3-6 \cosh \left (2 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)+\left (6 \cosh ^{-1}(c x)^2+3\right ) \sinh \left (2 \cosh ^{-1}(c x)\right )\right )}{c \sqrt {\frac {c x-1}{c x+1}} (c x+1)}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-c^{2} d x^{2} + d} {\left (b^{2} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname {arcosh}\left (c x\right ) + a^{2}\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.30, size = 528, normalized size = 2.59 \[ \frac {a^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {a^{2} d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c \,\mathrm {arccosh}\left (c x \right ) x^{2}}{2 \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} x^{3}}{4 \left (c x +1\right ) \left (c x -1\right )}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x}{4 \left (c x +1\right ) \left (c x -1\right )}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{3}}{6 \sqrt {c x -1}\, \sqrt {c x +1}\, c}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} \mathrm {arccosh}\left (c x \right )^{2} x^{3}}{2 \left (c x +1\right ) \left (c x -1\right )}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2} x}{2 \left (c x +1\right ) \left (c x -1\right )}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )}{4 \sqrt {c x +1}\, \sqrt {c x -1}\, c}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2}}{2 \sqrt {c x -1}\, \sqrt {c x +1}\, c}+\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} \mathrm {arccosh}\left (c x \right ) x^{3}}{\left (c x +1\right ) \left (c x -1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c \,x^{2}}{2 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x}{\left (c x +1\right ) \left (c x -1\right )}+\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}}{4 \sqrt {c x +1}\, \sqrt {c x -1}\, c} \]
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 \[ \frac {1}{2} \, {\left (\sqrt {-c^{2} d x^{2} + d} x + \frac {\sqrt {d} \arcsin \left (c x\right )}{c}\right )} a^{2} + \int \sqrt {-c^{2} d x^{2} + d} b^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2} + 2 \, \sqrt {-c^{2} d x^{2} + d} a b \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )\,{d x} \]
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 \[ \int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2} \,d x \]
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 \[ \int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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