Optimal. Leaf size=186 \[ \frac {2 b x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d}-\frac {2 b c x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2}{27} b^2 x^2 \sqrt {d-c^2 d x^2}-\frac {14 b^2 \sqrt {d-c^2 d x^2}}{27 c^2} \]
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Rubi [A] time = 0.37, antiderivative size = 194, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5798, 5718, 5680, 12, 460, 74} \[ -\frac {2 b c x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {(1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2}+\frac {2}{27} b^2 x^2 \sqrt {d-c^2 d x^2}-\frac {14 b^2 \sqrt {d-c^2 d x^2}}{27 c^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 74
Rule 460
Rule 5680
Rule 5718
Rule 5798
Rubi steps
\begin {align*} \int x \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 x \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-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2}-\frac {\left (2 b \sqrt {d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{3 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {2 b x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {(1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2}+\frac {\left (2 b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (-3+c^2 x^2\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {2 b x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {(1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2}+\frac {\left (2 b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (-3+c^2 x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{9 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {2}{27} b^2 x^2 \sqrt {d-c^2 d x^2}+\frac {2 b x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {(1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2}-\frac {\left (14 b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{27 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {14 b^2 \sqrt {d-c^2 d x^2}}{27 c^2}+\frac {2}{27} b^2 x^2 \sqrt {d-c^2 d x^2}+\frac {2 b x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {(1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 181, normalized size = 0.97 \[ \frac {\sqrt {d-c^2 d x^2} \left (9 a^2 \left (c^2 x^2-1\right )^2-6 a b c x \sqrt {c x-1} \sqrt {c x+1} \left (c^2 x^2-3\right )+6 b \cosh ^{-1}(c x) \left (3 a \left (c^2 x^2-1\right )^2+b c x \sqrt {c x-1} \sqrt {c x+1} \left (3-c^2 x^2\right )\right )+9 b^2 \left (c^2 x^2-1\right )^2 \cosh ^{-1}(c x)^2+2 b^2 \left (c^4 x^4-8 c^2 x^2+7\right )\right )}{27 c^2 \left (c^2 x^2-1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 280, normalized size = 1.51 \[ \frac {9 \, {\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} - 6 \, {\left (a b c^{3} x^{3} - 3 \, a b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 6 \, {\left ({\left (b^{2} c^{3} x^{3} - 3 \, b^{2} c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 3 \, {\left (a b c^{4} x^{4} - 2 \, a b c^{2} x^{2} + a b\right )} \sqrt {-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{4} x^{4} - 2 \, {\left (9 \, a^{2} + 8 \, b^{2}\right )} c^{2} x^{2} + 9 \, a^{2} + 14 \, b^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{27 \, {\left (c^{4} x^{2} - c^{2}\right )}} \]
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 = 0.36, size = 726, normalized size = 3.90 \[ -\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}-3 \sqrt {c x +1}\, \sqrt {c x -1}\, x c +1\right ) \left (9 \mathrm {arccosh}\left (c x \right )^{2}-6 \,\mathrm {arccosh}\left (c x \right )+2\right )}{216 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \left (\mathrm {arccosh}\left (c x \right )^{2}-2 \,\mathrm {arccosh}\left (c x \right )+2\right )}{8 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \left (\mathrm {arccosh}\left (c x \right )^{2}+2 \,\mathrm {arccosh}\left (c x \right )+2\right )}{8 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}+4 c^{4} x^{4}+3 \sqrt {c x +1}\, \sqrt {c x -1}\, x c -5 c^{2} x^{2}+1\right ) \left (9 \mathrm {arccosh}\left (c x \right )^{2}+6 \,\mathrm {arccosh}\left (c x \right )+2\right )}{216 \left (c x +1\right ) c^{2} \left (c x -1\right )}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}-3 \sqrt {c x +1}\, \sqrt {c x -1}\, x c +1\right ) \left (-1+3 \,\mathrm {arccosh}\left (c x \right )\right )}{72 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \left (-1+\mathrm {arccosh}\left (c x \right )\right )}{8 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \left (1+\mathrm {arccosh}\left (c x \right )\right )}{8 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}+4 c^{4} x^{4}+3 \sqrt {c x +1}\, \sqrt {c x -1}\, x c -5 c^{2} x^{2}+1\right ) \left (1+3 \,\mathrm {arccosh}\left (c x \right )\right )}{72 \left (c x +1\right ) c^{2} \left (c x -1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.24, size = 204, normalized size = 1.10 \[ \frac {2}{27} \, b^{2} {\left (\frac {\sqrt {c^{2} x^{2} - 1} \sqrt {-d} d x^{2} - \frac {7 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} d}{c^{2}}}{d} - \frac {3 \, {\left (c^{2} \sqrt {-d} d x^{3} - 3 \, \sqrt {-d} d x\right )} \operatorname {arcosh}\left (c x\right )}{c d}\right )} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} b^{2} \operatorname {arcosh}\left (c x\right )^{2}}{3 \, c^{2} d} - \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a b \operatorname {arcosh}\left (c x\right )}{3 \, c^{2} d} - \frac {2 \, {\left (c^{2} \sqrt {-d} d x^{3} - 3 \, \sqrt {-d} d x\right )} a b}{9 \, c d} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a^{2}}{3 \, c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2} \,d x \]
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 \[ \int x \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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