Optimal. Leaf size=421 \[ -\frac {2 b x^5 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{25 c \sqrt {d-c^2 d x^2}}-\frac {x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 d}-\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^6 d}-\frac {16 a b x \sqrt {c x-1} \sqrt {c x+1}}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {4 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4 d}-\frac {8 b x^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{45 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x^4 (1-c x) (c x+1)}{125 c^2 \sqrt {d-c^2 d x^2}}-\frac {4144 b^2 (1-c x) (c x+1)}{3375 c^6 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {272 b^2 x^2 (1-c x) (c x+1)}{3375 c^4 \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 1.13, antiderivative size = 445, normalized size of antiderivative = 1.06, number of steps used = 17, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {5798, 5759, 5718, 5654, 74, 5662, 100, 12} \[ -\frac {16 a b x \sqrt {c x-1} \sqrt {c x+1}}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {2 b x^5 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{25 c \sqrt {d-c^2 d x^2}}-\frac {x^4 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 \sqrt {d-c^2 d x^2}}-\frac {8 b x^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{45 c^3 \sqrt {d-c^2 d x^2}}-\frac {4 x^2 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4 \sqrt {d-c^2 d x^2}}-\frac {8 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^6 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x^4 (1-c x) (c x+1)}{125 c^2 \sqrt {d-c^2 d x^2}}-\frac {272 b^2 x^2 (1-c x) (c x+1)}{3375 c^4 \sqrt {d-c^2 d x^2}}-\frac {4144 b^2 (1-c x) (c x+1)}{3375 c^6 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{15 c^5 \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 74
Rule 100
Rule 5654
Rule 5662
Rule 5718
Rule 5759
Rule 5798
Rubi steps
\begin {align*} \int \frac {x^5 \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^5 \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^4 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{5 c^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x^4 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{5 c \sqrt {d-c^2 d x^2}}\\ &=-\frac {2 b x^5 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c \sqrt {d-c^2 d x^2}}-\frac {4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4 \sqrt {d-c^2 d x^2}}-\frac {x^4 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^5}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{25 \sqrt {d-c^2 d x^2}}+\frac {\left (8 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{15 c^4 \sqrt {d-c^2 d x^2}}-\frac {\left (8 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{15 c^3 \sqrt {d-c^2 d x^2}}\\ &=-\frac {2 b^2 x^4 (1-c x) (1+c x)}{125 c^2 \sqrt {d-c^2 d x^2}}-\frac {8 b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{45 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b x^5 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^6 \sqrt {d-c^2 d x^2}}-\frac {4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4 \sqrt {d-c^2 d x^2}}-\frac {x^4 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 \sqrt {d-c^2 d x^2}}-\frac {\left (16 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{15 c^5 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {4 x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{125 c^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{45 c^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {8 b^2 x^2 (1-c x) (1+c x)}{135 c^4 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x^4 (1-c x) (1+c x)}{125 c^2 \sqrt {d-c^2 d x^2}}-\frac {8 b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{45 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b x^5 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^6 \sqrt {d-c^2 d x^2}}-\frac {4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4 \sqrt {d-c^2 d x^2}}-\frac {x^4 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 \sqrt {d-c^2 d x^2}}-\frac {\left (16 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \cosh ^{-1}(c x) \, dx}{15 c^5 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{135 c^4 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{125 c^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {272 b^2 x^2 (1-c x) (1+c x)}{3375 c^4 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x^4 (1-c x) (1+c x)}{125 c^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {8 b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{45 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b x^5 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^6 \sqrt {d-c^2 d x^2}}-\frac {4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4 \sqrt {d-c^2 d x^2}}-\frac {x^4 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{375 c^4 \sqrt {d-c^2 d x^2}}+\frac {\left (16 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{135 c^4 \sqrt {d-c^2 d x^2}}+\frac {\left (16 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{15 c^4 \sqrt {d-c^2 d x^2}}\\ &=-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {32 b^2 (1-c x) (1+c x)}{27 c^6 \sqrt {d-c^2 d x^2}}-\frac {272 b^2 x^2 (1-c x) (1+c x)}{3375 c^4 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x^4 (1-c x) (1+c x)}{125 c^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {8 b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{45 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b x^5 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^6 \sqrt {d-c^2 d x^2}}-\frac {4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4 \sqrt {d-c^2 d x^2}}-\frac {x^4 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 \sqrt {d-c^2 d x^2}}+\frac {\left (16 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{375 c^4 \sqrt {d-c^2 d x^2}}\\ &=-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {4144 b^2 (1-c x) (1+c x)}{3375 c^6 \sqrt {d-c^2 d x^2}}-\frac {272 b^2 x^2 (1-c x) (1+c x)}{3375 c^4 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x^4 (1-c x) (1+c x)}{125 c^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {8 b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{45 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b x^5 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^6 \sqrt {d-c^2 d x^2}}-\frac {4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4 \sqrt {d-c^2 d x^2}}-\frac {x^4 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.55, size = 255, normalized size = 0.61 \[ \frac {\sqrt {d-c^2 d x^2} \left (-225 a^2 \left (3 c^6 x^6+c^4 x^4+4 c^2 x^2-8\right )+30 a b c x \sqrt {c x-1} \sqrt {c x+1} \left (9 c^4 x^4+20 c^2 x^2+120\right )+30 b \cosh ^{-1}(c x) \left (b c x \sqrt {c x-1} \sqrt {c x+1} \left (9 c^4 x^4+20 c^2 x^2+120\right )-15 a \left (3 c^6 x^6+c^4 x^4+4 c^2 x^2-8\right )\right )-2 b^2 \left (27 c^6 x^6+109 c^4 x^4+1936 c^2 x^2-2072\right )-225 b^2 \left (3 c^6 x^6+c^4 x^4+4 c^2 x^2-8\right ) \cosh ^{-1}(c x)^2\right )}{3375 c^6 d (c x-1) (c x+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 348, normalized size = 0.83 \[ -\frac {225 \, {\left (3 \, b^{2} c^{6} x^{6} + b^{2} c^{4} x^{4} + 4 \, b^{2} c^{2} x^{2} - 8 \, b^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} - 30 \, {\left (9 \, a b c^{5} x^{5} + 20 \, a b c^{3} x^{3} + 120 \, a b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 30 \, {\left ({\left (9 \, b^{2} c^{5} x^{5} + 20 \, b^{2} c^{3} x^{3} + 120 \, b^{2} c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 15 \, {\left (3 \, a b c^{6} x^{6} + a b c^{4} x^{4} + 4 \, a b c^{2} x^{2} - 8 \, a b\right )} \sqrt {-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (27 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{6} x^{6} + {\left (225 \, a^{2} + 218 \, b^{2}\right )} c^{4} x^{4} + 4 \, {\left (225 \, a^{2} + 968 \, b^{2}\right )} c^{2} x^{2} - 1800 \, a^{2} - 4144 \, b^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{3375 \, {\left (c^{8} d x^{2} - c^{6} d\right )}} \]
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 = 0.80, size = 1314, normalized size = 3.12 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 407, normalized size = 0.97 \[ -\frac {1}{15} \, {\left (\frac {3 \, \sqrt {-c^{2} d x^{2} + d} x^{4}}{c^{2} d} + \frac {4 \, \sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {-c^{2} d x^{2} + d}}{c^{6} d}\right )} b^{2} \operatorname {arcosh}\left (c x\right )^{2} - \frac {2}{15} \, {\left (\frac {3 \, \sqrt {-c^{2} d x^{2} + d} x^{4}}{c^{2} d} + \frac {4 \, \sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {-c^{2} d x^{2} + d}}{c^{6} d}\right )} a b \operatorname {arcosh}\left (c x\right ) - \frac {1}{15} \, {\left (\frac {3 \, \sqrt {-c^{2} d x^{2} + d} x^{4}}{c^{2} d} + \frac {4 \, \sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {-c^{2} d x^{2} + d}}{c^{6} d}\right )} a^{2} - \frac {2}{3375} \, b^{2} {\left (\frac {27 \, \sqrt {c^{2} x^{2} - 1} c^{2} \sqrt {-d} x^{4} + 136 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} x^{2} + \frac {2072 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d}}{c^{2}}}{c^{4} d} - \frac {15 \, {\left (9 \, c^{4} \sqrt {-d} x^{5} + 20 \, c^{2} \sqrt {-d} x^{3} + 120 \, \sqrt {-d} x\right )} \operatorname {arcosh}\left (c x\right )}{c^{5} d}\right )} + \frac {2 \, {\left (9 \, c^{4} \sqrt {-d} x^{5} + 20 \, c^{2} \sqrt {-d} x^{3} + 120 \, \sqrt {-d} x\right )} a b}{225 \, c^{5} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^5\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{\sqrt {d-c^2\,d\,x^2}} \,d x \]
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 \[ \int \frac {x^{5} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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