Optimal. Leaf size=272 \[ \frac {8 b x \sqrt {d-c^2 d x^2}}{105 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b x^3 \sqrt {d-c^2 d x^2}}{315 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x^5 \sqrt {d-c^2 d x^2}}{175 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^6 d}+\frac {2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^6 d^3} \]
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Rubi [A]
time = 0.12, antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {272, 45, 5922,
12} \begin {gather*} -\frac {\left (d-c^2 d x^2\right )^{7/2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^6 d}-\frac {b c x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x^5 \sqrt {d-c^2 d x^2}}{175 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 b x \sqrt {d-c^2 d x^2}}{105 c^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {4 b x^3 \sqrt {d-c^2 d x^2}}{315 c^3 \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 272
Rule 5922
Rubi steps
\begin {align*} \int x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {\sqrt {d-c^2 d x^2} \int x^5 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {8 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 c^6}-\frac {4 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 c^4}-\frac {x^4 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {-8-4 c^2 x^2-3 c^4 x^4+15 c^6 x^6}{105 c^6} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {8 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 c^6}-\frac {4 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 c^4}-\frac {x^4 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}-\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int \left (-8-4 c^2 x^2-3 c^4 x^4+15 c^6 x^6\right ) \, dx}{105 c^5 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {8 b x \sqrt {d-c^2 d x^2}}{105 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b x^3 \sqrt {d-c^2 d x^2}}{315 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x^5 \sqrt {d-c^2 d x^2}}{175 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {8 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 c^6}-\frac {4 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 c^4}-\frac {x^4 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 152, normalized size = 0.56 \begin {gather*} \frac {\sqrt {d-c^2 d x^2} \left (b \left (8 x+\frac {4 c^2 x^3}{3}+\frac {3 c^4 x^5}{5}-\frac {15 c^6 x^7}{7}\right )+15 c^3 x^4 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {4 (-1+c x)^{3/2} (1+c x)^{3/2} \left (2+3 c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c}\right )}{105 c^5 \sqrt {-1+c x} \sqrt {1+c x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(987\) vs.
\(2(228)=456\).
time = 4.22, size = 988, normalized size = 3.63
method | result | size |
default | \(a \left (-\frac {x^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{7 c^{2} d}+\frac {-\frac {4 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{35 c^{2} d}-\frac {8 \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{105 d \,c^{4}}}{c^{2}}\right )+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (64 c^{8} x^{8}-144 x^{6} c^{6}+64 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{7} c^{7}+104 c^{4} x^{4}-112 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{5} c^{5}-25 c^{2} x^{2}+56 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}-7 \sqrt {c x +1}\, \sqrt {c x -1}\, x c +1\right ) \left (-1+7 \,\mathrm {arccosh}\left (c x \right )\right )}{6272 \left (c x +1\right ) c^{6} \left (c x -1\right )}+\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (16 x^{6} c^{6}-28 c^{4} x^{4}+16 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{5} c^{5}+13 c^{2} x^{2}-20 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}+5 \sqrt {c x +1}\, \sqrt {c x -1}\, x c -1\right ) \left (-1+5 \,\mathrm {arccosh}\left (c x \right )\right )}{3200 \left (c x +1\right ) c^{6} \left (c x -1\right )}+\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 )}{1152 \left (c x +1\right ) c^{6} \left (c x -1\right )}-\frac {5 \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 )}{128 \left (c x +1\right ) c^{6} \left (c x -1\right )}-\frac {5 \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 )}{128 \left (c x +1\right ) c^{6} \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 )}{1152 \left (c x +1\right ) c^{6} \left (c x -1\right )}+\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-16 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{5} c^{5}+16 x^{6} c^{6}+20 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}-28 c^{4} x^{4}-5 \sqrt {c x +1}\, \sqrt {c x -1}\, x c +13 c^{2} x^{2}-1\right ) \left (1+5 \,\mathrm {arccosh}\left (c x \right )\right )}{3200 \left (c x +1\right ) c^{6} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-64 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{7} c^{7}+64 c^{8} x^{8}+112 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{5} c^{5}-144 x^{6} c^{6}-56 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}+104 c^{4} x^{4}+7 \sqrt {c x +1}\, \sqrt {c x -1}\, x c -25 c^{2} x^{2}+1\right ) \left (1+7 \,\mathrm {arccosh}\left (c x \right )\right )}{6272 \left (c x +1\right ) c^{6} \left (c x -1\right )}\right )\) | \(988\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 205, normalized size = 0.75 \begin {gather*} -\frac {1}{105} \, {\left (\frac {15 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}}{c^{2} d} + \frac {12 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{4} d} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{6} d}\right )} b \operatorname {arcosh}\left (c x\right ) - \frac {1}{105} \, {\left (\frac {15 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}}{c^{2} d} + \frac {12 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{4} d} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{6} d}\right )} a - \frac {{\left (225 \, c^{6} \sqrt {-d} x^{7} - 63 \, c^{4} \sqrt {-d} x^{5} - 140 \, c^{2} \sqrt {-d} x^{3} - 840 \, \sqrt {-d} x\right )} b}{11025 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 203, normalized size = 0.75 \begin {gather*} \frac {105 \, {\left (15 \, b c^{8} x^{8} - 18 \, b c^{6} x^{6} - b c^{4} x^{4} - 4 \, b c^{2} x^{2} + 8 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (225 \, b c^{7} x^{7} - 63 \, b c^{5} x^{5} - 140 \, b c^{3} x^{3} - 840 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 105 \, {\left (15 \, a c^{8} x^{8} - 18 \, a c^{6} x^{6} - a c^{4} x^{4} - 4 \, a c^{2} x^{2} + 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{11025 \, {\left (c^{8} x^{2} - c^{6}\right )}} \end {gather*}
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 \begin {gather*} \int x^{5} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int x^5\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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