Optimal. Leaf size=328 \[ -\frac {b c d \sqrt {d-c^2 d x^2}}{72 x^8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b c^3 d \sqrt {d-c^2 d x^2}}{189 x^6 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{420 x^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^7 d \sqrt {d-c^2 d x^2}}{315 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{9 d x^9}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{63 d x^7}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{315 d x^5}+\frac {8 b c^9 d \sqrt {d-c^2 d x^2} \log (x)}{315 \sqrt {-1+c x} \sqrt {1+c x}} \]
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Rubi [A]
time = 0.15, antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {277, 270, 5922,
12, 1265, 907} \begin {gather*} -\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{9 d x^9}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{63 d x^7}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{315 d x^5}-\frac {b c d \sqrt {d-c^2 d x^2}}{72 x^8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 b c^9 d \log (x) \sqrt {d-c^2 d x^2}}{315 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c^7 d \sqrt {d-c^2 d x^2}}{315 x^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{420 x^4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b c^3 d \sqrt {d-c^2 d x^2}}{189 x^6 \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 277
Rule 907
Rule 1265
Rule 5922
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^{10}} \, dx &=-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^{10}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^7}-\frac {c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^5}-\frac {4 c^6 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x^3}-\frac {8 c^8 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^2 \left (35+20 c^2 x^2+8 c^4 x^4\right )}{315 x^9} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^7}-\frac {c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^5}-\frac {4 c^6 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x^3}-\frac {8 c^8 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^2 \left (35+20 c^2 x^2+8 c^4 x^4\right )}{x^9} \, dx}{315 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^7}-\frac {c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^5}-\frac {4 c^6 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x^3}-\frac {8 c^8 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\left (1-c^2 x\right )^2 \left (35+20 c^2 x+8 c^4 x^2\right )}{x^5} \, dx,x,x^2\right )}{630 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^7}-\frac {c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^5}-\frac {4 c^6 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x^3}-\frac {8 c^8 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {35}{x^5}-\frac {50 c^2}{x^4}+\frac {3 c^4}{x^3}+\frac {4 c^6}{x^2}+\frac {8 c^8}{x}\right ) \, dx,x,x^2\right )}{630 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d \sqrt {d-c^2 d x^2}}{72 x^8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b c^3 d \sqrt {d-c^2 d x^2}}{189 x^6 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{420 x^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^7 d \sqrt {d-c^2 d x^2}}{315 x^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^7}-\frac {c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^5}-\frac {4 c^6 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x^3}-\frac {8 c^8 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}+\frac {8 b c^9 d \sqrt {d-c^2 d x^2} \log (x)}{315 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 154, normalized size = 0.47 \begin {gather*} -\frac {d \sqrt {d-c^2 d x^2} \left (840 (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+96 c^2 x^2 (-1+c x)^{5/2} (1+c x)^{5/2} \left (5+2 c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+b c x \left (105-200 c^2 x^2+18 c^4 x^4+48 c^6 x^6-192 c^8 x^8 \log (x)\right )\right )}{7560 x^9 \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. \(4261\) vs.
\(2(276)=552\).
time = 7.26, size = 4262, normalized size = 12.99
method | result | size |
default | \(\text {Expression too large to display}\) | \(4262\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 225, normalized size = 0.69 \begin {gather*} \frac {1}{7560} \, {\left (192 \, c^{8} \sqrt {-d} d \log \left (x\right ) - \frac {48 \, c^{6} \sqrt {-d} d x^{6} + 18 \, c^{4} \sqrt {-d} d x^{4} - 200 \, c^{2} \sqrt {-d} d x^{2} + 105 \, \sqrt {-d} d}{x^{8}}\right )} b c - \frac {1}{315} \, b {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{4}}{d x^{5}} + \frac {20 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{7}} + \frac {35 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{9}}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {1}{315} \, a {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{4}}{d x^{5}} + \frac {20 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{7}} + \frac {35 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{9}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 720, normalized size = 2.20 \begin {gather*} \left [-\frac {24 \, {\left (8 \, b c^{10} d x^{10} - 4 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 53 \, b c^{4} d x^{4} + 85 \, b c^{2} d x^{2} - 35 \, b d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 96 \, {\left (b c^{11} d x^{11} - b c^{9} d x^{9}\right )} \sqrt {-d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} - \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{4} - 1\right )} \sqrt {-d} - d}{c^{2} x^{4} - x^{2}}\right ) + {\left (48 \, b c^{7} d x^{7} + 18 \, b c^{5} d x^{5} - {\left (48 \, b c^{7} + 18 \, b c^{5} - 200 \, b c^{3} + 105 \, b c\right )} d x^{9} - 200 \, b c^{3} d x^{3} + 105 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 24 \, {\left (8 \, a c^{10} d x^{10} - 4 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 53 \, a c^{4} d x^{4} + 85 \, a c^{2} d x^{2} - 35 \, a d\right )} \sqrt {-c^{2} d x^{2} + d}}{7560 \, {\left (c^{2} x^{11} - x^{9}\right )}}, \frac {192 \, {\left (b c^{11} d x^{11} - b c^{9} d x^{9}\right )} \sqrt {d} \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{2} + 1\right )} \sqrt {d}}{c^{2} d x^{4} - {\left (c^{2} + 1\right )} d x^{2} + d}\right ) - 24 \, {\left (8 \, b c^{10} d x^{10} - 4 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 53 \, b c^{4} d x^{4} + 85 \, b c^{2} d x^{2} - 35 \, b d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (48 \, b c^{7} d x^{7} + 18 \, b c^{5} d x^{5} - {\left (48 \, b c^{7} + 18 \, b c^{5} - 200 \, b c^{3} + 105 \, b c\right )} d x^{9} - 200 \, b c^{3} d x^{3} + 105 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 24 \, {\left (8 \, a c^{10} d x^{10} - 4 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 53 \, a c^{4} d x^{4} + 85 \, a c^{2} d x^{2} - 35 \, a d\right )} \sqrt {-c^{2} d x^{2} + d}}{7560 \, {\left (c^{2} x^{11} - x^{9}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^{10}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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