Optimal. Leaf size=279 \[ -\frac {b c \sqrt {d-c^2 d x^2}}{42 x^6 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 \sqrt {d-c^2 d x^2}}{140 x^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^5 \sqrt {d-c^2 d x^2}}{105 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{7 d x^7}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{35 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{105 d x^3}-\frac {8 b c^7 \sqrt {d-c^2 d x^2} \log (x)}{105 \sqrt {-1+c x} \sqrt {1+c x}} \]
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Rubi [A]
time = 0.11, antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {277, 270, 5922,
12, 14} \begin {gather*} -\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{7 d x^7}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{35 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{105 d x^3}-\frac {b c \sqrt {d-c^2 d x^2}}{42 x^6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {8 b c^7 \log (x) \sqrt {d-c^2 d x^2}}{105 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b c^5 \sqrt {d-c^2 d x^2}}{105 x^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 \sqrt {d-c^2 d x^2}}{140 x^4 \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 270
Rule 277
Rule 5922
Rubi steps
\begin {align*} \int \frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x^8} \, dx &=\frac {\sqrt {d-c^2 d x^2} \int \frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{x^8} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 x^7}+\frac {c^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 x^5}+\frac {4 c^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^3}+\frac {8 c^6 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {-15+3 c^2 x^2+4 c^4 x^4+8 c^6 x^6}{105 x^7} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 x^7}+\frac {c^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 x^5}+\frac {4 c^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^3}+\frac {8 c^6 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {-15+3 c^2 x^2+4 c^4 x^4+8 c^6 x^6}{x^7} \, dx}{105 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 x^7}+\frac {c^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 x^5}+\frac {4 c^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^3}+\frac {8 c^6 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \left (-\frac {15}{x^7}+\frac {3 c^2}{x^5}+\frac {4 c^4}{x^3}+\frac {8 c^6}{x}\right ) \, dx}{105 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c \sqrt {d-c^2 d x^2}}{42 x^6 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 \sqrt {d-c^2 d x^2}}{140 x^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^5 \sqrt {d-c^2 d x^2}}{105 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 x^7}+\frac {c^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 x^5}+\frac {4 c^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^3}+\frac {8 c^6 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x}-\frac {8 b c^7 \sqrt {d-c^2 d x^2} \log (x)}{105 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 146, normalized size = 0.52 \begin {gather*} \frac {\sqrt {d-c^2 d x^2} \left (60 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )+16 c^2 x^2 (-1+c x)^{3/2} (1+c x)^{3/2} \left (3+2 c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-b c x \left (10-3 c^2 x^2-8 c^4 x^4+32 c^6 x^6 \log (x)\right )\right )}{420 x^7 \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. \(2536\) vs.
\(2(235)=470\).
time = 7.58, size = 2537, normalized size = 9.09
method | result | size |
default | \(\text {Expression too large to display}\) | \(2537\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 207, normalized size = 0.74 \begin {gather*} -\frac {1}{420} \, {\left (32 \, c^{6} \sqrt {-d} \log \left (x\right ) - \frac {8 \, c^{4} \sqrt {-d} x^{4} + 3 \, c^{2} \sqrt {-d} x^{2} - 10 \, \sqrt {-d}}{x^{6}}\right )} b c - \frac {1}{105} \, {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{4}}{d x^{3}} + \frac {12 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2}}{d x^{5}} + \frac {15 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{d x^{7}}\right )} b \operatorname {arcosh}\left (c x\right ) - \frac {1}{105} \, {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{4}}{d x^{3}} + \frac {12 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2}}{d x^{5}} + \frac {15 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{d x^{7}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 615, normalized size = 2.20 \begin {gather*} \left [\frac {4 \, {\left (8 \, b c^{8} x^{8} - 4 \, b c^{6} x^{6} - b c^{4} x^{4} - 18 \, b c^{2} x^{2} + 15 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 16 \, {\left (b c^{9} x^{9} - b c^{7} x^{7}\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 (8 \, b c^{5} x^{5} - {\left (8 \, b c^{5} + 3 \, b c^{3} - 10 \, b c\right )} x^{7} + 3 \, b c^{3} x^{3} - 10 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 4 \, {\left (8 \, a c^{8} x^{8} - 4 \, a c^{6} x^{6} - a c^{4} x^{4} - 18 \, a c^{2} x^{2} + 15 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{420 \, {\left (c^{2} x^{9} - x^{7}\right )}}, -\frac {32 \, {\left (b c^{9} x^{9} - b c^{7} x^{7}\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 ) - 4 \, {\left (8 \, b c^{8} x^{8} - 4 \, b c^{6} x^{6} - b c^{4} x^{4} - 18 \, b c^{2} x^{2} + 15 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (8 \, b c^{5} x^{5} - {\left (8 \, b c^{5} + 3 \, b c^{3} - 10 \, b c\right )} x^{7} + 3 \, b c^{3} x^{3} - 10 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 4 \, {\left (8 \, a c^{8} x^{8} - 4 \, a c^{6} x^{6} - a c^{4} x^{4} - 18 \, a c^{2} x^{2} + 15 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{420 \, {\left (c^{2} x^{9} - x^{7}\right )}}\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 \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{x^{8}}\, 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 \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2}}{x^8} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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