Optimal. Leaf size=409 \[ -\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{11 d x^{11}}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{33 d x^9}-\frac {16 c^6 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 d x^5}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{231 d x^7}-\frac {b c d \sqrt {d-c^2 d x^2}}{110 x^{10} \sqrt {c x-1} \sqrt {c x+1}}+\frac {16 b c^{11} d \log (x) \sqrt {d-c^2 d x^2}}{1155 \sqrt {c x-1} \sqrt {c x+1}}-\frac {4 b c^9 d \sqrt {d-c^2 d x^2}}{1155 x^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^7 d \sqrt {d-c^2 d x^2}}{770 x^4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{1386 x^6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d \sqrt {d-c^2 d x^2}}{66 x^8 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 0.61, antiderivative size = 480, normalized size of antiderivative = 1.17, number of steps used = 6, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {5798, 97, 12, 103, 95, 5733, 1799, 1620} \[ -\frac {16 c^{10} d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x}-\frac {8 c^8 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x^3}-\frac {2 c^6 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{385 x^5}-\frac {c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^7}+\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{33 x^9}-\frac {d (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{11 x^{11}}-\frac {4 b c^9 d \sqrt {d-c^2 d x^2}}{1155 x^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^7 d \sqrt {d-c^2 d x^2}}{770 x^4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{1386 x^6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d \sqrt {d-c^2 d x^2}}{66 x^8 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d \sqrt {d-c^2 d x^2}}{110 x^{10} \sqrt {c x-1} \sqrt {c x+1}}+\frac {16 b c^{11} d \log (x) \sqrt {d-c^2 d x^2}}{1155 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 97
Rule 103
Rule 1620
Rule 1799
Rule 5733
Rule 5798
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^{12}} \, 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^{12}} \, 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 )}{33 x^9}-\frac {c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^7}-\frac {2 c^6 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{385 x^5}-\frac {8 c^8 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x^3}-\frac {16 c^{10} d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{11 x^{11}}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^2 \left (105+70 c^2 x^2+40 c^4 x^4+16 c^6 x^6\right )}{1155 x^{11}} \, 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 )}{33 x^9}-\frac {c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^7}-\frac {2 c^6 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{385 x^5}-\frac {8 c^8 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x^3}-\frac {16 c^{10} d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{11 x^{11}}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^2 \left (105+70 c^2 x^2+40 c^4 x^4+16 c^6 x^6\right )}{x^{11}} \, dx}{1155 \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 )}{33 x^9}-\frac {c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^7}-\frac {2 c^6 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{385 x^5}-\frac {8 c^8 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x^3}-\frac {16 c^{10} d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{11 x^{11}}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1-c^2 x\right )^2 \left (105+70 c^2 x+40 c^4 x^2+16 c^6 x^3\right )}{x^6} \, dx,x,x^2\right )}{2310 \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 )}{33 x^9}-\frac {c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^7}-\frac {2 c^6 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{385 x^5}-\frac {8 c^8 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x^3}-\frac {16 c^{10} d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{11 x^{11}}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {105}{x^6}-\frac {140 c^2}{x^5}+\frac {5 c^4}{x^4}+\frac {6 c^6}{x^3}+\frac {8 c^8}{x^2}+\frac {16 c^{10}}{x}\right ) \, dx,x,x^2\right )}{2310 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d \sqrt {d-c^2 d x^2}}{110 x^{10} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d \sqrt {d-c^2 d x^2}}{66 x^8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{1386 x^6 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^7 d \sqrt {d-c^2 d x^2}}{770 x^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b c^9 d \sqrt {d-c^2 d x^2}}{1155 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 )}{33 x^9}-\frac {c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{231 x^7}-\frac {2 c^6 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{385 x^5}-\frac {8 c^8 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x^3}-\frac {16 c^{10} d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{1155 x}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{11 x^{11}}+\frac {16 b c^{11} d \sqrt {d-c^2 d x^2} \log (x)}{1155 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 170, normalized size = 0.42 \[ -\frac {d \sqrt {d-c^2 d x^2} \left (12 c^2 x^2 (c x-1)^{5/2} \left (8 c^4 x^4+20 c^2 x^2+35\right ) (c x+1)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+630 (c x-1)^{5/2} (c x+1)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+b c x \left (-96 c^{10} x^{10} \log (x)+24 c^8 x^8+9 c^6 x^6+5 c^4 x^4-105 c^2 x^2+63\right )\right )}{6930 x^{11} \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 792, normalized size = 1.94 \[ \left [-\frac {6 \, {\left (16 \, b c^{12} d x^{12} - 8 \, b c^{10} d x^{10} - 2 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 145 \, b c^{4} d x^{4} + 245 \, b c^{2} d x^{2} - 105 \, b d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 48 \, {\left (b c^{13} d x^{13} - b c^{11} d x^{11}\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 (24 \, b c^{9} d x^{9} + 9 \, b c^{7} d x^{7} - {\left (24 \, b c^{9} + 9 \, b c^{7} + 5 \, b c^{5} - 105 \, b c^{3} + 63 \, b c\right )} d x^{11} + 5 \, b c^{5} d x^{5} - 105 \, b c^{3} d x^{3} + 63 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 6 \, {\left (16 \, a c^{12} d x^{12} - 8 \, a c^{10} d x^{10} - 2 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 145 \, a c^{4} d x^{4} + 245 \, a c^{2} d x^{2} - 105 \, a d\right )} \sqrt {-c^{2} d x^{2} + d}}{6930 \, {\left (c^{2} x^{13} - x^{11}\right )}}, \frac {96 \, {\left (b c^{13} d x^{13} - b c^{11} d x^{11}\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 ) - 6 \, {\left (16 \, b c^{12} d x^{12} - 8 \, b c^{10} d x^{10} - 2 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 145 \, b c^{4} d x^{4} + 245 \, b c^{2} d x^{2} - 105 \, b d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (24 \, b c^{9} d x^{9} + 9 \, b c^{7} d x^{7} - {\left (24 \, b c^{9} + 9 \, b c^{7} + 5 \, b c^{5} - 105 \, b c^{3} + 63 \, b c\right )} d x^{11} + 5 \, b c^{5} d x^{5} - 105 \, b c^{3} d x^{3} + 63 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 6 \, {\left (16 \, a c^{12} d x^{12} - 8 \, a c^{10} d x^{10} - 2 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 145 \, a c^{4} d x^{4} + 245 \, a c^{2} d x^{2} - 105 \, a d\right )} \sqrt {-c^{2} d x^{2} + d}}{6930 \, {\left (c^{2} x^{13} - x^{11}\right )}}\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 = 1.24, size = 5518, normalized size = 13.49 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 287, normalized size = 0.70 \[ \frac {1}{6930} \, {\left (96 \, c^{10} \sqrt {-d} d \log \relax (x) - \frac {24 \, c^{8} \sqrt {-d} d x^{8} + 9 \, c^{6} \sqrt {-d} d x^{6} + 5 \, c^{4} \sqrt {-d} d x^{4} - 105 \, c^{2} \sqrt {-d} d x^{2} + 63 \, \sqrt {-d} d}{x^{10}}\right )} b c - \frac {1}{1155} \, {\left (\frac {16 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{6}}{d x^{5}} + \frac {40 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{4}}{d x^{7}} + \frac {70 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{9}} + \frac {105 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{11}}\right )} b \operatorname {arcosh}\left (c x\right ) - \frac {1}{1155} \, {\left (\frac {16 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{6}}{d x^{5}} + \frac {40 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{4}}{d x^{7}} + \frac {70 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{9}} + \frac {105 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{11}}\right )} a \]
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 \[ \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^{12}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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