Optimal. Leaf size=256 \[ -\frac {1}{11} c^6 d^3 x^{11} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{7} c^2 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {b d^3 (c x-1)^{11/2} (c x+1)^{11/2}}{121 c^5}+\frac {4 b d^3 (c x-1)^{9/2} (c x+1)^{9/2}}{297 c^5}+\frac {b d^3 (c x-1)^{7/2} (c x+1)^{7/2}}{1617 c^5}-\frac {2 b d^3 (c x-1)^{5/2} (c x+1)^{5/2}}{1925 c^5}+\frac {8 b d^3 (c x-1)^{3/2} (c x+1)^{3/2}}{3465 c^5}-\frac {16 b d^3 \sqrt {c x-1} \sqrt {c x+1}}{1155 c^5} \]
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Rubi [A] time = 0.44, antiderivative size = 326, normalized size of antiderivative = 1.27, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {270, 5731, 12, 1610, 1799, 1620} \[ -\frac {1}{11} c^6 d^3 x^{11} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{7} c^2 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {b d^3 \left (1-c^2 x^2\right )^6}{121 c^5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {4 b d^3 \left (1-c^2 x^2\right )^5}{297 c^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^3 \left (1-c^2 x^2\right )^4}{1617 c^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b d^3 \left (1-c^2 x^2\right )^3}{1925 c^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 b d^3 \left (1-c^2 x^2\right )^2}{3465 c^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {16 b d^3 \left (1-c^2 x^2\right )}{1155 c^5 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 1610
Rule 1620
Rule 1799
Rule 5731
Rubi steps
\begin {align*} \int x^4 \left (d-c^2 d x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {1}{5} d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{7} c^2 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{11} c^6 d^3 x^{11} \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {d^3 x^5 \left (231-495 c^2 x^2+385 c^4 x^4-105 c^6 x^6\right )}{1155 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{5} d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{7} c^2 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{11} c^6 d^3 x^{11} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^3\right ) \int \frac {x^5 \left (231-495 c^2 x^2+385 c^4 x^4-105 c^6 x^6\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{1155}\\ &=\frac {1}{5} d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{7} c^2 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{11} c^6 d^3 x^{11} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \int \frac {x^5 \left (231-495 c^2 x^2+385 c^4 x^4-105 c^6 x^6\right )}{\sqrt {-1+c^2 x^2}} \, dx}{1155 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{5} d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{7} c^2 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{11} c^6 d^3 x^{11} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (231-495 c^2 x+385 c^4 x^2-105 c^6 x^3\right )}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{2310 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{5} d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{7} c^2 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{11} c^6 d^3 x^{11} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {16}{c^4 \sqrt {-1+c^2 x}}-\frac {8 \sqrt {-1+c^2 x}}{c^4}+\frac {6 \left (-1+c^2 x\right )^{3/2}}{c^4}-\frac {5 \left (-1+c^2 x\right )^{5/2}}{c^4}-\frac {140 \left (-1+c^2 x\right )^{7/2}}{c^4}-\frac {105 \left (-1+c^2 x\right )^{9/2}}{c^4}\right ) \, dx,x,x^2\right )}{2310 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {16 b d^3 \left (1-c^2 x^2\right )}{1155 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {8 b d^3 \left (1-c^2 x^2\right )^2}{3465 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d^3 \left (1-c^2 x^2\right )^3}{1925 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^3 \left (1-c^2 x^2\right )^4}{1617 c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b d^3 \left (1-c^2 x^2\right )^5}{297 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^3 \left (1-c^2 x^2\right )^6}{121 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{5} d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{7} c^2 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{11} c^6 d^3 x^{11} \left (a+b \cosh ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.28, size = 147, normalized size = 0.57 \[ -\frac {d^3 \left (3465 a c^5 x^5 \left (105 c^6 x^6-385 c^4 x^4+495 c^2 x^2-231\right )+3465 b c^5 x^5 \left (105 c^6 x^6-385 c^4 x^4+495 c^2 x^2-231\right ) \cosh ^{-1}(c x)+b \sqrt {c x-1} \sqrt {c x+1} \left (-33075 c^{10} x^{10}+111475 c^8 x^8-117625 c^6 x^6+18933 c^4 x^4+25244 c^2 x^2+50488\right )\right )}{4002075 c^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 201, normalized size = 0.79 \[ -\frac {363825 \, a c^{11} d^{3} x^{11} - 1334025 \, a c^{9} d^{3} x^{9} + 1715175 \, a c^{7} d^{3} x^{7} - 800415 \, a c^{5} d^{3} x^{5} + 3465 \, {\left (105 \, b c^{11} d^{3} x^{11} - 385 \, b c^{9} d^{3} x^{9} + 495 \, b c^{7} d^{3} x^{7} - 231 \, b c^{5} d^{3} x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (33075 \, b c^{10} d^{3} x^{10} - 111475 \, b c^{8} d^{3} x^{8} + 117625 \, b c^{6} d^{3} x^{6} - 18933 \, b c^{4} d^{3} x^{4} - 25244 \, b c^{2} d^{3} x^{2} - 50488 \, b d^{3}\right )} \sqrt {c^{2} x^{2} - 1}}{4002075 \, c^{5}} \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 158, normalized size = 0.62 \[ \frac {-d^{3} a \left (\frac {1}{11} c^{11} x^{11}-\frac {1}{3} c^{9} x^{9}+\frac {3}{7} c^{7} x^{7}-\frac {1}{5} c^{5} x^{5}\right )-d^{3} b \left (\frac {\mathrm {arccosh}\left (c x \right ) c^{11} x^{11}}{11}-\frac {\mathrm {arccosh}\left (c x \right ) c^{9} x^{9}}{3}+\frac {3 \,\mathrm {arccosh}\left (c x \right ) c^{7} x^{7}}{7}-\frac {\mathrm {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (33075 c^{10} x^{10}-111475 c^{8} x^{8}+117625 c^{6} x^{6}-18933 c^{4} x^{4}-25244 c^{2} x^{2}-50488\right )}{4002075}\right )}{c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 465, normalized size = 1.82 \[ -\frac {1}{11} \, a c^{6} d^{3} x^{11} + \frac {1}{3} \, a c^{4} d^{3} x^{9} - \frac {3}{7} \, a c^{2} d^{3} x^{7} - \frac {1}{7623} \, {\left (693 \, x^{11} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {63 \, \sqrt {c^{2} x^{2} - 1} x^{10}}{c^{2}} + \frac {70 \, \sqrt {c^{2} x^{2} - 1} x^{8}}{c^{4}} + \frac {80 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{6}} + \frac {96 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{10}} + \frac {256 \, \sqrt {c^{2} x^{2} - 1}}{c^{12}}\right )} c\right )} b c^{6} d^{3} + \frac {1}{945} \, {\left (315 \, x^{9} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {35 \, \sqrt {c^{2} x^{2} - 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} - 1}}{c^{10}}\right )} c\right )} b c^{4} d^{3} + \frac {1}{5} \, a d^{3} x^{5} - \frac {3}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b c^{2} d^{3} + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d^{3} \]
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 \[ \int x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 39.28, size = 296, normalized size = 1.16 \[ \begin {cases} - \frac {a c^{6} d^{3} x^{11}}{11} + \frac {a c^{4} d^{3} x^{9}}{3} - \frac {3 a c^{2} d^{3} x^{7}}{7} + \frac {a d^{3} x^{5}}{5} - \frac {b c^{6} d^{3} x^{11} \operatorname {acosh}{\left (c x \right )}}{11} + \frac {b c^{5} d^{3} x^{10} \sqrt {c^{2} x^{2} - 1}}{121} + \frac {b c^{4} d^{3} x^{9} \operatorname {acosh}{\left (c x \right )}}{3} - \frac {91 b c^{3} d^{3} x^{8} \sqrt {c^{2} x^{2} - 1}}{3267} - \frac {3 b c^{2} d^{3} x^{7} \operatorname {acosh}{\left (c x \right )}}{7} + \frac {4705 b c d^{3} x^{6} \sqrt {c^{2} x^{2} - 1}}{160083} + \frac {b d^{3} x^{5} \operatorname {acosh}{\left (c x \right )}}{5} - \frac {6311 b d^{3} x^{4} \sqrt {c^{2} x^{2} - 1}}{1334025 c} - \frac {25244 b d^{3} x^{2} \sqrt {c^{2} x^{2} - 1}}{4002075 c^{3}} - \frac {50488 b d^{3} \sqrt {c^{2} x^{2} - 1}}{4002075 c^{5}} & \text {for}\: c \neq 0 \\\frac {d^{3} x^{5} \left (a + \frac {i \pi b}{2}\right )}{5} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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