Optimal. Leaf size=200 \[ -\frac {73 b d^2 x \sqrt {-1+c x} \sqrt {1+c x}}{3072 c^3}-\frac {73 b d^2 x^3 \sqrt {-1+c x} \sqrt {1+c x}}{4608 c}+\frac {43 b c d^2 x^5 \sqrt {-1+c x} \sqrt {1+c x}}{1152}-\frac {1}{64} b c^3 d^2 x^7 \sqrt {-1+c x} \sqrt {1+c x}-\frac {73 b d^2 \cosh ^{-1}(c x)}{3072 c^4}+\frac {1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{3} c^2 d^2 x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} c^4 d^2 x^8 \left (a+b \cosh ^{-1}(c x)\right ) \]
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Rubi [A]
time = 0.18, antiderivative size = 284, normalized size of antiderivative = 1.42, number of steps
used = 9, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {272, 45, 5921,
12, 534, 1281, 470, 327, 223, 212} \begin {gather*} \frac {1}{8} c^4 d^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{3} c^2 d^2 x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {43 b c d^2 x^5 \left (1-c^2 x^2\right )}{1152 \sqrt {c x-1} \sqrt {c x+1}}+\frac {73 b d^2 x^3 \left (1-c^2 x^2\right )}{4608 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {73 b d^2 \sqrt {c^2 x^2-1} \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{3072 c^4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {73 b d^2 x \left (1-c^2 x^2\right )}{3072 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d^2 x^7 \left (1-c^2 x^2\right )}{64 \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 212
Rule 223
Rule 272
Rule 327
Rule 470
Rule 534
Rule 1281
Rule 5921
Rubi steps
\begin {align*} \int x^3 \left (d-c^2 d x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{3} c^2 d^2 x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} c^4 d^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {d^2 x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{24 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{3} c^2 d^2 x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} c^4 d^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{24} \left (b c d^2\right ) \int \frac {x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{3} c^2 d^2 x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} c^4 d^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \int \frac {x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{\sqrt {-1+c^2 x^2}} \, dx}{24 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b c^3 d^2 x^7 \left (1-c^2 x^2\right )}{64 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{3} c^2 d^2 x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} c^4 d^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b d^2 \sqrt {-1+c^2 x^2}\right ) \int \frac {x^4 \left (48 c^2-43 c^4 x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{192 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {43 b c d^2 x^5 \left (1-c^2 x^2\right )}{1152 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 x^7 \left (1-c^2 x^2\right )}{64 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{3} c^2 d^2 x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} c^4 d^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (73 b c d^2 \sqrt {-1+c^2 x^2}\right ) \int \frac {x^4}{\sqrt {-1+c^2 x^2}} \, dx}{1152 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {73 b d^2 x^3 \left (1-c^2 x^2\right )}{4608 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {43 b c d^2 x^5 \left (1-c^2 x^2\right )}{1152 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 x^7 \left (1-c^2 x^2\right )}{64 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{3} c^2 d^2 x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} c^4 d^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (73 b d^2 \sqrt {-1+c^2 x^2}\right ) \int \frac {x^2}{\sqrt {-1+c^2 x^2}} \, dx}{1536 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {73 b d^2 x \left (1-c^2 x^2\right )}{3072 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {73 b d^2 x^3 \left (1-c^2 x^2\right )}{4608 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {43 b c d^2 x^5 \left (1-c^2 x^2\right )}{1152 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 x^7 \left (1-c^2 x^2\right )}{64 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{3} c^2 d^2 x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} c^4 d^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (73 b d^2 \sqrt {-1+c^2 x^2}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{3072 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {73 b d^2 x \left (1-c^2 x^2\right )}{3072 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {73 b d^2 x^3 \left (1-c^2 x^2\right )}{4608 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {43 b c d^2 x^5 \left (1-c^2 x^2\right )}{1152 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 x^7 \left (1-c^2 x^2\right )}{64 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{3} c^2 d^2 x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} c^4 d^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (73 b d^2 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{3072 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {73 b d^2 x \left (1-c^2 x^2\right )}{3072 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {73 b d^2 x^3 \left (1-c^2 x^2\right )}{4608 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {43 b c d^2 x^5 \left (1-c^2 x^2\right )}{1152 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 x^7 \left (1-c^2 x^2\right )}{64 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{3} c^2 d^2 x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} c^4 d^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac {73 b d^2 \sqrt {-1+c^2 x^2} \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{3072 c^4 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 141, normalized size = 0.70 \begin {gather*} \frac {d^2 \left (2304 a x^4-3072 a c^2 x^6+1152 a c^4 x^8-\frac {b x \sqrt {-1+c x} \sqrt {1+c x} \left (219+146 c^2 x^2-344 c^4 x^4+144 c^6 x^6\right )}{c^3}+384 b x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right ) \cosh ^{-1}(c x)-\frac {219 b \log \left (c x+\sqrt {-1+c x} \sqrt {1+c x}\right )}{c^4}\right )}{9216} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.77, size = 243, normalized size = 1.22
method | result | size |
derivativedivides | \(\frac {d^{2} a \left (\frac {\left (c^{2} x^{2}-1\right )^{4}}{8}+\frac {\left (c^{2} x^{2}-1\right )^{3}}{6}\right )+\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{8} x^{8}}{8}-\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{6} x^{6}}{3}+\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {b \,d^{2} \mathrm {arccosh}\left (c x \right )}{24}-\frac {d^{2} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{7} x^{7}}{64}+\frac {43 d^{2} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{5} x^{5}}{1152}-\frac {73 d^{2} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}}{4608}-\frac {73 b c \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{3072}+\frac {55 d^{2} b \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{3072 \sqrt {c^{2} x^{2}-1}}}{c^{4}}\) | \(243\) |
default | \(\frac {d^{2} a \left (\frac {\left (c^{2} x^{2}-1\right )^{4}}{8}+\frac {\left (c^{2} x^{2}-1\right )^{3}}{6}\right )+\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{8} x^{8}}{8}-\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{6} x^{6}}{3}+\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {b \,d^{2} \mathrm {arccosh}\left (c x \right )}{24}-\frac {d^{2} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{7} x^{7}}{64}+\frac {43 d^{2} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{5} x^{5}}{1152}-\frac {73 d^{2} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}}{4608}-\frac {73 b c \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{3072}+\frac {55 d^{2} b \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{3072 \sqrt {c^{2} x^{2}-1}}}{c^{4}}\) | \(243\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 346 vs.
\(2 (168) = 336\).
time = 0.28, size = 346, normalized size = 1.73 \begin {gather*} \frac {1}{8} \, a c^{4} d^{2} x^{8} - \frac {1}{3} \, a c^{2} d^{2} x^{6} + \frac {1}{3072} \, {\left (384 \, x^{8} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{7}}{c^{2}} + \frac {56 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{6}} + \frac {105 \, \sqrt {c^{2} x^{2} - 1} x}{c^{8}} + \frac {105 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{9}}\right )} c\right )} b c^{4} d^{2} + \frac {1}{4} \, a d^{2} x^{4} - \frac {1}{144} \, {\left (48 \, x^{6} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} - 1} x}{c^{6}} + \frac {15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{7}}\right )} c\right )} b c^{2} d^{2} + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b d^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 161, normalized size = 0.80 \begin {gather*} \frac {1152 \, a c^{8} d^{2} x^{8} - 3072 \, a c^{6} d^{2} x^{6} + 2304 \, a c^{4} d^{2} x^{4} + 3 \, {\left (384 \, b c^{8} d^{2} x^{8} - 1024 \, b c^{6} d^{2} x^{6} + 768 \, b c^{4} d^{2} x^{4} - 73 \, b d^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (144 \, b c^{7} d^{2} x^{7} - 344 \, b c^{5} d^{2} x^{5} + 146 \, b c^{3} d^{2} x^{3} + 219 \, b c d^{2} x\right )} \sqrt {c^{2} x^{2} - 1}}{9216 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.08, size = 224, normalized size = 1.12 \begin {gather*} \begin {cases} \frac {a c^{4} d^{2} x^{8}}{8} - \frac {a c^{2} d^{2} x^{6}}{3} + \frac {a d^{2} x^{4}}{4} + \frac {b c^{4} d^{2} x^{8} \operatorname {acosh}{\left (c x \right )}}{8} - \frac {b c^{3} d^{2} x^{7} \sqrt {c^{2} x^{2} - 1}}{64} - \frac {b c^{2} d^{2} x^{6} \operatorname {acosh}{\left (c x \right )}}{3} + \frac {43 b c d^{2} x^{5} \sqrt {c^{2} x^{2} - 1}}{1152} + \frac {b d^{2} x^{4} \operatorname {acosh}{\left (c x \right )}}{4} - \frac {73 b d^{2} x^{3} \sqrt {c^{2} x^{2} - 1}}{4608 c} - \frac {73 b d^{2} x \sqrt {c^{2} x^{2} - 1}}{3072 c^{3}} - \frac {73 b d^{2} \operatorname {acosh}{\left (c x \right )}}{3072 c^{4}} & \text {for}\: c \neq 0 \\\frac {d^{2} x^{4} \left (a + \frac {i \pi b}{2}\right )}{4} & \text {otherwise} \end {cases} \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^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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