Optimal. Leaf size=143 \[ -\frac {8 b d^2 \sqrt {-1+c x} \sqrt {1+c x}}{15 c}+\frac {4 b d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{45 c}-\frac {b d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{25 c}+d^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {2}{3} c^2 d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} c^4 d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right ) \]
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Rubi [A]
time = 0.10, antiderivative size = 177, normalized size of antiderivative = 1.24, number of steps
used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {200, 5894, 12,
534, 1261, 712} \begin {gather*} \frac {1}{5} c^4 d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {2}{3} c^2 d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {b d^2 \left (1-c^2 x^2\right )^3}{25 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {4 b d^2 \left (1-c^2 x^2\right )^2}{45 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 b d^2 \left (1-c^2 x^2\right )}{15 c \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 200
Rule 534
Rule 712
Rule 1261
Rule 5894
Rubi steps
\begin {align*} \int \left (d-c^2 d x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=d^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {2}{3} c^2 d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} c^4 d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {d^2 x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{15 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=d^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {2}{3} c^2 d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} c^4 d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{15} \left (b c d^2\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=d^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {2}{3} c^2 d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} c^4 d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{\sqrt {-1+c^2 x^2}} \, dx}{15 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=d^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {2}{3} c^2 d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} c^4 d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {15-10 c^2 x+3 c^4 x^2}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{30 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=d^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {2}{3} c^2 d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} c^4 d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {8}{\sqrt {-1+c^2 x}}-4 \sqrt {-1+c^2 x}+3 \left (-1+c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{30 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {8 b d^2 \left (1-c^2 x^2\right )}{15 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b d^2 \left (1-c^2 x^2\right )^2}{45 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \left (1-c^2 x^2\right )^3}{25 c \sqrt {-1+c x} \sqrt {1+c x}}+d^2 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {2}{3} c^2 d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} c^4 d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 99, normalized size = 0.69 \begin {gather*} \frac {d^2 \left (b \sqrt {-1+c x} \sqrt {1+c x} \left (-149+38 c^2 x^2-9 c^4 x^4\right )+15 a c x \left (15-10 c^2 x^2+3 c^4 x^4\right )+15 b c x \left (15-10 c^2 x^2+3 c^4 x^4\right ) \cosh ^{-1}(c x)\right )}{225 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.04, size = 102, normalized size = 0.71
method | result | size |
derivativedivides | \(\frac {d^{2} a \left (\frac {1}{5} c^{5} x^{5}-\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b \left (\frac {\mathrm {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {2 \,\mathrm {arccosh}\left (c x \right ) c^{3} x^{3}}{3}+c x \,\mathrm {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} x^{4}-38 c^{2} x^{2}+149\right )}{225}\right )}{c}\) | \(102\) |
default | \(\frac {d^{2} a \left (\frac {1}{5} c^{5} x^{5}-\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b \left (\frac {\mathrm {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {2 \,\mathrm {arccosh}\left (c x \right ) c^{3} x^{3}}{3}+c x \,\mathrm {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} x^{4}-38 c^{2} x^{2}+149\right )}{225}\right )}{c}\) | \(102\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 194, normalized size = 1.36 \begin {gather*} \frac {1}{5} \, a c^{4} d^{2} x^{5} + \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 c^{4} d^{2} - \frac {2}{3} \, a c^{2} d^{2} x^{3} - \frac {2}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b c^{2} d^{2} + a d^{2} x + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d^{2}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 133, normalized size = 0.93 \begin {gather*} \frac {45 \, a c^{5} d^{2} x^{5} - 150 \, a c^{3} d^{2} x^{3} + 225 \, a c d^{2} x + 15 \, {\left (3 \, b c^{5} d^{2} x^{5} - 10 \, b c^{3} d^{2} x^{3} + 15 \, b c d^{2} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (9 \, b c^{4} d^{2} x^{4} - 38 \, b c^{2} d^{2} x^{2} + 149 \, b d^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{225 \, c} \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 = 0.34, size = 172, normalized size = 1.20 \begin {gather*} \begin {cases} \frac {a c^{4} d^{2} x^{5}}{5} - \frac {2 a c^{2} d^{2} x^{3}}{3} + a d^{2} x + \frac {b c^{4} d^{2} x^{5} \operatorname {acosh}{\left (c x \right )}}{5} - \frac {b c^{3} d^{2} x^{4} \sqrt {c^{2} x^{2} - 1}}{25} - \frac {2 b c^{2} d^{2} x^{3} \operatorname {acosh}{\left (c x \right )}}{3} + \frac {38 b c d^{2} x^{2} \sqrt {c^{2} x^{2} - 1}}{225} + b d^{2} x \operatorname {acosh}{\left (c x \right )} - \frac {149 b d^{2} \sqrt {c^{2} x^{2} - 1}}{225 c} & \text {for}\: c \neq 0 \\d^{2} x \left (a + \frac {i \pi b}{2}\right ) & \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 = -0.01 \begin {gather*} \int \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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