Optimal. Leaf size=181 \[ \frac {\left (15 a^4 c^2+10 a^2 c d+3 d^2\right ) \left (1-a^2 x^2\right )}{15 a^5 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {2 d \left (5 a^2 c+3 d\right ) \left (1-a^2 x^2\right )^2}{45 a^5 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {d^2 \left (1-a^2 x^2\right )^3}{25 a^5 \sqrt {-1+a x} \sqrt {1+a x}}+c^2 x \cosh ^{-1}(a x)+\frac {2}{3} c d x^3 \cosh ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cosh ^{-1}(a x) \]
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Rubi [A]
time = 0.14, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {200, 5908, 12,
534, 1261, 712} \begin {gather*} -\frac {2 d \left (1-a^2 x^2\right )^2 \left (5 a^2 c+3 d\right )}{45 a^5 \sqrt {a x-1} \sqrt {a x+1}}+\frac {d^2 \left (1-a^2 x^2\right )^3}{25 a^5 \sqrt {a x-1} \sqrt {a x+1}}+\frac {\left (1-a^2 x^2\right ) \left (15 a^4 c^2+10 a^2 c d+3 d^2\right )}{15 a^5 \sqrt {a x-1} \sqrt {a x+1}}+c^2 x \cosh ^{-1}(a x)+\frac {2}{3} c d x^3 \cosh ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cosh ^{-1}(a x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 200
Rule 534
Rule 712
Rule 1261
Rule 5908
Rubi steps
\begin {align*} \int \left (c+d x^2\right )^2 \cosh ^{-1}(a x) \, dx &=c^2 x \cosh ^{-1}(a x)+\frac {2}{3} c d x^3 \cosh ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cosh ^{-1}(a x)-a \int \frac {x \left (15 c^2+10 c d x^2+3 d^2 x^4\right )}{15 \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=c^2 x \cosh ^{-1}(a x)+\frac {2}{3} c d x^3 \cosh ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cosh ^{-1}(a x)-\frac {1}{15} a \int \frac {x \left (15 c^2+10 c d x^2+3 d^2 x^4\right )}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=c^2 x \cosh ^{-1}(a x)+\frac {2}{3} c d x^3 \cosh ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cosh ^{-1}(a x)-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \int \frac {x \left (15 c^2+10 c d x^2+3 d^2 x^4\right )}{\sqrt {-1+a^2 x^2}} \, dx}{15 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=c^2 x \cosh ^{-1}(a x)+\frac {2}{3} c d x^3 \cosh ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cosh ^{-1}(a x)-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {15 c^2+10 c d x+3 d^2 x^2}{\sqrt {-1+a^2 x}} \, dx,x,x^2\right )}{30 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=c^2 x \cosh ^{-1}(a x)+\frac {2}{3} c d x^3 \cosh ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cosh ^{-1}(a x)-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {15 a^4 c^2+10 a^2 c d+3 d^2}{a^4 \sqrt {-1+a^2 x}}+\frac {2 d \left (5 a^2 c+3 d\right ) \sqrt {-1+a^2 x}}{a^4}+\frac {3 d^2 \left (-1+a^2 x\right )^{3/2}}{a^4}\right ) \, dx,x,x^2\right )}{30 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {\left (15 a^4 c^2+10 a^2 c d+3 d^2\right ) \left (1-a^2 x^2\right )}{15 a^5 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {2 d \left (5 a^2 c+3 d\right ) \left (1-a^2 x^2\right )^2}{45 a^5 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {d^2 \left (1-a^2 x^2\right )^3}{25 a^5 \sqrt {-1+a x} \sqrt {1+a x}}+c^2 x \cosh ^{-1}(a x)+\frac {2}{3} c d x^3 \cosh ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cosh ^{-1}(a x)\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 103, normalized size = 0.57 \begin {gather*} -\frac {\sqrt {-1+a x} \sqrt {1+a x} \left (24 d^2+4 a^2 d \left (25 c+3 d x^2\right )+a^4 \left (225 c^2+50 c d x^2+9 d^2 x^4\right )\right )}{225 a^5}+\left (c^2 x+\frac {2}{3} c d x^3+\frac {d^2 x^5}{5}\right ) \cosh ^{-1}(a x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.66, size = 113, normalized size = 0.62
method | result | size |
derivativedivides | \(\frac {\mathrm {arccosh}\left (a x \right ) c^{2} a x +\frac {2 a \,\mathrm {arccosh}\left (a x \right ) c d \,x^{3}}{3}+\frac {a \,\mathrm {arccosh}\left (a x \right ) d^{2} x^{5}}{5}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (9 d^{2} a^{4} x^{4}+50 c \,a^{4} d \,x^{2}+225 a^{4} c^{2}+12 a^{2} d^{2} x^{2}+100 a^{2} c d +24 d^{2}\right )}{225 a^{4}}}{a}\) | \(113\) |
default | \(\frac {\mathrm {arccosh}\left (a x \right ) c^{2} a x +\frac {2 a \,\mathrm {arccosh}\left (a x \right ) c d \,x^{3}}{3}+\frac {a \,\mathrm {arccosh}\left (a x \right ) d^{2} x^{5}}{5}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (9 d^{2} a^{4} x^{4}+50 c \,a^{4} d \,x^{2}+225 a^{4} c^{2}+12 a^{2} d^{2} x^{2}+100 a^{2} c d +24 d^{2}\right )}{225 a^{4}}}{a}\) | \(113\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 154, normalized size = 0.85 \begin {gather*} -\frac {1}{225} \, {\left (\frac {9 \, \sqrt {a^{2} x^{2} - 1} d^{2} x^{4}}{a^{2}} + \frac {50 \, \sqrt {a^{2} x^{2} - 1} c d x^{2}}{a^{2}} + \frac {225 \, \sqrt {a^{2} x^{2} - 1} c^{2}}{a^{2}} + \frac {12 \, \sqrt {a^{2} x^{2} - 1} d^{2} x^{2}}{a^{4}} + \frac {100 \, \sqrt {a^{2} x^{2} - 1} c d}{a^{4}} + \frac {24 \, \sqrt {a^{2} x^{2} - 1} d^{2}}{a^{6}}\right )} a + \frac {1}{15} \, {\left (3 \, d^{2} x^{5} + 10 \, c d x^{3} + 15 \, c^{2} x\right )} \operatorname {arcosh}\left (a x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 121, normalized size = 0.67 \begin {gather*} \frac {15 \, {\left (3 \, a^{5} d^{2} x^{5} + 10 \, a^{5} c d x^{3} + 15 \, a^{5} c^{2} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (9 \, a^{4} d^{2} x^{4} + 225 \, a^{4} c^{2} + 100 \, a^{2} c d + 2 \, {\left (25 \, a^{4} c d + 6 \, a^{2} d^{2}\right )} x^{2} + 24 \, d^{2}\right )} \sqrt {a^{2} x^{2} - 1}}{225 \, a^{5}} \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 = 0.33, size = 199, normalized size = 1.10 \begin {gather*} \begin {cases} c^{2} x \operatorname {acosh}{\left (a x \right )} + \frac {2 c d x^{3} \operatorname {acosh}{\left (a x \right )}}{3} + \frac {d^{2} x^{5} \operatorname {acosh}{\left (a x \right )}}{5} - \frac {c^{2} \sqrt {a^{2} x^{2} - 1}}{a} - \frac {2 c d x^{2} \sqrt {a^{2} x^{2} - 1}}{9 a} - \frac {d^{2} x^{4} \sqrt {a^{2} x^{2} - 1}}{25 a} - \frac {4 c d \sqrt {a^{2} x^{2} - 1}}{9 a^{3}} - \frac {4 d^{2} x^{2} \sqrt {a^{2} x^{2} - 1}}{75 a^{3}} - \frac {8 d^{2} \sqrt {a^{2} x^{2} - 1}}{75 a^{5}} & \text {for}\: a \neq 0 \\\frac {i \pi \left (c^{2} x + \frac {2 c d x^{3}}{3} + \frac {d^{2} x^{5}}{5}\right )}{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 134, normalized size = 0.74 \begin {gather*} \frac {1}{15} \, {\left (3 \, d^{2} x^{5} + 10 \, c d x^{3} + 15 \, c^{2} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - \frac {{\left (15 \, a^{4} c^{2} + 10 \, a^{2} c d + 3 \, d^{2}\right )} \sqrt {a^{2} x^{2} - 1}}{15 \, a^{5}} - \frac {50 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} a^{2} c d + 9 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {5}{2}} d^{2} + 30 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} d^{2}}{225 \, a^{5}} \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 \mathrm {acosh}\left (a\,x\right )\,{\left (d\,x^2+c\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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