Optimal. Leaf size=64 \[ 2 b^2 x-\frac {2 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d} \]
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Rubi [A]
time = 0.08, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5995, 5879,
5915, 8} \begin {gather*} -\frac {2 b \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )}{d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d}+2 b^2 x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 5879
Rule 5915
Rule 5995
Rubi steps
\begin {align*} \int \left (a+b \cosh ^{-1}(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int \left (a+b \cosh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d}-\frac {(2 b) \text {Subst}\left (\int \frac {x \left (a+b \cosh ^{-1}(x)\right )}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d}+\frac {\left (2 b^2\right ) \text {Subst}(\int 1 \, dx,x,c+d x)}{d}\\ &=2 b^2 x-\frac {2 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 105, normalized size = 1.64 \begin {gather*} \frac {a^2 (c+d x)+2 b^2 (c+d x)-2 a b \sqrt {-1+c+d x} \sqrt {1+c+d x}-2 b \left (-a (c+d x)+b \sqrt {-1+c+d x} \sqrt {1+c+d x}\right ) \cosh ^{-1}(c+d x)+b^2 (c+d x) \cosh ^{-1}(c+d x)^2}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 16.32, size = 100, normalized size = 1.56
method | result | size |
derivativedivides | \(\frac {\left (d x +c \right ) a^{2}+b^{2} \left (\mathrm {arccosh}\left (d x +c \right )^{2} \left (d x +c \right )-2 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+2 d x +2 c \right )+2 a b \left (\left (d x +c \right ) \mathrm {arccosh}\left (d x +c \right )-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{d}\) | \(100\) |
default | \(\frac {\left (d x +c \right ) a^{2}+b^{2} \left (\mathrm {arccosh}\left (d x +c \right )^{2} \left (d x +c \right )-2 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+2 d x +2 c \right )+2 a b \left (\left (d x +c \right ) \mathrm {arccosh}\left (d x +c \right )-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{d}\) | \(100\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 141 vs.
\(2 (60) = 120\).
time = 0.40, size = 141, normalized size = 2.20 \begin {gather*} \frac {{\left (a^{2} + 2 \, b^{2}\right )} d x + {\left (b^{2} d x + b^{2} c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{2} - 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} a b + 2 \, {\left (a b d x + a b c - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} b^{2}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 143 vs.
\(2 (60) = 120\).
time = 0.12, size = 143, normalized size = 2.23 \begin {gather*} \begin {cases} a^{2} x + \frac {2 a b c \operatorname {acosh}{\left (c + d x \right )}}{d} + 2 a b x \operatorname {acosh}{\left (c + d x \right )} - \frac {2 a b \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{d} + \frac {b^{2} c \operatorname {acosh}^{2}{\left (c + d x \right )}}{d} + b^{2} x \operatorname {acosh}^{2}{\left (c + d x \right )} + 2 b^{2} x - \frac {2 b^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {acosh}{\left (c \right )}\right )^{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.02 \begin {gather*} \int {\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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