Optimal. Leaf size=106 \[ \frac {(d x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{d (1+m)}-\frac {b c (d x)^{2+m} \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{d^2 (1+m) (2+m) \sqrt {-1+c x} \sqrt {1+c x}} \]
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Rubi [A]
time = 0.04, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5883, 127, 372,
371} \begin {gather*} \frac {(d x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{d (m+1)}-\frac {b c \sqrt {1-c^2 x^2} (d x)^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{d^2 (m+1) (m+2) \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 127
Rule 371
Rule 372
Rule 5883
Rubi steps
\begin {align*} \int (d x)^m \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {(d x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{d (1+m)}-\frac {(b c) \int \frac {(d x)^{1+m}}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{d (1+m)}\\ &=\frac {(d x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{d (1+m)}-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {(d x)^{1+m}}{\sqrt {-1+c^2 x^2}} \, dx}{d (1+m) \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(d x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{d (1+m)}-\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \int \frac {(d x)^{1+m}}{\sqrt {1-c^2 x^2}} \, dx}{d (1+m) \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(d x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{d (1+m)}-\frac {b c (d x)^{2+m} \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{d^2 (1+m) (2+m) \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 87, normalized size = 0.82 \begin {gather*} \frac {x (d x)^m \left (a+b \cosh ^{-1}(c x)-\frac {b c x \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{(2+m) \sqrt {-1+c x} \sqrt {1+c x}}\right )}{1+m} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 9.59, size = 0, normalized size = 0.00 \[\int \left (d x \right )^{m} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )\, dx\]
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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d x\right )^{m} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )\, dx \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 = -0.01 \begin {gather*} \int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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