Optimal. Leaf size=87 \[ \frac {(d x)^{1+m} \left (a+b \text {sech}^{-1}(c x)\right )}{d (1+m)}+\frac {b (d x)^{1+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};c^2 x^2\right )}{d (1+m)^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6418, 126, 371}
\begin {gather*} \frac {(d x)^{m+1} \left (a+b \text {sech}^{-1}(c x)\right )}{d (m+1)}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} (d x)^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};c^2 x^2\right )}{d (m+1)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 126
Rule 371
Rule 6418
Rubi steps
\begin {align*} \int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=\frac {(d x)^{1+m} \left (a+b \text {sech}^{-1}(c x)\right )}{d (1+m)}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {(d x)^m}{\sqrt {1-c x} \sqrt {1+c x}} \, dx}{1+m}\\ &=\frac {(d x)^{1+m} \left (a+b \text {sech}^{-1}(c x)\right )}{d (1+m)}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {(d x)^m}{\sqrt {1-c^2 x^2}} \, dx}{1+m}\\ &=\frac {(d x)^{1+m} \left (a+b \text {sech}^{-1}(c x)\right )}{d (1+m)}+\frac {b (d x)^{1+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};c^2 x^2\right )}{d (1+m)^2}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 97, normalized size = 1.11 \begin {gather*} \frac {x (d x)^m \left ((1+m) (-1+c x) \left (a+b \text {sech}^{-1}(c x)\right )-b \sqrt {\frac {1-c x}{1+c x}} \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};c^2 x^2\right )\right )}{(1+m)^2 (-1+c x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.33, size = 0, normalized size = 0.00 \[\int \left (d x \right )^{m} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )\, dx\]
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 {asech}{\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 (d\,x\right )}^m\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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