Optimal. Leaf size=161 \[ \frac {(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 f \left (1-c^2 x^2\right )}-\frac {b c (f x)^{2+m} \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {3}{2},\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{2 d^2 f^2 (2+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(1-m) \text {Int}\left (\frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2},x\right )}{2 d} \]
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Rubi [A]
time = 0.14, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx &=\frac {(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 f \left (1-c^2 x^2\right )}+\frac {(b c) \int \frac {(f x)^{1+m}}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^2 f}+\frac {(1-m) \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{2 d}\\ &=\frac {(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 f \left (1-c^2 x^2\right )}+\frac {(1-m) \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{2 d}+\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+m}}{\left (-1+c^2 x^2\right )^{3/2}} \, dx}{2 d^2 f \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 f \left (1-c^2 x^2\right )}+\frac {(1-m) \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{2 d}-\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \int \frac {(f x)^{1+m}}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 d^2 f \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 f \left (1-c^2 x^2\right )}-\frac {b c (f x)^{2+m} \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {3}{2},\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{2 d^2 f^2 (2+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(1-m) \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{2 d}\\ \end {align*}
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Mathematica [A]
time = 7.87, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (f x \right )^{m} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}{\left (-c^{2} d \,x^{2}+d \right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
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 [A]
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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a \left (f x\right )^{m}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b \left (f x\right )^{m} \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
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 [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (f\,x\right )}^m}{{\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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