Optimal. Leaf size=278 \[ -\frac {b c (f x)^{2+m} \sqrt {d-c^2 d x^2}}{f^2 (2+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(f x)^{1+m} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{f (2+m)}+\frac {(f x)^{1+m} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};c^2 x^2\right )}{f \left (2+3 m+m^2\right ) \sqrt {1-c x} \sqrt {1+c x}}-\frac {b c (f x)^{2+m} \sqrt {d-c^2 d x^2} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )}{f^2 (1+m) (2+m)^2 \sqrt {-1+c x} \sqrt {1+c x}} \]
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Rubi [A]
time = 0.24, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {5926, 5949, 32}
\begin {gather*} -\frac {b c \sqrt {d-c^2 d x^2} (f x)^{m+2} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;c^2 x^2\right )}{f^2 (m+1) (m+2)^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {\sqrt {d-c^2 d x^2} (f x)^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{f \left (m^2+3 m+2\right ) \sqrt {1-c x} \sqrt {c x+1}}+\frac {\sqrt {d-c^2 d x^2} (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{f (m+2)}-\frac {b c \sqrt {d-c^2 d x^2} (f x)^{m+2}}{f^2 (m+2)^2 \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 5926
Rule 5949
Rubi steps
\begin {align*} \int (f x)^m \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {\sqrt {d-c^2 d x^2} \int (f x)^m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(f x)^{1+m} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{f (2+m)}-\frac {\sqrt {d-c^2 d x^2} \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{(2+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int (f x)^{1+m} \, dx}{f (2+m) \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c (f x)^{2+m} \sqrt {d-c^2 d x^2}}{f^2 (2+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(f x)^{1+m} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{f (2+m)}+\frac {(f x)^{1+m} \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};c^2 x^2\right )}{f \left (2+3 m+m^2\right ) (1-c x) (1+c x)}-\frac {b c (f x)^{2+m} \sqrt {d-c^2 d x^2} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )}{f^2 (1+m) (2+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 223, normalized size = 0.80 \begin {gather*} \frac {x (f x)^m \sqrt {d-c^2 d x^2} \left ((1+m) \left (-b c x \sqrt {-1+c x} \sqrt {1+c x}+a (2+m) \left (-1+c^2 x^2\right )+b (2+m) \left (-1+c^2 x^2\right ) \cosh ^{-1}(c x)\right )-(2+m) \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right ) \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};c^2 x^2\right )-b c x \sqrt {-1+c x} \sqrt {1+c x} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )\right )}{(1+m) (2+m)^2 (-1+c x) (1+c x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (f x \right )^{m} \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )\, dx\]
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 (f x\right )^{m} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 = -0.00 \begin {gather*} \int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2}\,{\left (f\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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