Integrand size = 29, antiderivative size = 278 \[ \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=-\frac {b c (f x)^{2+m} \sqrt {d-c^2 d x^2}}{f^2 (2+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f (2+m)}+\frac {(f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{f \left (2+3 m+m^2\right ) \sqrt {1-c x} \sqrt {1+c x}}-\frac {b c (f x)^{2+m} \sqrt {d-c^2 d x^2} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )}{f^2 (1+m) (2+m)^2 \sqrt {-1+c x} \sqrt {1+c x}} \]
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Time = 0.24 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {5926, 5949, 32} \[ \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=-\frac {b c \sqrt {d-c^2 d x^2} (f x)^{m+2} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;c^2 x^2\right )}{f^2 (m+1) (m+2)^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {\sqrt {d-c^2 d x^2} (f x)^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},c^2 x^2\right ) (a+b \text {arccosh}(c x))}{f \left (m^2+3 m+2\right ) \sqrt {1-c x} \sqrt {c x+1}}+\frac {\sqrt {d-c^2 d x^2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \sqrt {d-c^2 d x^2} (f x)^{m+2}}{f^2 (m+2)^2 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rule 32
Rule 5926
Rule 5949
Rubi steps \begin{align*} \text {integral}& = \frac {(f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f (2+m)}-\frac {\sqrt {d-c^2 d x^2} \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\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} (a+b \text {arccosh}(c x))}{f (2+m)}+\frac {(f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\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}} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.80 \[ \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {x (f x)^m \sqrt {d-c^2 d x^2} \left ((1+m) \left (-b c x \sqrt {-1+c x} \sqrt {1+c x}+a (2+m) \left (-1+c^2 x^2\right )+b (2+m) \left (-1+c^2 x^2\right ) \text {arccosh}(c x)\right )-(2+m) \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )-b c x \sqrt {-1+c x} \sqrt {1+c x} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )\right )}{(1+m) (2+m)^2 (-1+c x) (1+c x)} \]
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\[\int \left (f x \right )^{m} \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )d x\]
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\[ \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]
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\[ \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int \left (f x\right )^{m} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )\, dx \]
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\[ \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]
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Exception generated. \[ \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2}\,{\left (f\,x\right )}^m \,d x \]
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