Integrand size = 25, antiderivative size = 184 \[ \int (f x)^m \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\frac {b c d (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x}}{f^2 (3+m)^2}+\frac {d (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}-\frac {c^2 d (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}-\frac {b c d (7+3 m) (f x)^{2+m} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{f^2 (1+m) (2+m) (3+m)^2 \sqrt {-1+c x} \sqrt {1+c x}} \]
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Time = 0.18 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {14, 5921, 12, 471, 127, 372, 371} \[ \int (f x)^m \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {c^2 d (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {d (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}-\frac {b c d (3 m+7) \sqrt {1-c^2 x^2} (f x)^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{f^2 (m+1) (m+2) (m+3)^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c d \sqrt {c x-1} \sqrt {c x+1} (f x)^{m+2}}{f^2 (m+3)^2} \]
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Rule 12
Rule 14
Rule 127
Rule 371
Rule 372
Rule 471
Rule 5921
Rubi steps \begin{align*} \text {integral}& = \frac {d (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}-\frac {c^2 d (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}-(b c) \int \frac {d (f x)^{1+m} \left (3+m-c^2 (1+m) x^2\right )}{f (1+m) (3+m) \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = \frac {d (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}-\frac {c^2 d (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}-\frac {(b c d) \int \frac {(f x)^{1+m} \left (3+m-c^2 (1+m) x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{f \left (3+4 m+m^2\right )} \\ & = \frac {b c d (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x}}{f^2 (3+m)^2}+\frac {d (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}-\frac {c^2 d (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}-\frac {(b c d (7+3 m)) \int \frac {(f x)^{1+m}}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{f (1+m) (3+m)^2} \\ & = \frac {b c d (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x}}{f^2 (3+m)^2}+\frac {d (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}-\frac {c^2 d (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}-\frac {\left (b c d (7+3 m) \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+m}}{\sqrt {-1+c^2 x^2}} \, dx}{f (1+m) (3+m)^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b c d (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x}}{f^2 (3+m)^2}+\frac {d (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}-\frac {c^2 d (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}-\frac {\left (b c d (7+3 m) \sqrt {1-c^2 x^2}\right ) \int \frac {(f x)^{1+m}}{\sqrt {1-c^2 x^2}} \, dx}{f (1+m) (3+m)^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b c d (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x}}{f^2 (3+m)^2}+\frac {d (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}-\frac {c^2 d (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}-\frac {b c d (7+3 m) (f x)^{2+m} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{f^2 (1+m) (2+m) (3+m)^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.04 \[ \int (f x)^m \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=d x (f x)^m \left (\frac {a+b \text {arccosh}(c x)}{1+m}-\frac {c^2 x^2 (a+b \text {arccosh}(c x))}{3+m}-\frac {b c x \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{\left (2+3 m+m^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 x^3 \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4+m}{2},\frac {6+m}{2},c^2 x^2\right )}{\left (12+7 m+m^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}\right ) \]
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\[\int \left (f x \right )^{m} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )d x\]
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\[ \int (f x)^m \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\int { -{\left (c^{2} d x^{2} - d\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]
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\[ \int (f x)^m \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=- d \left (\int \left (- a \left (f x\right )^{m}\right )\, dx + \int \left (- b \left (f x\right )^{m} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int a c^{2} x^{2} \left (f x\right )^{m}\, dx + \int b c^{2} x^{2} \left (f x\right )^{m} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]
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\[ \int (f x)^m \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\int { -{\left (c^{2} d x^{2} - d\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]
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Exception generated. \[ \int (f x)^m \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int (f x)^m \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x)) \, dx=\int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right )\,{\left (f\,x\right )}^m \,d x \]
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