Integrand size = 29, antiderivative size = 455 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {3 b c d (f x)^{2+m} \sqrt {d-c^2 d x^2}}{f^2 (2+m)^2 (4+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d (f x)^{2+m} \sqrt {d-c^2 d x^2}}{f^2 (2+m) (4+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d (f x)^{4+m} \sqrt {d-c^2 d x^2}}{f^4 (4+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 d (f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f \left (8+6 m+m^2\right )}+\frac {(f x)^{1+m} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{f (4+m)}+\frac {3 d (f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{f (4+m) \left (2+3 m+m^2\right ) \sqrt {1-c x} \sqrt {1+c x}}-\frac {3 b c d (f x)^{2+m} \sqrt {d-c^2 d x^2} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )}{f^2 (1+m) (2+m)^2 (4+m) \sqrt {-1+c x} \sqrt {1+c x}} \]
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Time = 0.36 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {5930, 5926, 5949, 32, 74, 14} \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {3 b c d \sqrt {d-c^2 d x^2} (f x)^{m+2} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;c^2 x^2\right )}{f^2 (m+1) (m+2)^2 (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 d \sqrt {d-c^2 d x^2} (f x)^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},c^2 x^2\right ) (a+b \text {arccosh}(c x))}{f (m+4) \left (m^2+3 m+2\right ) \sqrt {1-c x} \sqrt {c x+1}}+\frac {3 d \sqrt {d-c^2 d x^2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f \left (m^2+6 m+8\right )}+\frac {\left (d-c^2 d x^2\right )^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}-\frac {b c d \sqrt {d-c^2 d x^2} (f x)^{m+2}}{f^2 (m+2) (m+4) \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 b c d \sqrt {d-c^2 d x^2} (f x)^{m+2}}{f^2 (m+2)^2 (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d \sqrt {d-c^2 d x^2} (f x)^{m+4}}{f^4 (m+4)^2 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rule 14
Rule 32
Rule 74
Rule 5926
Rule 5930
Rule 5949
Rubi steps \begin{align*} \text {integral}& = \frac {(f x)^{1+m} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{f (4+m)}+\frac {(3 d) \int (f x)^m \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx}{4+m}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int (f x)^{1+m} (-1+c x) (1+c x) \, dx}{f (4+m) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {3 d (f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f \left (8+6 m+m^2\right )}+\frac {(f x)^{1+m} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{f (4+m)}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int (f x)^{1+m} \left (-1+c^2 x^2\right ) \, dx}{f (4+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 d \sqrt {d-c^2 d x^2}\right ) \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{(2+m) (4+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b c d \sqrt {d-c^2 d x^2}\right ) \int (f x)^{1+m} \, dx}{f (2+m) (4+m) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {3 b c d (f x)^{2+m} \sqrt {d-c^2 d x^2}}{f^2 (2+m)^2 (4+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 d (f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f \left (8+6 m+m^2\right )}+\frac {(f x)^{1+m} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{f (4+m)}+\frac {3 d (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 (1+m) (2+m) (4+m) \sqrt {1-c x} \sqrt {1+c x}}-\frac {3 b c d (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 (4+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \left (-(f x)^{1+m}+\frac {c^2 (f x)^{3+m}}{f^2}\right ) \, dx}{f (4+m) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {3 b c d (f x)^{2+m} \sqrt {d-c^2 d x^2}}{f^2 (2+m)^2 (4+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d (f x)^{2+m} \sqrt {d-c^2 d x^2}}{f^2 (2+m) (4+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d (f x)^{4+m} \sqrt {d-c^2 d x^2}}{f^4 (4+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 d (f x)^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f \left (8+6 m+m^2\right )}+\frac {(f x)^{1+m} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{f (4+m)}+\frac {3 d (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 (1+m) (2+m) (4+m) \sqrt {1-c x} \sqrt {1+c x}}-\frac {3 b c d (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 (4+m) \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.60 \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {d x (f x)^m \sqrt {d-c^2 d x^2} \left (\frac {3 b c x}{(2+m)^2}+b c x \left (\frac {1}{2+m}-\frac {c^2 x^2}{4+m}\right )-\frac {3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{2+m}+(-1+c x)^{3/2} (1+c x)^{3/2} (a+b \text {arccosh}(c x))+\frac {3 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{(1+m) (2+m) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c x \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )}{(1+m) (2+m)^2}\right )}{(4+m) \sqrt {-1+c x} \sqrt {1+c x}} \]
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\[\int \left (f x \right )^{m} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )d x\]
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\[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]
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Timed out. \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\text {Timed out} \]
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\[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]
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Exception generated. \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int (f x)^m \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}\,{\left (f\,x\right )}^m \,d x \]
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