Integrand size = 27, antiderivative size = 429 \[ \int (f x)^m \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {b c d^3 \left (2271+1329 m+284 m^2+27 m^3+m^4\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{f^2 (3+m)^2 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^3 (9+m) (13+2 m) (f x)^{4+m} \left (1-c^2 x^2\right )}{f^4 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^3 (f x)^{6+m} \left (1-c^2 x^2\right )}{f^6 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^3 (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}-\frac {3 c^2 d^3 (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}+\frac {3 c^4 d^3 (f x)^{5+m} (a+b \text {arccosh}(c x))}{f^5 (5+m)}-\frac {c^6 d^3 (f x)^{7+m} (a+b \text {arccosh}(c x))}{f^7 (7+m)}-\frac {3 b c d^3 \left (2161+1813 m+455 m^2+35 m^3\right ) (f x)^{2+m} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{f^2 (1+m) (2+m) (3+m)^2 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}} \]
[Out]
Time = 1.71 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {276, 5921, 12, 1624, 1823, 1281, 470, 372, 371} \[ \int (f x)^m \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {c^6 d^3 (f x)^{m+7} (a+b \text {arccosh}(c x))}{f^7 (m+7)}+\frac {3 c^4 d^3 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}-\frac {3 c^2 d^3 (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {d^3 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}-\frac {3 b c d^3 \left (35 m^3+455 m^2+1813 m+2161\right ) \sqrt {1-c^2 x^2} (f x)^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{f^2 (m+1) (m+2) (m+3)^2 (m+5)^2 (m+7)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^3 \left (m^4+27 m^3+284 m^2+1329 m+2271\right ) \left (1-c^2 x^2\right ) (f x)^{m+2}}{f^2 (m+3)^2 (m+5)^2 (m+7)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d^3 \left (1-c^2 x^2\right ) (f x)^{m+6}}{f^6 (m+7)^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d^3 (m+9) (2 m+13) \left (1-c^2 x^2\right ) (f x)^{m+4}}{f^4 (m+5)^2 (m+7)^2 \sqrt {c x-1} \sqrt {c x+1}} \]
[In]
[Out]
Rule 12
Rule 276
Rule 371
Rule 372
Rule 470
Rule 1281
Rule 1624
Rule 1823
Rule 5921
Rubi steps \begin{align*} \text {integral}& = \frac {d^3 (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}-\frac {3 c^2 d^3 (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}+\frac {3 c^4 d^3 (f x)^{5+m} (a+b \text {arccosh}(c x))}{f^5 (5+m)}-\frac {c^6 d^3 (f x)^{7+m} (a+b \text {arccosh}(c x))}{f^7 (7+m)}-(b c) \int \frac {d^3 (f x)^{1+m} \left (\frac {1}{1+m}-\frac {3 c^2 x^2}{3+m}+\frac {3 c^4 x^4}{5+m}-\frac {c^6 x^6}{7+m}\right )}{f \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = \frac {d^3 (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}-\frac {3 c^2 d^3 (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}+\frac {3 c^4 d^3 (f x)^{5+m} (a+b \text {arccosh}(c x))}{f^5 (5+m)}-\frac {c^6 d^3 (f x)^{7+m} (a+b \text {arccosh}(c x))}{f^7 (7+m)}-\frac {\left (b c d^3\right ) \int \frac {(f x)^{1+m} \left (\frac {1}{1+m}-\frac {3 c^2 x^2}{3+m}+\frac {3 c^4 x^4}{5+m}-\frac {c^6 x^6}{7+m}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{f} \\ & = \frac {d^3 (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}-\frac {3 c^2 d^3 (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}+\frac {3 c^4 d^3 (f x)^{5+m} (a+b \text {arccosh}(c x))}{f^5 (5+m)}-\frac {c^6 d^3 (f x)^{7+m} (a+b \text {arccosh}(c x))}{f^7 (7+m)}-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+m} \left (\frac {1}{1+m}-\frac {3 c^2 x^2}{3+m}+\frac {3 c^4 x^4}{5+m}-\frac {c^6 x^6}{7+m}\right )}{\sqrt {-1+c^2 x^2}} \, dx}{f \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c^5 d^3 (f x)^{6+m} \left (1-c^2 x^2\right )}{f^6 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^3 (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}-\frac {3 c^2 d^3 (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}+\frac {3 c^4 d^3 (f x)^{5+m} (a+b \text {arccosh}(c x))}{f^5 (5+m)}-\frac {c^6 d^3 (f x)^{7+m} (a+b \text {arccosh}(c x))}{f^7 (7+m)}-\frac {\left (b d^3 \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+m} \left (\frac {c^2 (7+m)}{1+m}-\frac {3 c^4 (7+m) x^2}{3+m}+\frac {c^6 (9+m) (13+2 m) x^4}{(5+m) (7+m)}\right )}{\sqrt {-1+c^2 x^2}} \, dx}{c f (7+m) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b c^3 d^3 (9+m) (13+2 m) (f x)^{4+m} \left (1-c^2 x^2\right )}{f^4 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^3 (f x)^{6+m} \left (1-c^2 x^2\right )}{f^6 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^3 (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}-\frac {3 c^2 d^3 (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}+\frac {3 c^4 d^3 (f x)^{5+m} (a+b \text {arccosh}(c x))}{f^5 (5+m)}-\frac {c^6 d^3 (f x)^{7+m} (a+b \text {arccosh}(c x))}{f^7 (7+m)}-\frac {\left (b d^3 \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+m} \left (\frac {c^4 (5+m) (7+m)}{1+m}-\frac {c^6 \left (2271+1329 m+284 m^2+27 m^3+m^4\right ) x^2}{(3+m) (5+m) (7+m)}\right )}{\sqrt {-1+c^2 x^2}} \, dx}{c^3 f (5+m) (7+m) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c d^3 \left (2271+1329 m+284 m^2+27 m^3+m^4\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{f^2 (3+m)^2 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^3 (9+m) (13+2 m) (f x)^{4+m} \left (1-c^2 x^2\right )}{f^4 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^3 (f x)^{6+m} \left (1-c^2 x^2\right )}{f^6 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^3 (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}-\frac {3 c^2 d^3 (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}+\frac {3 c^4 d^3 (f x)^{5+m} (a+b \text {arccosh}(c x))}{f^5 (5+m)}-\frac {c^6 d^3 (f x)^{7+m} (a+b \text {arccosh}(c x))}{f^7 (7+m)}-\frac {\left (3 b c d^3 \left (2161+1813 m+455 m^2+35 m^3\right ) \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 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c d^3 \left (2271+1329 m+284 m^2+27 m^3+m^4\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{f^2 (3+m)^2 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^3 (9+m) (13+2 m) (f x)^{4+m} \left (1-c^2 x^2\right )}{f^4 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^3 (f x)^{6+m} \left (1-c^2 x^2\right )}{f^6 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^3 (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}-\frac {3 c^2 d^3 (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}+\frac {3 c^4 d^3 (f x)^{5+m} (a+b \text {arccosh}(c x))}{f^5 (5+m)}-\frac {c^6 d^3 (f x)^{7+m} (a+b \text {arccosh}(c x))}{f^7 (7+m)}-\frac {\left (3 b c d^3 \left (2161+1813 m+455 m^2+35 m^3\right ) \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 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c d^3 \left (2271+1329 m+284 m^2+27 m^3+m^4\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{f^2 (3+m)^2 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^3 (9+m) (13+2 m) (f x)^{4+m} \left (1-c^2 x^2\right )}{f^4 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^3 (f x)^{6+m} \left (1-c^2 x^2\right )}{f^6 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^3 (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}-\frac {3 c^2 d^3 (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}+\frac {3 c^4 d^3 (f x)^{5+m} (a+b \text {arccosh}(c x))}{f^5 (5+m)}-\frac {c^6 d^3 (f x)^{7+m} (a+b \text {arccosh}(c x))}{f^7 (7+m)}-\frac {3 b c d^3 \left (2161+1813 m+455 m^2+35 m^3\right ) (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 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}
Time = 0.79 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.90 \[ \int (f x)^m \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=d^3 x (f x)^m \left (\frac {a+b \text {arccosh}(c x)}{1+m}-\frac {3 c^2 x^2 (a+b \text {arccosh}(c x))}{3+m}+\frac {3 c^4 x^4 (a+b \text {arccosh}(c x))}{5+m}-\frac {c^6 x^6 (a+b \text {arccosh}(c x))}{7+m}+\frac {b c^7 x^7 \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},4+\frac {m}{2},5+\frac {m}{2},c^2 x^2\right )}{(7+m) (8+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{\left (2+3 m+m^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 x^3 \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4+m}{2},\frac {6+m}{2},c^2 x^2\right )}{\left (12+7 m+m^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c^5 x^5 \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {6+m}{2},\frac {8+m}{2},c^2 x^2\right )}{(5+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}\right ) \]
[In]
[Out]
\[\int \left (f x \right )^{m} \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )d x\]
[In]
[Out]
\[ \int (f x)^m \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\int { -{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]
[In]
[Out]
Timed out. \[ \int (f x)^m \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int (f x)^m \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\int { -{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]
[In]
[Out]
Exception generated. \[ \int (f x)^m \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int (f x)^m \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3\,{\left (f\,x\right )}^m \,d x \]
[In]
[Out]