Integrand size = 20, antiderivative size = 182 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{3 d \left (c^2 d+e\right ) \sqrt {d+e x^2}}+\frac {x (a+b \text {arccosh}(c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {2 b \sqrt {1-c^2 x^2} \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {e} \sqrt {-1+c x} \sqrt {1+c x}} \]
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Time = 0.13 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {198, 197, 5908, 12, 533, 585, 79, 65, 223, 212} \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {x (a+b \text {arccosh}(c x))}{3 d \left (d+e x^2\right )^{3/2}}-\frac {2 b \sqrt {c^2 x^2-1} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {e} \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c \left (1-c^2 x^2\right )}{3 d \sqrt {c x-1} \sqrt {c x+1} \left (c^2 d+e\right ) \sqrt {d+e x^2}} \]
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Rule 12
Rule 65
Rule 79
Rule 197
Rule 198
Rule 212
Rule 223
Rule 533
Rule 585
Rule 5908
Rubi steps \begin{align*} \text {integral}& = \frac {x (a+b \text {arccosh}(c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d+e x^2}}-(b c) \int \frac {x \left (3 d+2 e x^2\right )}{3 d^2 \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )^{3/2}} \, dx \\ & = \frac {x (a+b \text {arccosh}(c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d+e x^2}}-\frac {(b c) \int \frac {x \left (3 d+2 e x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )^{3/2}} \, dx}{3 d^2} \\ & = \frac {x (a+b \text {arccosh}(c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d+e x^2}}-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {x \left (3 d+2 e x^2\right )}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {x (a+b \text {arccosh}(c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d+e x^2}}-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {3 d+2 e x}{\sqrt {-1+c^2 x} (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 d^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b c \left (1-c^2 x^2\right )}{3 d \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \sqrt {d+e x^2}}+\frac {x (a+b \text {arccosh}(c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d+e x^2}}-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{3 d^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b c \left (1-c^2 x^2\right )}{3 d \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \sqrt {d+e x^2}}+\frac {x (a+b \text {arccosh}(c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d+e x^2}}-\frac {\left (2 b \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {d+\frac {e}{c^2}+\frac {e x^2}{c^2}}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{3 c d^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b c \left (1-c^2 x^2\right )}{3 d \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \sqrt {d+e x^2}}+\frac {x (a+b \text {arccosh}(c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d+e x^2}}-\frac {\left (2 b \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\frac {e x^2}{c^2}} \, dx,x,\frac {\sqrt {-1+c^2 x^2}}{\sqrt {d+e x^2}}\right )}{3 c d^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b c \left (1-c^2 x^2\right )}{3 d \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \sqrt {d+e x^2}}+\frac {x (a+b \text {arccosh}(c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d+e x^2}}-\frac {2 b \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {e} \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.87 (sec) , antiderivative size = 633, normalized size of antiderivative = 3.48 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {-\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}{d \left (c^2 d+e\right )}+\frac {a x \left (3 d+2 e x^2\right )}{d^2}+\frac {b x \left (3 d+2 e x^2\right ) \text {arccosh}(c x)}{d^2}+\frac {4 b (-1+c x)^{3/2} \sqrt {\frac {\left (c \sqrt {d}-i \sqrt {e}\right ) (1+c x)}{\left (c \sqrt {d}+i \sqrt {e}\right ) (-1+c x)}} \left (d+e x^2\right ) \left (\frac {c \left (-i c \sqrt {d}+\sqrt {e}\right ) \left (i \sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {1+\frac {i c \sqrt {d}}{\sqrt {e}}-c x+\frac {i \sqrt {e} x}{\sqrt {d}}}{1-c x}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{2-2 c x}}\right ),\frac {4 i c \sqrt {d} \sqrt {e}}{\left (c \sqrt {d}+i \sqrt {e}\right )^2}\right )}{-1+c x}+c \sqrt {d} \left (-c \sqrt {d}+i \sqrt {e}\right ) \sqrt {\frac {\left (c^2 d+e\right ) \left (d+e x^2\right )}{d e (-1+c x)^2}} \sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{1-c x}} \operatorname {EllipticPi}\left (\frac {2 c \sqrt {d}}{c \sqrt {d}+i \sqrt {e}},\arcsin \left (\sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{2-2 c x}}\right ),\frac {4 i c \sqrt {d} \sqrt {e}}{\left (c \sqrt {d}+i \sqrt {e}\right )^2}\right )\right )}{c d^2 \left (c^2 d+e\right ) \sqrt {1+c x} \sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{1-c x}}}}{3 \left (d+e x^2\right )^{3/2}} \]
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\[\int \frac {a +b \,\operatorname {arccosh}\left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (150) = 300\).
Time = 0.33 (sec) , antiderivative size = 724, normalized size of antiderivative = 3.98 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\left [\frac {{\left (b c^{2} d^{3} + {\left (b c^{2} d e^{2} + b e^{3}\right )} x^{4} + b d^{2} e + 2 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x^{2}\right )} \sqrt {e} \log \left (8 \, c^{4} e^{2} x^{4} + c^{4} d^{2} - 6 \, c^{2} d e + 8 \, {\left (c^{4} d e - c^{2} e^{2}\right )} x^{2} - 4 \, {\left (2 \, c^{3} e x^{2} + c^{3} d - c e\right )} \sqrt {c^{2} x^{2} - 1} \sqrt {e x^{2} + d} \sqrt {e} + e^{2}\right ) + 2 \, {\left (2 \, {\left (b c^{2} d e^{2} + b e^{3}\right )} x^{3} + 3 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x\right )} \sqrt {e x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, {\left (2 \, {\left (a c^{2} d e^{2} + a e^{3}\right )} x^{3} + 3 \, {\left (a c^{2} d^{2} e + a d e^{2}\right )} x - {\left (b c d e^{2} x^{2} + b c d^{2} e\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d}}{6 \, {\left (c^{2} d^{5} e + d^{4} e^{2} + {\left (c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{4} + 2 \, {\left (c^{2} d^{4} e^{2} + d^{3} e^{3}\right )} x^{2}\right )}}, \frac {{\left (b c^{2} d^{3} + {\left (b c^{2} d e^{2} + b e^{3}\right )} x^{4} + b d^{2} e + 2 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x^{2}\right )} \sqrt {-e} \arctan \left (\frac {{\left (2 \, c^{2} e x^{2} + c^{2} d - e\right )} \sqrt {c^{2} x^{2} - 1} \sqrt {e x^{2} + d} \sqrt {-e}}{2 \, {\left (c^{3} e^{2} x^{4} - c d e + {\left (c^{3} d e - c e^{2}\right )} x^{2}\right )}}\right ) + {\left (2 \, {\left (b c^{2} d e^{2} + b e^{3}\right )} x^{3} + 3 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x\right )} \sqrt {e x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (2 \, {\left (a c^{2} d e^{2} + a e^{3}\right )} x^{3} + 3 \, {\left (a c^{2} d^{2} e + a d e^{2}\right )} x - {\left (b c d e^{2} x^{2} + b c d^{2} e\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d}}{3 \, {\left (c^{2} d^{5} e + d^{4} e^{2} + {\left (c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{4} + 2 \, {\left (c^{2} d^{4} e^{2} + d^{3} e^{3}\right )} x^{2}\right )}}\right ] \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{\left (d + e x^{2}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]
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