Integrand size = 25, antiderivative size = 127 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {b x \sqrt {-1+c x} \sqrt {1+c x}}{6 c d \left (d-c^2 d x^2\right )^{3/2}}+\frac {a+b \text {arccosh}(c x)}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{6 c^2 d^2 \sqrt {d-c^2 d x^2}} \]
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Time = 0.07 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {5914, 41, 205, 213} \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {a+b \text {arccosh}(c x)}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}(c x)}{6 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x \sqrt {c x-1} \sqrt {c x+1}}{6 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}} \]
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Rule 41
Rule 205
Rule 213
Rule 5914
Rubi steps \begin{align*} \text {integral}& = \frac {a+b \text {arccosh}(c x)}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{(-1+c x)^2 (1+c x)^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {a+b \text {arccosh}(c x)}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{\left (-1+c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b x \sqrt {-1+c x} \sqrt {1+c x}}{6 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{6 c d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b x \sqrt {-1+c x} \sqrt {1+c x}}{6 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{6 c^2 d^2 \sqrt {d-c^2 d x^2}} \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.28 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (-2 b c x+2 b c^3 x^3+4 a \sqrt {-1+c x} \sqrt {1+c x}+4 b \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)+b \left (-1+c^2 x^2\right )^2 \log (-1+c x)-b \log (1+c x)+2 b c^2 x^2 \log (1+c x)-b c^4 x^4 \log (1+c x)\right )}{12 c^2 d^3 (-1+c x)^{5/2} (1+c x)^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(219\) vs. \(2(107)=214\).
Time = 1.14 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.73
method | result | size |
default | \(\frac {a}{3 c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +2 \,\operatorname {arccosh}\left (c x \right )\right )}{6 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{2}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{6 d^{3} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )}{6 d^{3} c^{2} \left (c^{2} x^{2}-1\right )}\right )\) | \(220\) |
parts | \(\frac {a}{3 c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +2 \,\operatorname {arccosh}\left (c x \right )\right )}{6 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{2}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{6 d^{3} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )}{6 d^{3} c^{2} \left (c^{2} x^{2}-1\right )}\right )\) | \(220\) |
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none
Time = 0.30 (sec) , antiderivative size = 421, normalized size of antiderivative = 3.31 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\left [\frac {4 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} b c x + 8 \, \sqrt {-c^{2} d x^{2} + d} b \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {-d} \log \left (-\frac {c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} - 4 \, {\left (c^{3} x^{3} + c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} \sqrt {-d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right ) + 8 \, \sqrt {-c^{2} d x^{2} + d} a}{24 \, {\left (c^{6} d^{3} x^{4} - 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )}}, \frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} b c x - {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {d} \arctan \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} c \sqrt {d} x}{c^{4} d x^{4} - d}\right ) + 4 \, \sqrt {-c^{2} d x^{2} + d} b \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 4 \, \sqrt {-c^{2} d x^{2} + d} a}{12 \, {\left (c^{6} d^{3} x^{4} - 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )}}\right ] \]
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\[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]
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