Integrand size = 26, antiderivative size = 204 \[ \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {1}{4} b^2 x \sqrt {d-c^2 d x^2}+\frac {b^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{4 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {-1+c x} \sqrt {1+c x}} \]
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Time = 0.16 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {5895, 5893, 5883, 92, 54} \[ \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b^2 \text {arccosh}(c x) \sqrt {d-c^2 d x^2}}{4 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} b^2 x \sqrt {d-c^2 d x^2} \]
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Rule 54
Rule 92
Rule 5883
Rule 5893
Rule 5895
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int x (a+b \text {arccosh}(c x)) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 c^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {1}{4} b^2 x \sqrt {d-c^2 d x^2}-\frac {b c x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {1}{4} b^2 x \sqrt {d-c^2 d x^2}+\frac {b^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{4 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{6 b c \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}
Time = 1.16 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.15 \[ \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {1}{24} \left (12 a^2 x \sqrt {d-c^2 d x^2}-\frac {12 a^2 \sqrt {d} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )}{c}-\frac {6 a b \sqrt {d-c^2 d x^2} \left (2 \text {arccosh}(c x)^2+\cosh (2 \text {arccosh}(c x))-2 \text {arccosh}(c x) \sinh (2 \text {arccosh}(c x))\right )}{c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+\frac {b^2 \sqrt {d-c^2 d x^2} \left (-4 \text {arccosh}(c x)^3-6 \text {arccosh}(c x) \cosh (2 \text {arccosh}(c x))+\left (3+6 \text {arccosh}(c x)^2\right ) \sinh (2 \text {arccosh}(c x))\right )}{c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(526\) vs. \(2(172)=344\).
Time = 0.77 (sec) , antiderivative size = 527, normalized size of antiderivative = 2.58
method | result | size |
default | \(\frac {a^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {a^{2} d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{3}}{6 \sqrt {c x -1}\, \sqrt {c x +1}\, c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (2 \operatorname {arccosh}\left (c x \right )^{2}-2 \,\operatorname {arccosh}\left (c x \right )+1\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (2 \operatorname {arccosh}\left (c x \right )^{2}+2 \,\operatorname {arccosh}\left (c x \right )+1\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2}}{4 \sqrt {c x -1}\, \sqrt {c x +1}\, c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}\right )\) | \(527\) |
parts | \(\frac {a^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {a^{2} d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{3}}{6 \sqrt {c x -1}\, \sqrt {c x +1}\, c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (2 \operatorname {arccosh}\left (c x \right )^{2}-2 \,\operatorname {arccosh}\left (c x \right )+1\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (2 \operatorname {arccosh}\left (c x \right )^{2}+2 \,\operatorname {arccosh}\left (c x \right )+1\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2}}{4 \sqrt {c x -1}\, \sqrt {c x +1}\, c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c}\right )\) | \(527\) |
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\[ \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \,d x } \]
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\[ \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}\, dx \]
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\[ \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \,d x } \]
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Exception generated. \[ \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2} \,d x \]
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