Integrand size = 26, antiderivative size = 486 \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {245 b^2 d^2 x \sqrt {d-c^2 d x^2}}{1152}+\frac {65 b^2 d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 d^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2}+\frac {115 b^2 d^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{1152 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 b c d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{48 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{18 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{48 b c \sqrt {-1+c x} \sqrt {1+c x}} \]
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Time = 0.47 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5897, 5895, 5893, 5883, 92, 54, 5912, 5914, 38} \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{18 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{48 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{48 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {5 b c d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2+\frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {115 b^2 d^2 \text {arccosh}(c x) \sqrt {d-c^2 d x^2}}{1152 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{108} b^2 d^2 x (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2}+\frac {245 b^2 d^2 x \sqrt {d-c^2 d x^2}}{1152}+\frac {65 b^2 d^2 x (1-c x) (c x+1) \sqrt {d-c^2 d x^2}}{1728} \]
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Rule 38
Rule 54
Rule 92
Rule 5883
Rule 5893
Rule 5895
Rule 5897
Rule 5912
Rule 5914
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2+\frac {1}{6} (5 d) \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x (-1+c x)^2 (1+c x)^2 (a+b \text {arccosh}(c x)) \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2+\frac {1}{8} \left (5 d^2\right ) \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x \left (-1+c^2 x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x (-1+c x) (1+c x) (a+b \text {arccosh}(c x)) \, dx}{12 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{18 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{5/2} (1+c x)^{5/2} \, dx}{18 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x \left (-1+c^2 x^2\right ) (a+b \text {arccosh}(c x)) \, dx}{12 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x (a+b \text {arccosh}(c x)) \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {1}{108} b^2 d^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2}-\frac {5 b c d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{48 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{18 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{48 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx}{108 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx}{48 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {5}{32} b^2 d^2 x \sqrt {d-c^2 d x^2}+\frac {65 b^2 d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 d^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2}-\frac {5 b c d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{48 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{18 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{48 b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx}{144 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx}{64 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{32 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {245 b^2 d^2 x \sqrt {d-c^2 d x^2}}{1152}+\frac {65 b^2 d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 d^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2}+\frac {5 b^2 d^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{32 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 b c d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{48 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{18 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{48 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{288 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{128 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {245 b^2 d^2 x \sqrt {d-c^2 d x^2}}{1152}+\frac {65 b^2 d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 d^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2}+\frac {115 b^2 d^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{1152 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 b c d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{48 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{18 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{48 b c \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}
Time = 3.65 (sec) , antiderivative size = 740, normalized size of antiderivative = 1.52 \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {d^2 \left (9504 a^2 c x \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}+9504 a^2 c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}-7488 a^2 c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}-7488 a^2 c^4 x^4 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}+2304 a^2 c^5 x^5 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}+2304 a^2 c^6 x^6 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}-1440 b^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)^3-4320 a^2 \sqrt {d} \sqrt {\frac {-1+c x}{1+c x}} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-4320 a^2 c \sqrt {d} x \sqrt {\frac {-1+c x}{1+c x}} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-3240 a b \sqrt {d-c^2 d x^2} \cosh (2 \text {arccosh}(c x))+324 a b \sqrt {d-c^2 d x^2} \cosh (4 \text {arccosh}(c x))-24 a b \sqrt {d-c^2 d x^2} \cosh (6 \text {arccosh}(c x))+1620 b^2 \sqrt {d-c^2 d x^2} \sinh (2 \text {arccosh}(c x))-81 b^2 \sqrt {d-c^2 d x^2} \sinh (4 \text {arccosh}(c x))+4 b^2 \sqrt {d-c^2 d x^2} \sinh (6 \text {arccosh}(c x))-12 b \sqrt {d-c^2 d x^2} \text {arccosh}(c x) (270 b \cosh (2 \text {arccosh}(c x))-27 b \cosh (4 \text {arccosh}(c x))+2 b \cosh (6 \text {arccosh}(c x))-540 a \sinh (2 \text {arccosh}(c x))+108 a \sinh (4 \text {arccosh}(c x))-12 a \sinh (6 \text {arccosh}(c x)))+72 b \sqrt {d-c^2 d x^2} \text {arccosh}(c x)^2 (-60 a+45 b \sinh (2 \text {arccosh}(c x))-9 b \sinh (4 \text {arccosh}(c x))+b \sinh (6 \text {arccosh}(c x)))\right )}{13824 c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1736\) vs. \(2(422)=844\).
Time = 1.05 (sec) , antiderivative size = 1737, normalized size of antiderivative = 3.57
method | result | size |
default | \(\text {Expression too large to display}\) | \(1737\) |
parts | \(\text {Expression too large to display}\) | \(1737\) |
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\[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \,d x } \]
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Timed out. \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Timed out} \]
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\[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \,d x } \]
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Exception generated. \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]
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