Integrand size = 29, antiderivative size = 355 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {15 b^2 x (1-c x) (1+c x)}{64 c^4 \sqrt {d-c^2 d x^2}}-\frac {b^2 x^3 (1-c x) (1+c x)}{32 c^2 \sqrt {d-c^2 d x^2}}+\frac {15 b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{64 c^5 \sqrt {d-c^2 d x^2}}-\frac {3 b x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{8 c^3 \sqrt {d-c^2 d x^2}}-\frac {b x^4 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{8 c \sqrt {d-c^2 d x^2}}-\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{8 c^4 d}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 c^2 d}+\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^3}{8 b c^5 \sqrt {d-c^2 d x^2}} \]
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Time = 0.32 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {5938, 5892, 5883, 92, 54, 102, 12} \[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {b x^4 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{8 c \sqrt {d-c^2 d x^2}}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 c^2 d}+\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^3}{8 b c^5 \sqrt {d-c^2 d x^2}}-\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{8 c^4 d}-\frac {3 b x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{8 c^3 \sqrt {d-c^2 d x^2}}+\frac {15 b^2 \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x)}{64 c^5 \sqrt {d-c^2 d x^2}}-\frac {b^2 x^3 (1-c x) (c x+1)}{32 c^2 \sqrt {d-c^2 d x^2}}-\frac {15 b^2 x (1-c x) (c x+1)}{64 c^4 \sqrt {d-c^2 d x^2}} \]
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Rule 12
Rule 54
Rule 92
Rule 102
Rule 5883
Rule 5892
Rule 5938
Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 c^2 d}+\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{4 c^2}-\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x^3 (a+b \text {arccosh}(c x)) \, dx}{2 c \sqrt {d-c^2 d x^2}} \\ & = -\frac {b x^4 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{8 c \sqrt {d-c^2 d x^2}}-\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{8 c^4 d}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 c^2 d}+\frac {3 \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{8 c^4}+\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^4}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 \sqrt {d-c^2 d x^2}}-\frac {\left (3 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x (a+b \text {arccosh}(c x)) \, dx}{4 c^3 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b^2 x^3 (1-c x) (1+c x)}{32 c^2 \sqrt {d-c^2 d x^2}}-\frac {3 b x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{8 c^3 \sqrt {d-c^2 d x^2}}-\frac {b x^4 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{8 c \sqrt {d-c^2 d x^2}}-\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{8 c^4 d}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 c^2 d}+\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^3}{8 b c^5 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {3 x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{32 c^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 c^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {3 b^2 x (1-c x) (1+c x)}{16 c^4 \sqrt {d-c^2 d x^2}}-\frac {b^2 x^3 (1-c x) (1+c x)}{32 c^2 \sqrt {d-c^2 d x^2}}-\frac {3 b x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{8 c^3 \sqrt {d-c^2 d x^2}}-\frac {b x^4 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{8 c \sqrt {d-c^2 d x^2}}-\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{8 c^4 d}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 c^2 d}+\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^3}{8 b c^5 \sqrt {d-c^2 d x^2}}+\frac {\left (3 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{16 c^4 \sqrt {d-c^2 d x^2}}+\frac {\left (3 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{32 c^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {15 b^2 x (1-c x) (1+c x)}{64 c^4 \sqrt {d-c^2 d x^2}}-\frac {b^2 x^3 (1-c x) (1+c x)}{32 c^2 \sqrt {d-c^2 d x^2}}+\frac {3 b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{16 c^5 \sqrt {d-c^2 d x^2}}-\frac {3 b x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{8 c^3 \sqrt {d-c^2 d x^2}}-\frac {b x^4 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{8 c \sqrt {d-c^2 d x^2}}-\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{8 c^4 d}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 c^2 d}+\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^3}{8 b c^5 \sqrt {d-c^2 d x^2}}+\frac {\left (3 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{64 c^4 \sqrt {d-c^2 d x^2}} \\ & = -\frac {15 b^2 x (1-c x) (1+c x)}{64 c^4 \sqrt {d-c^2 d x^2}}-\frac {b^2 x^3 (1-c x) (1+c x)}{32 c^2 \sqrt {d-c^2 d x^2}}+\frac {15 b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{64 c^5 \sqrt {d-c^2 d x^2}}-\frac {3 b x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{8 c^3 \sqrt {d-c^2 d x^2}}-\frac {b x^4 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{8 c \sqrt {d-c^2 d x^2}}-\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{8 c^4 d}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 c^2 d}+\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^3}{8 b c^5 \sqrt {d-c^2 d x^2}} \\ \end{align*}
Time = 1.54 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.83 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {32 a^2 c \sqrt {d} x \left (-1+c^2 x^2\right ) \left (3+2 c^2 x^2\right )-96 a^2 \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+b^2 \sqrt {d} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (32 \text {arccosh}(c x)^3-4 \text {arccosh}(c x) (16 \cosh (2 \text {arccosh}(c x))+\cosh (4 \text {arccosh}(c x)))+32 \sinh (2 \text {arccosh}(c x))+\sinh (4 \text {arccosh}(c x))+8 \text {arccosh}(c x)^2 (8 \sinh (2 \text {arccosh}(c x))+\sinh (4 \text {arccosh}(c x)))\right )-4 a b \sqrt {d} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) (16 \cosh (2 \text {arccosh}(c x))+\cosh (4 \text {arccosh}(c x))-4 \text {arccosh}(c x) (6 \text {arccosh}(c x)+8 \sinh (2 \text {arccosh}(c x))+\sinh (4 \text {arccosh}(c x))))}{256 c^5 \sqrt {d} \sqrt {d-c^2 d x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1091\) vs. \(2(307)=614\).
Time = 0.93 (sec) , antiderivative size = 1092, normalized size of antiderivative = 3.08
method | result | size |
default | \(\text {Expression too large to display}\) | \(1092\) |
parts | \(\text {Expression too large to display}\) | \(1092\) |
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\[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{4}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]
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\[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^{4} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
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\[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{4}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]
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\[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{4}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]
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Timed out. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{\sqrt {d-c^2\,d\,x^2}} \,d x \]
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