Integrand size = 29, antiderivative size = 421 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {4144 b^2 (1-c x) (1+c x)}{3375 c^6 \sqrt {d-c^2 d x^2}}-\frac {272 b^2 x^2 (1-c x) (1+c x)}{3375 c^4 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x^4 (1-c x) (1+c x)}{125 c^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {8 b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{45 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b x^5 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^6 d}-\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^4 d}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{5 c^2 d} \]
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Time = 0.40 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {5938, 5914, 5879, 75, 5883, 102, 12} \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {2 b x^5 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{25 c \sqrt {d-c^2 d x^2}}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{5 c^2 d}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^6 d}-\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^4 d}-\frac {8 b x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{45 c^3 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {c x-1} \sqrt {c x+1}}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x)}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x^4 (1-c x) (c x+1)}{125 c^2 \sqrt {d-c^2 d x^2}}-\frac {4144 b^2 (1-c x) (c x+1)}{3375 c^6 \sqrt {d-c^2 d x^2}}-\frac {272 b^2 x^2 (1-c x) (c x+1)}{3375 c^4 \sqrt {d-c^2 d x^2}} \]
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Rule 12
Rule 75
Rule 102
Rule 5879
Rule 5883
Rule 5914
Rule 5938
Rubi steps \begin{align*} \text {integral}& = -\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{5 c^2 d}+\frac {4 \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{5 c^2}-\frac {\left (2 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x^4 (a+b \text {arccosh}(c x)) \, dx}{5 c \sqrt {d-c^2 d x^2}} \\ & = -\frac {2 b x^5 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c \sqrt {d-c^2 d x^2}}-\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^4 d}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{5 c^2 d}+\frac {8 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{15 c^4}+\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^5}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{25 \sqrt {d-c^2 d x^2}}-\frac {\left (8 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x^2 (a+b \text {arccosh}(c x)) \, dx}{15 c^3 \sqrt {d-c^2 d x^2}} \\ & = -\frac {2 b^2 x^4 (1-c x) (1+c x)}{125 c^2 \sqrt {d-c^2 d x^2}}-\frac {8 b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{45 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b x^5 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^6 d}-\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^4 d}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{5 c^2 d}-\frac {\left (16 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int (a+b \text {arccosh}(c x)) \, dx}{15 c^5 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {4 x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{125 c^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{45 c^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {8 b^2 x^2 (1-c x) (1+c x)}{135 c^4 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x^4 (1-c x) (1+c x)}{125 c^2 \sqrt {d-c^2 d x^2}}-\frac {8 b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{45 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b x^5 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^6 d}-\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^4 d}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{5 c^2 d}-\frac {\left (16 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \text {arccosh}(c x) \, dx}{15 c^5 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{135 c^4 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{125 c^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {272 b^2 x^2 (1-c x) (1+c x)}{3375 c^4 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x^4 (1-c x) (1+c x)}{125 c^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {8 b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{45 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b x^5 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^6 d}-\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^4 d}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{5 c^2 d}+\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{375 c^4 \sqrt {d-c^2 d x^2}}+\frac {\left (16 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{135 c^4 \sqrt {d-c^2 d x^2}}+\frac {\left (16 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{15 c^4 \sqrt {d-c^2 d x^2}} \\ & = -\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {32 b^2 (1-c x) (1+c x)}{27 c^6 \sqrt {d-c^2 d x^2}}-\frac {272 b^2 x^2 (1-c x) (1+c x)}{3375 c^4 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x^4 (1-c x) (1+c x)}{125 c^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {8 b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{45 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b x^5 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^6 d}-\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^4 d}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{5 c^2 d}+\frac {\left (16 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{375 c^4 \sqrt {d-c^2 d x^2}} \\ & = -\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {4144 b^2 (1-c x) (1+c x)}{3375 c^6 \sqrt {d-c^2 d x^2}}-\frac {272 b^2 x^2 (1-c x) (1+c x)}{3375 c^4 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x^4 (1-c x) (1+c x)}{125 c^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {8 b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{45 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b x^5 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^6 d}-\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{15 c^4 d}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{5 c^2 d} \\ \end{align*}
Time = 0.48 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.61 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (30 a b c x \sqrt {-1+c x} \sqrt {1+c x} \left (120+20 c^2 x^2+9 c^4 x^4\right )-225 a^2 \left (-8+4 c^2 x^2+c^4 x^4+3 c^6 x^6\right )-2 b^2 \left (-2072+1936 c^2 x^2+109 c^4 x^4+27 c^6 x^6\right )+30 b \left (b c x \sqrt {-1+c x} \sqrt {1+c x} \left (120+20 c^2 x^2+9 c^4 x^4\right )-15 a \left (-8+4 c^2 x^2+c^4 x^4+3 c^6 x^6\right )\right ) \text {arccosh}(c x)-225 b^2 \left (-8+4 c^2 x^2+c^4 x^4+3 c^6 x^6\right ) \text {arccosh}(c x)^2\right )}{3375 c^6 d (-1+c x) (1+c x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1313\) vs. \(2(365)=730\).
Time = 0.91 (sec) , antiderivative size = 1314, normalized size of antiderivative = 3.12
method | result | size |
default | \(\text {Expression too large to display}\) | \(1314\) |
parts | \(\text {Expression too large to display}\) | \(1314\) |
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Time = 0.27 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.83 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {225 \, {\left (3 \, b^{2} c^{6} x^{6} + b^{2} c^{4} x^{4} + 4 \, b^{2} c^{2} x^{2} - 8 \, b^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} - 30 \, {\left (9 \, a b c^{5} x^{5} + 20 \, a b c^{3} x^{3} + 120 \, a b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 30 \, {\left ({\left (9 \, b^{2} c^{5} x^{5} + 20 \, b^{2} c^{3} x^{3} + 120 \, b^{2} c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 15 \, {\left (3 \, a b c^{6} x^{6} + a b c^{4} x^{4} + 4 \, a b c^{2} x^{2} - 8 \, a b\right )} \sqrt {-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (27 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{6} x^{6} + {\left (225 \, a^{2} + 218 \, b^{2}\right )} c^{4} x^{4} + 4 \, {\left (225 \, a^{2} + 968 \, b^{2}\right )} c^{2} x^{2} - 1800 \, a^{2} - 4144 \, b^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{3375 \, {\left (c^{8} d x^{2} - c^{6} d\right )}} \]
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\[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^{5} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 407, normalized size of antiderivative = 0.97 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {1}{15} \, {\left (\frac {3 \, \sqrt {-c^{2} d x^{2} + d} x^{4}}{c^{2} d} + \frac {4 \, \sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {-c^{2} d x^{2} + d}}{c^{6} d}\right )} b^{2} \operatorname {arcosh}\left (c x\right )^{2} - \frac {2}{15} \, {\left (\frac {3 \, \sqrt {-c^{2} d x^{2} + d} x^{4}}{c^{2} d} + \frac {4 \, \sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {-c^{2} d x^{2} + d}}{c^{6} d}\right )} a b \operatorname {arcosh}\left (c x\right ) - \frac {1}{15} \, {\left (\frac {3 \, \sqrt {-c^{2} d x^{2} + d} x^{4}}{c^{2} d} + \frac {4 \, \sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {-c^{2} d x^{2} + d}}{c^{6} d}\right )} a^{2} - \frac {2}{3375} \, b^{2} {\left (\frac {27 \, \sqrt {c^{2} x^{2} - 1} c^{2} \sqrt {-d} x^{4} + 136 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} x^{2} + \frac {2072 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d}}{c^{2}}}{c^{4} d} - \frac {15 \, {\left (9 \, c^{4} \sqrt {-d} x^{5} + 20 \, c^{2} \sqrt {-d} x^{3} + 120 \, \sqrt {-d} x\right )} \operatorname {arcosh}\left (c x\right )}{c^{5} d}\right )} + \frac {2 \, {\left (9 \, c^{4} \sqrt {-d} x^{5} + 20 \, c^{2} \sqrt {-d} x^{3} + 120 \, \sqrt {-d} x\right )} a b}{225 \, c^{5} d} \]
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Exception generated. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^5\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{\sqrt {d-c^2\,d\,x^2}} \,d x \]
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