Integrand size = 29, antiderivative size = 495 \[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {37384 b^2 d \sqrt {d-c^2 d x^2}}{385875 c^4}+\frac {3358 b^2 d x^2 \sqrt {d-c^2 d x^2}}{385875 c^2}+\frac {484 b^2 d x^4 \sqrt {d-c^2 d x^2}}{42875}-\frac {2}{343} b^2 c^2 d x^6 \sqrt {d-c^2 d x^2}+\frac {4 a b d x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b^2 d x \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{35 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{105 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{175 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{35 c^4}-\frac {d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \]
[Out]
Time = 0.79 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {5930, 5926, 5939, 5915, 5879, 75, 5883, 102, 12, 5912, 14, 5921, 471} \[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{35 c^2}-\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{175 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {2 b d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{105 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{35 c^4}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{49 \sqrt {c x-1} \sqrt {c x+1}}+\frac {4 a b d x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {4 b^2 d x \text {arccosh}(c x) \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3358 b^2 d x^2 \sqrt {d-c^2 d x^2}}{385875 c^2}-\frac {2}{343} b^2 c^2 d x^6 \sqrt {d-c^2 d x^2}+\frac {484 b^2 d x^4 \sqrt {d-c^2 d x^2}}{42875}-\frac {37384 b^2 d \sqrt {d-c^2 d x^2}}{385875 c^4} \]
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Rule 12
Rule 14
Rule 75
Rule 102
Rule 471
Rule 5879
Rule 5883
Rule 5912
Rule 5915
Rule 5921
Rule 5926
Rule 5930
Rule 5939
Rubi steps \begin{align*} \text {integral}& = \frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{7} (3 d) \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx+\frac {\left (2 b c d \sqrt {d-c^2 d x^2}\right ) \int x^4 (-1+c x) (1+c x) (a+b \text {arccosh}(c x)) \, dx}{7 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {\left (3 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{35 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (6 b c d \sqrt {d-c^2 d x^2}\right ) \int x^4 (a+b \text {arccosh}(c x)) \, dx}{35 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b c d \sqrt {d-c^2 d x^2}\right ) \int x^4 \left (-1+c^2 x^2\right ) (a+b \text {arccosh}(c x)) \, dx}{7 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{175 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {\left (2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{35 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b d \sqrt {d-c^2 d x^2}\right ) \int x^2 (a+b \text {arccosh}(c x)) \, dx}{35 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (6 b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{175 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5 \left (-7+5 c^2 x^2\right )}{35 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{7 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {6}{875} b^2 d x^4 \sqrt {d-c^2 d x^2}+\frac {2 b d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{105 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{175 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{35 c^4}-\frac {d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {\left (6 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {4 x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{875 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{105 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (4 b d \sqrt {d-c^2 d x^2}\right ) \int (a+b \text {arccosh}(c x)) \, dx}{35 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5 \left (-7+5 c^2 x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{245 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {2 b^2 d x^2 \sqrt {d-c^2 d x^2}}{315 c^2}+\frac {6}{875} b^2 d x^4 \sqrt {d-c^2 d x^2}-\frac {2}{343} b^2 c^2 d x^6 \sqrt {d-c^2 d x^2}+\frac {4 a b d x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{105 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{175 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{35 c^4}-\frac {d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {\left (24 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{875 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (4 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \text {arccosh}(c x) \, dx}{35 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{315 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (38 b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{1715 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {22 b^2 d x^2 \sqrt {d-c^2 d x^2}}{7875 c^2}+\frac {484 b^2 d x^4 \sqrt {d-c^2 d x^2}}{42875}-\frac {2}{343} b^2 c^2 d x^6 \sqrt {d-c^2 d x^2}+\frac {4 a b d x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b^2 d x \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{35 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{105 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{175 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{35 c^4}-\frac {d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {\left (38 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {4 x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8575 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (8 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{875 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (4 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{315 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (4 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{35 c^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {8 b^2 d \sqrt {d-c^2 d x^2}}{63 c^4}+\frac {22 b^2 d x^2 \sqrt {d-c^2 d x^2}}{7875 c^2}+\frac {484 b^2 d x^4 \sqrt {d-c^2 d x^2}}{42875}-\frac {2}{343} b^2 c^2 d x^6 \sqrt {d-c^2 d x^2}+\frac {4 a b d x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b^2 d x \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{35 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{105 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{175 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{35 c^4}-\frac {d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {\left (152 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8575 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (16 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{875 c^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {856 b^2 d \sqrt {d-c^2 d x^2}}{7875 c^4}+\frac {3358 b^2 d x^2 \sqrt {d-c^2 d x^2}}{385875 c^2}+\frac {484 b^2 d x^4 \sqrt {d-c^2 d x^2}}{42875}-\frac {2}{343} b^2 c^2 d x^6 \sqrt {d-c^2 d x^2}+\frac {4 a b d x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b^2 d x \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{35 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{105 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{175 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{35 c^4}-\frac {d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {\left (152 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{25725 c^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {856 b^2 d \sqrt {d-c^2 d x^2}}{7875 c^4}+\frac {3358 b^2 d x^2 \sqrt {d-c^2 d x^2}}{385875 c^2}+\frac {484 b^2 d x^4 \sqrt {d-c^2 d x^2}}{42875}-\frac {2}{343} b^2 c^2 d x^6 \sqrt {d-c^2 d x^2}+\frac {4 a b d x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b^2 d x \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{35 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{105 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{175 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{35 c^4}-\frac {d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {\left (304 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{25725 c^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {37384 b^2 d \sqrt {d-c^2 d x^2}}{385875 c^4}+\frac {3358 b^2 d x^2 \sqrt {d-c^2 d x^2}}{385875 c^2}+\frac {484 b^2 d x^4 \sqrt {d-c^2 d x^2}}{42875}-\frac {2}{343} b^2 c^2 d x^6 \sqrt {d-c^2 d x^2}+\frac {4 a b d x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b^2 d x \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{35 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{105 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{175 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{35 c^4}-\frac {d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \\ \end{align*}
Time = 0.51 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.53 \[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {d \sqrt {d-c^2 d x^2} \left (11025 a^2 \left (-1+c^2 x^2\right )^3 \left (2+5 c^2 x^2\right )-210 a b c x \sqrt {-1+c x} \sqrt {1+c x} \left (210+35 c^2 x^2-168 c^4 x^4+75 c^6 x^6\right )+2 b^2 \left (-18692+20371 c^2 x^2+499 c^4 x^4-3303 c^6 x^6+1125 c^8 x^8\right )-210 b \left (-105 a \left (-1+c^2 x^2\right )^3 \left (2+5 c^2 x^2\right )+b c x \sqrt {-1+c x} \sqrt {1+c x} \left (210+35 c^2 x^2-168 c^4 x^4+75 c^6 x^6\right )\right ) \text {arccosh}(c x)+11025 b^2 \left (-1+c^2 x^2\right )^3 \left (2+5 c^2 x^2\right ) \text {arccosh}(c x)^2\right )}{385875 c^4 \left (-1+c^2 x^2\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1951\) vs. \(2(423)=846\).
Time = 0.83 (sec) , antiderivative size = 1952, normalized size of antiderivative = 3.94
method | result | size |
default | \(\text {Expression too large to display}\) | \(1952\) |
parts | \(\text {Expression too large to display}\) | \(1952\) |
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Time = 0.28 (sec) , antiderivative size = 432, normalized size of antiderivative = 0.87 \[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {11025 \, {\left (5 \, b^{2} c^{8} d x^{8} - 13 \, b^{2} c^{6} d x^{6} + 9 \, b^{2} c^{4} d x^{4} + b^{2} c^{2} d x^{2} - 2 \, b^{2} d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} - 210 \, {\left (75 \, a b c^{7} d x^{7} - 168 \, a b c^{5} d x^{5} + 35 \, a b c^{3} d x^{3} + 210 \, a b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 210 \, {\left ({\left (75 \, b^{2} c^{7} d x^{7} - 168 \, b^{2} c^{5} d x^{5} + 35 \, b^{2} c^{3} d x^{3} + 210 \, b^{2} c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 105 \, {\left (5 \, a b c^{8} d x^{8} - 13 \, a b c^{6} d x^{6} + 9 \, a b c^{4} d x^{4} + a b c^{2} d x^{2} - 2 \, a b d\right )} \sqrt {-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (1125 \, {\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{8} d x^{8} - 9 \, {\left (15925 \, a^{2} + 734 \, b^{2}\right )} c^{6} d x^{6} + {\left (99225 \, a^{2} + 998 \, b^{2}\right )} c^{4} d x^{4} + {\left (11025 \, a^{2} + 40742 \, b^{2}\right )} c^{2} d x^{2} - 2 \, {\left (11025 \, a^{2} + 18692 \, b^{2}\right )} d\right )} \sqrt {-c^{2} d x^{2} + d}}{385875 \, {\left (c^{6} x^{2} - c^{4}\right )}} \]
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Timed out. \[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Timed out} \]
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Time = 0.31 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.78 \[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {1}{35} \, {\left (\frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} b^{2} \operatorname {arcosh}\left (c x\right )^{2} - \frac {2}{35} \, {\left (\frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} a b \operatorname {arcosh}\left (c x\right ) - \frac {1}{35} \, {\left (\frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} a^{2} - \frac {2}{385875} \, b^{2} {\left (\frac {1125 \, \sqrt {c^{2} x^{2} - 1} c^{4} \sqrt {-d} d x^{6} - 2178 \, \sqrt {c^{2} x^{2} - 1} c^{2} \sqrt {-d} d x^{4} - 1679 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} d x^{2} + \frac {18692 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} d}{c^{2}}}{c^{2}} - \frac {105 \, {\left (75 \, c^{6} \sqrt {-d} d x^{7} - 168 \, c^{4} \sqrt {-d} d x^{5} + 35 \, c^{2} \sqrt {-d} d x^{3} + 210 \, \sqrt {-d} d x\right )} \operatorname {arcosh}\left (c x\right )}{c^{3}}\right )} + \frac {2 \, {\left (75 \, c^{6} \sqrt {-d} d x^{7} - 168 \, c^{4} \sqrt {-d} d x^{5} + 35 \, c^{2} \sqrt {-d} d x^{3} + 210 \, \sqrt {-d} d x\right )} a b}{3675 \, c^{3}} \]
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Exception generated. \[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\int x^3\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]
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