Optimal. Leaf size=495 \[ -\frac {d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^2}-\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{175 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {2 b d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^4}+\frac {4 a b d x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3358 b^2 d x^2 \sqrt {d-c^2 d x^2}}{385875 c^2}-\frac {2}{343} b^2 c^2 d x^6 \sqrt {d-c^2 d x^2}+\frac {484 b^2 d x^4 \sqrt {d-c^2 d x^2}}{42875}-\frac {37384 b^2 d \sqrt {d-c^2 d x^2}}{385875 c^4}+\frac {4 b^2 d x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{35 c^3 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 1.67, antiderivative size = 507, normalized size of antiderivative = 1.02, number of steps used = 26, number of rules used = 13, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {5798, 5745, 5743, 5759, 5718, 5654, 74, 5662, 100, 12, 14, 5731, 460} \[ \frac {4 a b d x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt {c x-1} \sqrt {c x+1}}-\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{175 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{7} d x^4 (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {2 b d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^2}-\frac {2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^4}-\frac {2}{343} b^2 c^2 d x^6 \sqrt {d-c^2 d x^2}+\frac {484 b^2 d x^4 \sqrt {d-c^2 d x^2}}{42875}+\frac {3358 b^2 d x^2 \sqrt {d-c^2 d x^2}}{385875 c^2}-\frac {37384 b^2 d \sqrt {d-c^2 d x^2}}{385875 c^4}+\frac {4 b^2 d x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{35 c^3 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 74
Rule 100
Rule 460
Rule 5654
Rule 5662
Rule 5718
Rule 5731
Rule 5743
Rule 5745
Rule 5759
Rule 5798
Rubi steps
\begin {align*} \int x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int x^3 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{7} d x^4 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {\left (3 d \sqrt {d-c^2 d x^2}\right ) \int x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{7 \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 ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{7 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {2 b c d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{7} d x^4 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\left (3 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )^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 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{35 \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 {16 b c d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{175 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{7} d x^4 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\left (2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )^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 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{35 c \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 {\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 {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 {2 b d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{175 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^4}-\frac {d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{7} d x^4 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^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 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{35 c^3 \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 {2 b^2 d x^2 \sqrt {d-c^2 d x^2}}{315 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 {2 b d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{175 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^4}-\frac {d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{7} d x^4 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^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 (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 \cosh ^{-1}(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 {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} \cosh ^{-1}(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} \left (a+b \cosh ^{-1}(c x)\right )}{105 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{175 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^4}-\frac {d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{7} d x^4 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^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 (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 {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} \cosh ^{-1}(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} \left (a+b \cosh ^{-1}(c x)\right )}{105 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{175 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^4}-\frac {d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{7} d x^4 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^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 {\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} \cosh ^{-1}(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} \left (a+b \cosh ^{-1}(c x)\right )}{105 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{175 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^4}-\frac {d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{7} d x^4 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^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} \cosh ^{-1}(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} \left (a+b \cosh ^{-1}(c x)\right )}{105 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{175 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^4}-\frac {d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{7} d x^4 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2\\ \end {align*}
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Mathematica [A] time = 0.58, size = 262, normalized size = 0.53 \[ -\frac {d \sqrt {d-c^2 d x^2} \left (11025 a^2 \left (5 c^2 x^2+2\right ) \left (c^2 x^2-1\right )^3-210 a b c x \sqrt {c x-1} \sqrt {c x+1} \left (75 c^6 x^6-168 c^4 x^4+35 c^2 x^2+210\right )-210 b \cosh ^{-1}(c x) \left (b c x \sqrt {c x-1} \sqrt {c x+1} \left (75 c^6 x^6-168 c^4 x^4+35 c^2 x^2+210\right )-105 a \left (c^2 x^2-1\right )^3 \left (5 c^2 x^2+2\right )\right )+11025 b^2 \left (5 c^2 x^2+2\right ) \left (c^2 x^2-1\right )^3 \cosh ^{-1}(c x)^2+2 b^2 \left (1125 c^8 x^8-3303 c^6 x^6+499 c^4 x^4+20371 c^2 x^2-18692\right )\right )}{385875 c^4 \left (c^2 x^2-1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 432, normalized size = 0.87 \[ -\frac {11025 \, {\left (5 \, b^{2} c^{8} d x^{8} - 13 \, b^{2} c^{6} d x^{6} + 9 \, b^{2} c^{4} d x^{4} + b^{2} c^{2} d x^{2} - 2 \, b^{2} d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} - 210 \, {\left (75 \, a b c^{7} d x^{7} - 168 \, a b c^{5} d x^{5} + 35 \, a b c^{3} d x^{3} + 210 \, a b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 210 \, {\left ({\left (75 \, b^{2} c^{7} d x^{7} - 168 \, b^{2} c^{5} d x^{5} + 35 \, b^{2} c^{3} d x^{3} + 210 \, b^{2} c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 105 \, {\left (5 \, a b c^{8} d x^{8} - 13 \, a b c^{6} d x^{6} + 9 \, a b c^{4} d x^{4} + a b c^{2} d x^{2} - 2 \, a b d\right )} \sqrt {-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (1125 \, {\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{8} d x^{8} - 9 \, {\left (15925 \, a^{2} + 734 \, b^{2}\right )} c^{6} d x^{6} + {\left (99225 \, a^{2} + 998 \, b^{2}\right )} c^{4} d x^{4} + {\left (11025 \, a^{2} + 40742 \, b^{2}\right )} c^{2} d x^{2} - 2 \, {\left (11025 \, a^{2} + 18692 \, b^{2}\right )} d\right )} \sqrt {-c^{2} d x^{2} + d}}{385875 \, {\left (c^{6} x^{2} - c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.77, size = 1952, normalized size = 3.94 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.20, size = 388, normalized size = 0.78 \[ -\frac {1}{35} \, {\left (\frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} b^{2} \operatorname {arcosh}\left (c x\right )^{2} - \frac {2}{35} \, {\left (\frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} a b \operatorname {arcosh}\left (c x\right ) - \frac {1}{35} \, {\left (\frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} a^{2} - \frac {2}{385875} \, b^{2} {\left (\frac {1125 \, \sqrt {c^{2} x^{2} - 1} c^{4} \sqrt {-d} d x^{6} - 2178 \, \sqrt {c^{2} x^{2} - 1} c^{2} \sqrt {-d} d x^{4} - 1679 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} d x^{2} + \frac {18692 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} d}{c^{2}}}{c^{2}} - \frac {105 \, {\left (75 \, c^{6} \sqrt {-d} d x^{7} - 168 \, c^{4} \sqrt {-d} d x^{5} + 35 \, c^{2} \sqrt {-d} d x^{3} + 210 \, \sqrt {-d} d x\right )} \operatorname {arcosh}\left (c x\right )}{c^{3}}\right )} + \frac {2 \, {\left (75 \, c^{6} \sqrt {-d} d x^{7} - 168 \, c^{4} \sqrt {-d} d x^{5} + 35 \, c^{2} \sqrt {-d} d x^{3} + 210 \, \sqrt {-d} d x\right )} a b}{3675 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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