Optimal. Leaf size=348 \[ \frac {2 b d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 d}-\frac {4 b c d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b c^3 d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b^2 d \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{125 c^2 (1-c x) (c x+1)}-\frac {8 b^2 d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{225 c^2 (1-c x) (c x+1)}-\frac {16 b^2 d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{75 c^2 (1-c x) (c x+1)} \]
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Rubi [A] time = 0.55, antiderivative size = 361, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {5798, 5718, 194, 5680, 12, 520, 1247, 698} \[ \frac {2 b c^3 d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt {c x-1} \sqrt {c x+1}}-\frac {4 b c d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {d (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2}-\frac {2 b^2 d \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{125 c^2 (1-c x) (c x+1)}-\frac {8 b^2 d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{225 c^2 (1-c x) (c x+1)}-\frac {16 b^2 d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{75 c^2 (1-c x) (c x+1)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 194
Rule 520
Rule 698
Rule 1247
Rule 5680
Rule 5718
Rule 5798
Rubi steps
\begin {align*} \int x \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 (-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 {d (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2}+\frac {\left (2 b d \sqrt {d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{5 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {2 b d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b c d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2}-\frac {\left (2 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{15 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{5 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {2 b d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b c d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2}-\frac {\left (2 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{75 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {2 b d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b c d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2}-\frac {\left (2 b^2 d \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{\sqrt {-1+c^2 x^2}} \, dx}{75 (-1+c x) (1+c x)}\\ &=\frac {2 b d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b c d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2}-\frac {\left (b^2 d \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {15-10 c^2 x+3 c^4 x^2}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{75 (-1+c x) (1+c x)}\\ &=\frac {2 b d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b c d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2}-\frac {\left (b^2 d \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {8}{\sqrt {-1+c^2 x}}-4 \sqrt {-1+c^2 x}+3 \left (-1+c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{75 (-1+c x) (1+c x)}\\ &=-\frac {16 b^2 d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{75 c^2 (1-c x) (1+c x)}-\frac {8 b^2 d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{225 c^2 (1-c x) (1+c x)}-\frac {2 b^2 d \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{125 c^2 (1-c x) (1+c x)}+\frac {2 b d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b c d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2}\\ \end {align*}
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Mathematica [A] time = 0.53, size = 208, normalized size = 0.60 \[ -\frac {d \sqrt {d-c^2 d x^2} \left (225 a^2 \left (c^2 x^2-1\right )^3-30 a b c x \sqrt {c x-1} \sqrt {c x+1} \left (3 c^4 x^4-10 c^2 x^2+15\right )-30 b \cosh ^{-1}(c x) \left (b c x \sqrt {c x-1} \sqrt {c x+1} \left (3 c^4 x^4-10 c^2 x^2+15\right )-15 a \left (c^2 x^2-1\right )^3\right )+225 b^2 \left (c^2 x^2-1\right )^3 \cosh ^{-1}(c x)^2+2 b^2 \left (9 c^6 x^6-47 c^4 x^4+187 c^2 x^2-149\right )\right )}{1125 c^2 \left (c^2 x^2-1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 367, normalized size = 1.05 \[ -\frac {225 \, {\left (b^{2} c^{6} d x^{6} - 3 \, b^{2} c^{4} d x^{4} + 3 \, b^{2} c^{2} d x^{2} - b^{2} d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} - 30 \, {\left (3 \, a b c^{5} d x^{5} - 10 \, a b c^{3} d x^{3} + 15 \, a b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 30 \, {\left ({\left (3 \, b^{2} c^{5} d x^{5} - 10 \, b^{2} c^{3} d x^{3} + 15 \, b^{2} c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 15 \, {\left (a b c^{6} d x^{6} - 3 \, a b c^{4} d x^{4} + 3 \, a b c^{2} d x^{2} - a b d\right )} \sqrt {-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (9 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{6} d x^{6} - {\left (675 \, a^{2} + 94 \, b^{2}\right )} c^{4} d x^{4} + {\left (675 \, a^{2} + 374 \, b^{2}\right )} c^{2} d x^{2} - {\left (225 \, a^{2} + 298 \, b^{2}\right )} d\right )} \sqrt {-c^{2} d x^{2} + d}}{1125 \, {\left (c^{4} x^{2} - c^{2}\right )}} \]
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 = 0.45, size = 1270, normalized size = 3.65 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.75, size = 278, normalized size = 0.80 \[ -\frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} b^{2} \operatorname {arcosh}\left (c x\right )^{2}}{5 \, c^{2} d} - \frac {2}{1125} \, b^{2} {\left (\frac {9 \, \sqrt {c^{2} x^{2} - 1} c^{2} \sqrt {-d} d^{2} x^{4} - 38 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} d^{2} x^{2} + \frac {149 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} d^{2}}{c^{2}}}{d} - \frac {15 \, {\left (3 \, c^{4} \sqrt {-d} d^{2} x^{5} - 10 \, c^{2} \sqrt {-d} d^{2} x^{3} + 15 \, \sqrt {-d} d^{2} x\right )} \operatorname {arcosh}\left (c x\right )}{c d}\right )} - \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a b \operatorname {arcosh}\left (c x\right )}{5 \, c^{2} d} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a^{2}}{5 \, c^{2} d} + \frac {2 \, {\left (3 \, c^{4} \sqrt {-d} d^{2} x^{5} - 10 \, c^{2} \sqrt {-d} d^{2} x^{3} + 15 \, \sqrt {-d} d^{2} x\right )} a b}{75 \, c d} \]
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 \[ \int x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]
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 \[ \int x \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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