Optimal. Leaf size=371 \[ \frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2}}{256 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2}}{768 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2}}{288 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{256 b c^3 \sqrt {-1+c x} \sqrt {1+c x}} \]
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Rubi [A]
time = 0.46, antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5930, 5926,
5939, 5893, 30, 74, 14, 272, 45} \begin {gather*} -\frac {5 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{256 b c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2}}{256 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2}}{768 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {c x-1} \sqrt {c x+1}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2}}{288 \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 45
Rule 74
Rule 272
Rule 5893
Rule 5926
Rule 5930
Rule 5939
Rubi steps
\begin {align*} \int x^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (-1+c^2 x^2\right )^2 \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int x \left (-1+c^2 x\right )^2 \, dx,x,x^2\right )}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (-1+c^2 x^2\right ) \, dx}{48 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{64 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (x-2 c^2 x^2+c^4 x^3\right ) \, dx,x,x^2\right )}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{64 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-x^3+c^2 x^5\right ) \, dx}{48 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2}}{768 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2}}{288 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{128 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b d^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{128 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2}}{256 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2}}{768 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2}}{288 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{256 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A]
time = 3.14, size = 415, normalized size = 1.12 \begin {gather*} \frac {192 a c d^2 x \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2} \left (-15+118 c^2 x^2-136 c^4 x^4+48 c^6 x^6\right )-2880 a d^{5/2} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {ArcTan}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-576 b d^2 \sqrt {d-c^2 d x^2} \left (8 \cosh ^{-1}(c x)^2+\cosh \left (4 \cosh ^{-1}(c x)\right )-4 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )-64 b d^2 \sqrt {d-c^2 d x^2} \left (-72 \cosh ^{-1}(c x)^2+18 \cosh \left (2 \cosh ^{-1}(c x)\right )-9 \cosh \left (4 \cosh ^{-1}(c x)\right )-2 \cosh \left (6 \cosh ^{-1}(c x)\right )+12 \cosh ^{-1}(c x) \left (-3 \sinh \left (2 \cosh ^{-1}(c x)\right )+3 \sinh \left (4 \cosh ^{-1}(c x)\right )+\sinh \left (6 \cosh ^{-1}(c x)\right )\right )\right )+b d^2 \sqrt {d-c^2 d x^2} \left (-1440 \cosh ^{-1}(c x)^2+576 \cosh \left (2 \cosh ^{-1}(c x)\right )-144 \cosh \left (4 \cosh ^{-1}(c x)\right )-64 \cosh \left (6 \cosh ^{-1}(c x)\right )-9 \cosh \left (8 \cosh ^{-1}(c x)\right )+24 \cosh ^{-1}(c x) \left (-48 \sinh \left (2 \cosh ^{-1}(c x)\right )+24 \sinh \left (4 \cosh ^{-1}(c x)\right )+16 \sinh \left (6 \cosh ^{-1}(c x)\right )+3 \sinh \left (8 \cosh ^{-1}(c x)\right )\right )\right )}{73728 c^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1288\) vs.
\(2(315)=630\).
time = 3.81, size = 1289, normalized size = 3.47
method | result | size |
default | \(-\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{8 c^{2} d}+\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{48 c^{2}}+\frac {5 a d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{192 c^{2}}+\frac {5 a \,d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{128 c^{2}}+\frac {5 a \,d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{128 c^{2} \sqrt {c^{2} d}}+b \left (-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2} d^{2}}{256 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (128 c^{9} x^{9}-320 c^{7} x^{7}+128 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{8} c^{8}+272 c^{5} x^{5}-256 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{6} c^{6}-88 c^{3} x^{3}+160 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{4} c^{4}+8 c x -32 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+8 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{16384 \left (c x +1\right ) c^{3} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (32 c^{7} x^{7}-64 c^{5} x^{5}+32 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{6} c^{6}+38 c^{3} x^{3}-48 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{4} c^{4}-6 c x +18 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+6 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{2304 \left (c x +1\right ) c^{3} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{4} c^{4}+4 c x -8 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+4 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{1024 \left (c x +1\right ) c^{3} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) \left (1+4 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{1024 \left (c x +1\right ) c^{3} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-32 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{6} c^{6}+32 c^{7} x^{7}+48 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{4} c^{4}-64 c^{5} x^{5}-18 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}+38 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-6 c x \right ) \left (1+6 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{2304 \left (c x +1\right ) c^{3} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-128 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{8} c^{8}+128 c^{9} x^{9}+256 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{6} c^{6}-320 c^{7} x^{7}-160 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{4} c^{4}+272 c^{5} x^{5}+32 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-88 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+8 c x \right ) \left (1+8 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{16384 \left (c x +1\right ) c^{3} \left (c x -1\right )}\right )\) | \(1289\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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