Optimal. Leaf size=454 \[ \frac {3 b d^2 x^2 \sqrt {d-c^2 d x^2}}{512 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 x^4 \sqrt {d-c^2 d x^2}}{512 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{256 c^4}-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^2}+\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{512 b c^5 \sqrt {-1+c x} \sqrt {1+c x}} \]
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Rubi [A]
time = 0.60, antiderivative size = 454, normalized size of antiderivative = 1.00, number of steps
used = 16, 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 {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^2}+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{512 b c^5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{256 c^4}-\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 x^4 \sqrt {d-c^2 d x^2}}{512 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b d^2 x^2 \sqrt {d-c^2 d x^2}}{512 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \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^4 \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^4 (-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}{10} d^2 x^5 (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 (d^2 \sqrt {d-c^2 d x^2}\right ) \int x^4 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^5 \left (-1+c^2 x^2\right )^2 \, dx}{10 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{16} d^2 x^5 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{10} d^2 x^5 (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 (3 d^2 \sqrt {d-c^2 d x^2}\right ) \int x^4 \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^2 \left (-1+c^2 x\right )^2 \, dx,x,x^2\right )}{20 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^5 \left (-1+c^2 x^2\right ) \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{16} d^2 x^5 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{10} d^2 x^5 (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 (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^5 \, dx}{32 \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^2-2 c^2 x^3+c^4 x^4\right ) \, dx,x,x^2\right )}{20 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-x^5+c^2 x^7\right ) \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^2}+\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{16} d^2 x^5 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{10} d^2 x^5 (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 (3 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}{128 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{128 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b d^2 x^4 \sqrt {d-c^2 d x^2}}{512 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{256 c^4}-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^2}+\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{16} d^2 x^5 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{10} d^2 x^5 (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 (3 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}{256 c^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b d^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{256 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {3 b d^2 x^2 \sqrt {d-c^2 d x^2}}{512 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 x^4 \sqrt {d-c^2 d x^2}}{512 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{256 c^4}-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^2}+\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{16} d^2 x^5 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{10} d^2 x^5 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{512 b c^5 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A]
time = 5.04, size = 500, normalized size = 1.10 \begin {gather*} \frac {2880 a c d^2 x \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2} \left (-15-10 c^2 x^2+248 c^4 x^4-336 c^6 x^6+128 c^8 x^8\right )-43200 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 )+1600 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 )+100 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 )+b d^2 \sqrt {d-c^2 d x^2} \left (-50400 \cosh ^{-1}(c x)^2+25200 \cosh \left (2 \cosh ^{-1}(c x)\right )-3600 \cosh \left (4 \cosh ^{-1}(c x)\right )-2600 \cosh \left (6 \cosh ^{-1}(c x)\right )-675 \cosh \left (8 \cosh ^{-1}(c x)\right )-72 \cosh \left (10 \cosh ^{-1}(c x)\right )+120 \cosh ^{-1}(c x) \left (-420 \sinh \left (2 \cosh ^{-1}(c x)\right )+120 \sinh \left (4 \cosh ^{-1}(c x)\right )+130 \sinh \left (6 \cosh ^{-1}(c x)\right )+45 \sinh \left (8 \cosh ^{-1}(c x)\right )+6 \sinh \left (10 \cosh ^{-1}(c x)\right )\right )\right )}{3686400 c^5 \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. \(1742\) vs.
\(2(386)=772\).
time = 4.54, size = 1743, normalized size = 3.84
method | result | size |
default | \(\text {Expression too large to display}\) | \(1743\) |
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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^4\,\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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