Optimal. Leaf size=278 \[ \frac {1}{6} x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{24 c^2}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c^5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^4}-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x^4 \sqrt {d-c^2 d x^2}}{96 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x^2 \sqrt {d-c^2 d x^2}}{32 c^3 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 0.78, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {5798, 5743, 5759, 5676, 30} \[ \frac {1}{6} x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{24 c^2}-\frac {x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^4}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c^5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x^4 \sqrt {d-c^2 d x^2}}{96 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x^2 \sqrt {d-c^2 d x^2}}{32 c^3 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 30
Rule 5676
Rule 5743
Rule 5759
Rule 5798
Rubi steps
\begin {align*} \int x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {\sqrt {d-c^2 d x^2} \int x^4 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{6 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int x^5 \, dx}{6 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{24 c^2}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{24 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b x^4 \sqrt {d-c^2 d x^2}}{96 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^4}-\frac {x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{24 c^2}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{16 c^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{16 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b x^2 \sqrt {d-c^2 d x^2}}{32 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x^4 \sqrt {d-c^2 d x^2}}{96 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^4}-\frac {x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{24 c^2}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c^5 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 1.22, size = 198, normalized size = 0.71 \[ \frac {-144 a \sqrt {d} \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )+48 a c x \left (8 c^4 x^4-2 c^2 x^2-3\right ) \sqrt {d-c^2 d x^2}+\frac {b \sqrt {d-c^2 d x^2} \left (-72 \cosh ^{-1}(c x)^2+18 \cosh \left (2 \cosh ^{-1}(c x)\right )-9 \cosh \left (4 \cosh ^{-1}(c x)\right )-2 \cosh \left (6 \cosh ^{-1}(c x)\right )+12 \cosh ^{-1}(c x) \left (-3 \sinh \left (2 \cosh ^{-1}(c x)\right )+3 \sinh \left (4 \cosh ^{-1}(c x)\right )+\sinh \left (6 \cosh ^{-1}(c x)\right )\right )\right )}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}}{2304 c^5} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{4} \operatorname {arcosh}\left (c x\right ) + a x^{4}\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]
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 \[ \int \sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.81, size = 449, normalized size = 1.62 \[ -\frac {a \,x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{6 c^{2} d}-\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{8 c^{4} d}+\frac {a x \sqrt {-c^{2} d \,x^{2}+d}}{16 c^{4}}+\frac {a d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{16 c^{4} \sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2}}{32 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{5}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} \mathrm {arccosh}\left (c x \right ) x^{7}}{6 \left (c x +1\right ) \left (c x -1\right )}-\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x^{5}}{24 \left (c x +1\right ) \left (c x -1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x^{3}}{48 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x}{16 \left (c x +1\right ) c^{4} \left (c x -1\right )}-\frac {25 b \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2304 \sqrt {c x +1}\, c^{5} \sqrt {c x -1}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c \,x^{6}}{36 \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{4}}{96 \sqrt {c x +1}\, c \sqrt {c x -1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{2}}{32 \sqrt {c x +1}\, c^{3} \sqrt {c x -1}} \]
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 \[ -\frac {1}{48} \, {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}}{c^{2} d} - \frac {3 \, \sqrt {-c^{2} d x^{2} + d} x}{c^{4}} + \frac {6 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x}{c^{4} d} - \frac {3 \, \sqrt {d} \arcsin \left (c x\right )}{c^{5}}\right )} a + b \int \sqrt {-c^{2} d x^{2} + d} x^{4} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )\,{d x} \]
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 \[ \int x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2} \,d x \]
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 \[ \int x^{4} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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