Optimal. Leaf size=486 \[ \frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{18 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{48 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{48 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5 b c d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{108} b^2 d^2 x (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2}+\frac {245 b^2 d^2 x \sqrt {d-c^2 d x^2}}{1152}+\frac {65 b^2 d^2 x (1-c x) (c x+1) \sqrt {d-c^2 d x^2}}{1728}+\frac {115 b^2 d^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{1152 c \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 0.87, antiderivative size = 517, normalized size of antiderivative = 1.06, number of steps used = 17, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5713, 5685, 5683, 5676, 5662, 90, 52, 5716, 38} \[ \frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{18 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{48 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{48 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} d^2 x (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5}{24} d^2 x (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5 b c d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{108} b^2 d^2 x (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2}+\frac {245 b^2 d^2 x \sqrt {d-c^2 d x^2}}{1152}+\frac {65 b^2 d^2 x (1-c x) (c x+1) \sqrt {d-c^2 d x^2}}{1728}+\frac {115 b^2 d^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{1152 c \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 38
Rule 52
Rule 90
Rule 5662
Rule 5676
Rule 5683
Rule 5685
Rule 5713
Rule 5716
Rubi steps
\begin {align*} \int \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{6} d^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{6 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x \left (-1+c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{18 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{24} d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{6} d^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{5/2} (1+c x)^{5/2} \, dx}{18 \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 \left (-1+c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{12 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{108} b^2 d^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2}+\frac {5 b d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{48 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{18 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5}{24} d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{6} d^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx}{108 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx}{48 \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 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {65 b^2 d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 d^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2}-\frac {5 b c d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{48 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{18 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5}{24} d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{6} d^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{48 b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx}{144 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx}{64 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {245 b^2 d^2 x \sqrt {d-c^2 d x^2}}{1152}+\frac {65 b^2 d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 d^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2}-\frac {5 b c d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{48 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{18 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5}{24} d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{6} d^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{48 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{288 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{128 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{32 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {245 b^2 d^2 x \sqrt {d-c^2 d x^2}}{1152}+\frac {65 b^2 d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 d^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2}+\frac {115 b^2 d^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{1152 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 b c d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{48 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{18 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5}{24} d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{6} d^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{48 b c \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 3.49, size = 740, normalized size = 1.52 \[ \frac {d^2 \left (9504 a^2 c^2 x^2 \sqrt {\frac {c x-1}{c x+1}} \sqrt {d-c^2 d x^2}+9504 a^2 c x \sqrt {\frac {c x-1}{c x+1}} \sqrt {d-c^2 d x^2}-4320 a^2 c \sqrt {d} x \sqrt {\frac {c x-1}{c x+1}} \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )-4320 a^2 \sqrt {d} \sqrt {\frac {c x-1}{c x+1}} \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )+2304 a^2 c^6 x^6 \sqrt {\frac {c x-1}{c x+1}} \sqrt {d-c^2 d x^2}+2304 a^2 c^5 x^5 \sqrt {\frac {c x-1}{c x+1}} \sqrt {d-c^2 d x^2}-7488 a^2 c^4 x^4 \sqrt {\frac {c x-1}{c x+1}} \sqrt {d-c^2 d x^2}-7488 a^2 c^3 x^3 \sqrt {\frac {c x-1}{c x+1}} \sqrt {d-c^2 d x^2}-3240 a b \sqrt {d-c^2 d x^2} \cosh \left (2 \cosh ^{-1}(c x)\right )+324 a b \sqrt {d-c^2 d x^2} \cosh \left (4 \cosh ^{-1}(c x)\right )-24 a b \sqrt {d-c^2 d x^2} \cosh \left (6 \cosh ^{-1}(c x)\right )-12 b \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x) \left (-540 a \sinh \left (2 \cosh ^{-1}(c x)\right )+108 a \sinh \left (4 \cosh ^{-1}(c x)\right )-12 a \sinh \left (6 \cosh ^{-1}(c x)\right )+270 b \cosh \left (2 \cosh ^{-1}(c x)\right )-27 b \cosh \left (4 \cosh ^{-1}(c x)\right )+2 b \cosh \left (6 \cosh ^{-1}(c x)\right )\right )+72 b \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)^2 \left (-60 a+45 b \sinh \left (2 \cosh ^{-1}(c x)\right )-9 b \sinh \left (4 \cosh ^{-1}(c x)\right )+b \sinh \left (6 \cosh ^{-1}(c x)\right )\right )-1440 b^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)^3+1620 b^2 \sqrt {d-c^2 d x^2} \sinh \left (2 \cosh ^{-1}(c x)\right )-81 b^2 \sqrt {d-c^2 d x^2} \sinh \left (4 \cosh ^{-1}(c x)\right )+4 b^2 \sqrt {d-c^2 d x^2} \sinh \left (6 \cosh ^{-1}(c x)\right )\right )}{13824 c \sqrt {\frac {c x-1}{c x+1}} (c x+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c^{4} d^{2} x^{4} - 2 \, a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} + {\left (b^{2} c^{4} d^{2} x^{4} - 2 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, {\left (a b c^{4} d^{2} x^{4} - 2 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-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 = 0.46, size = 1053, normalized size = 2.17 \[ \text {result too large to display} \]
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 \[ \frac {1}{48} \, {\left (8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x + 10 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x + 15 \, \sqrt {-c^{2} d x^{2} + d} d^{2} x + \frac {15 \, d^{\frac {5}{2}} \arcsin \left (c x\right )}{c}\right )} a^{2} + \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} b^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2} + 2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a b \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )\,{d x} \]
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 \[ \int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]
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 \[ \int \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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