Optimal. Leaf size=336 \[ \frac {15}{64} b^2 d x \sqrt {d-c^2 d x^2}+\frac {1}{32} b^2 d x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}+\frac {9 b^2 d \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{64 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3}{8} d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c \sqrt {-1+c x} \sqrt {1+c x}} \]
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Rubi [A]
time = 0.27, antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5897, 5895,
5893, 5883, 92, 54, 5912, 5914, 38} \begin {gather*} -\frac {d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {3}{8} d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {b d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 b c d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {15}{64} b^2 d x \sqrt {d-c^2 d x^2}+\frac {1}{32} b^2 d x (1-c x) (c x+1) \sqrt {d-c^2 d x^2}+\frac {9 b^2 d \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{64 c \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 38
Rule 54
Rule 92
Rule 5883
Rule 5893
Rule 5895
Rule 5897
Rule 5912
Rule 5914
Rubi steps
\begin {align*} \int \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 (-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 {1}{4} d x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {\left (3 d \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}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d \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}{2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3}{8} d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{4} d x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\left (3 d \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}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b^2 d \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b c d \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{32} b^2 d x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}-\frac {3 b c d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3}{8} d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{4} d x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx}{32 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {15}{64} b^2 d x \sqrt {d-c^2 d x^2}+\frac {1}{32} b^2 d x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}-\frac {3 b c d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3}{8} d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{4} d x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{64 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {15}{64} b^2 d x \sqrt {d-c^2 d x^2}+\frac {1}{32} b^2 d x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}+\frac {9 b^2 d \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{64 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3}{8} d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{4} d x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A]
time = 1.98, size = 374, normalized size = 1.11 \begin {gather*} \frac {-96 a^2 c d x \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (-5+2 c^2 x^2\right ) \sqrt {d-c^2 d x^2}-288 a^2 d^{3/2} \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 )-192 a b d \sqrt {d-c^2 d x^2} \left (\cosh \left (2 \cosh ^{-1}(c x)\right )+2 \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-\sinh \left (2 \cosh ^{-1}(c x)\right )\right )\right )-32 b^2 d \sqrt {d-c^2 d x^2} \left (4 \cosh ^{-1}(c x)^3+6 \cosh ^{-1}(c x) \cosh \left (2 \cosh ^{-1}(c x)\right )-3 \left (1+2 \cosh ^{-1}(c x)^2\right ) \sinh \left (2 \cosh ^{-1}(c x)\right )\right )+12 a b d \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 )+b^2 d \sqrt {d-c^2 d x^2} \left (32 \cosh ^{-1}(c x)^3+12 \cosh ^{-1}(c x) \cosh \left (4 \cosh ^{-1}(c x)\right )-3 \left (1+8 \cosh ^{-1}(c x)^2\right ) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )}{768 c \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. \(1060\) vs.
\(2(288)=576\).
time = 2.59, size = 1061, normalized size = 3.16
method | result | size |
default | \(\frac {x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} a^{2}}{4}+\frac {3 a^{2} d x \sqrt {-c^{2} d \,x^{2}+d}}{8}+\frac {3 a^{2} d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{3} d}{8 \sqrt {c x -1}\, \sqrt {c x +1}\, c}-\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 (8 \mathrm {arccosh}\left (c x \right )^{2}-4 \,\mathrm {arccosh}\left (c x \right )+1\right ) d}{512 \left (c x +1\right ) \left (c x -1\right ) c}+\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 (2 \mathrm {arccosh}\left (c x \right )^{2}-2 \,\mathrm {arccosh}\left (c x \right )+1\right ) d}{16 \left (c x +1\right ) \left (c x -1\right ) c}+\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 (2 \mathrm {arccosh}\left (c x \right )^{2}+2 \,\mathrm {arccosh}\left (c x \right )+1\right ) d}{16 \left (c x +1\right ) \left (c x -1\right ) c}-\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 (8 \mathrm {arccosh}\left (c x \right )^{2}+4 \,\mathrm {arccosh}\left (c x \right )+1\right ) d}{512 \left (c x +1\right ) \left (c x -1\right ) c}\right )+2 a b \left (-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2} d}{16 \sqrt {c x -1}\, \sqrt {c x +1}\, c}-\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}{256 \left (c x +1\right ) \left (c x -1\right ) c}+\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}{16 \left (c x +1\right ) \left (c x -1\right ) c}+\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}{16 \left (c x +1\right ) \left (c x -1\right ) c}-\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}{256 \left (c x +1\right ) \left (c x -1\right ) c}\right )\) | \(1061\) |
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \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 \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \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 {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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