Optimal. Leaf size=323 \[ -\frac {15 b^2 x \left (1+c^2 x^2\right )}{64 c^4 \sqrt {d+c^2 d x^2}}+\frac {b^2 x^3 \left (1+c^2 x^2\right )}{32 c^2 \sqrt {d+c^2 d x^2}}+\frac {15 b^2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{64 c^5 \sqrt {d+c^2 d x^2}}+\frac {3 b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c^3 \sqrt {d+c^2 d x^2}}-\frac {b x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c \sqrt {d+c^2 d x^2}}-\frac {3 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^4 d}+\frac {x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c^5 \sqrt {d+c^2 d x^2}} \]
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Rubi [A]
time = 0.30, antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {5812, 5783,
5776, 327, 221} \begin {gather*} -\frac {b x^4 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{8 c \sqrt {c^2 d x^2+d}}+\frac {x^3 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d}+\frac {\sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c^5 \sqrt {c^2 d x^2+d}}-\frac {3 x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^4 d}+\frac {3 b x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{8 c^3 \sqrt {c^2 d x^2+d}}+\frac {b^2 x^3 \left (c^2 x^2+1\right )}{32 c^2 \sqrt {c^2 d x^2+d}}+\frac {15 b^2 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)}{64 c^5 \sqrt {c^2 d x^2+d}}-\frac {15 b^2 x \left (c^2 x^2+1\right )}{64 c^4 \sqrt {c^2 d x^2+d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 327
Rule 5776
Rule 5783
Rule 5812
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}} \, dx &=\frac {x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d}-\frac {3 \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}} \, dx}{4 c^2}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int x^3 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{2 c \sqrt {d+c^2 d x^2}}\\ &=-\frac {b x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c \sqrt {d+c^2 d x^2}}-\frac {3 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^4 d}+\frac {x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d}+\frac {3 \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}} \, dx}{8 c^4}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^4}{\sqrt {1+c^2 x^2}} \, dx}{8 \sqrt {d+c^2 d x^2}}+\frac {\left (3 b \sqrt {1+c^2 x^2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{4 c^3 \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2 x^3 \left (1+c^2 x^2\right )}{32 c^2 \sqrt {d+c^2 d x^2}}+\frac {3 b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c^3 \sqrt {d+c^2 d x^2}}-\frac {b x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c \sqrt {d+c^2 d x^2}}-\frac {3 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^4 d}+\frac {x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d}+\frac {\left (3 \sqrt {1+c^2 x^2}\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{8 c^4 \sqrt {d+c^2 d x^2}}-\frac {\left (3 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{32 c^2 \sqrt {d+c^2 d x^2}}-\frac {\left (3 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{8 c^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {15 b^2 x \left (1+c^2 x^2\right )}{64 c^4 \sqrt {d+c^2 d x^2}}+\frac {b^2 x^3 \left (1+c^2 x^2\right )}{32 c^2 \sqrt {d+c^2 d x^2}}+\frac {3 b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c^3 \sqrt {d+c^2 d x^2}}-\frac {b x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c \sqrt {d+c^2 d x^2}}-\frac {3 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^4 d}+\frac {x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c^5 \sqrt {d+c^2 d x^2}}+\frac {\left (3 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{64 c^4 \sqrt {d+c^2 d x^2}}+\frac {\left (3 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{16 c^4 \sqrt {d+c^2 d x^2}}\\ &=-\frac {15 b^2 x \left (1+c^2 x^2\right )}{64 c^4 \sqrt {d+c^2 d x^2}}+\frac {b^2 x^3 \left (1+c^2 x^2\right )}{32 c^2 \sqrt {d+c^2 d x^2}}+\frac {15 b^2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{64 c^5 \sqrt {d+c^2 d x^2}}+\frac {3 b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c^3 \sqrt {d+c^2 d x^2}}-\frac {b x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c \sqrt {d+c^2 d x^2}}-\frac {3 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^4 d}+\frac {x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c^5 \sqrt {d+c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.55, size = 268, normalized size = 0.83 \begin {gather*} \frac {32 a^2 c \sqrt {d} x \left (1+c^2 x^2\right ) \left (-3+2 c^2 x^2\right )+96 a^2 \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+b^2 \sqrt {d} \sqrt {1+c^2 x^2} \left (32 \sinh ^{-1}(c x)^3-4 \sinh ^{-1}(c x) \left (-16 \cosh \left (2 \sinh ^{-1}(c x)\right )+\cosh \left (4 \sinh ^{-1}(c x)\right )\right )-32 \sinh \left (2 \sinh ^{-1}(c x)\right )+\sinh \left (4 \sinh ^{-1}(c x)\right )+8 \sinh ^{-1}(c x)^2 \left (-8 \sinh \left (2 \sinh ^{-1}(c x)\right )+\sinh \left (4 \sinh ^{-1}(c x)\right )\right )\right )+4 a b \sqrt {d} \sqrt {1+c^2 x^2} \left (16 \cosh \left (2 \sinh ^{-1}(c x)\right )-\cosh \left (4 \sinh ^{-1}(c x)\right )+4 \sinh ^{-1}(c x) \left (6 \sinh ^{-1}(c x)-8 \sinh \left (2 \sinh ^{-1}(c x)\right )+\sinh \left (4 \sinh ^{-1}(c x)\right )\right )\right )}{256 c^5 \sqrt {d} \sqrt {d+c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(991\) vs.
\(2(283)=566\).
time = 4.65, size = 992, normalized size = 3.07
method | result | size |
default | \(\frac {a^{2} x^{3} \sqrt {c^{2} d \,x^{2}+d}}{4 c^{2} d}-\frac {3 a^{2} x \sqrt {c^{2} d \,x^{2}+d}}{8 c^{4} d}+\frac {3 a^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 c^{4} \sqrt {c^{2} d}}+b^{2} \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{3}}{8 \sqrt {c^{2} x^{2}+1}\, c^{5} d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}+8 \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}+12 c^{3} x^{3}+8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (8 \arcsinh \left (c x \right )^{2}-4 \arcsinh \left (c x \right )+1\right )}{512 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}+2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (2 \arcsinh \left (c x \right )^{2}-2 \arcsinh \left (c x \right )+1\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}-2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (2 \arcsinh \left (c x \right )^{2}+2 \arcsinh \left (c x \right )+1\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}-8 \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}+12 c^{3} x^{3}-8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (8 \arcsinh \left (c x \right )^{2}+4 \arcsinh \left (c x \right )+1\right )}{512 c^{5} d \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2}}{16 \sqrt {c^{2} x^{2}+1}\, c^{5} d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}+8 \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}+12 c^{3} x^{3}+8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+4 \arcsinh \left (c x \right )\right )}{256 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}+2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \arcsinh \left (c x \right )\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}-2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \arcsinh \left (c x \right )\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}-8 \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}+12 c^{3} x^{3}-8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+4 \arcsinh \left (c x \right )\right )}{256 c^{5} d \left (c^{2} x^{2}+1\right )}\right )\) | \(992\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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 \frac {x^{4} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \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 \frac {x^4\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{\sqrt {d\,c^2\,x^2+d}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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