Integrand size = 21, antiderivative size = 348 \[ \int \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3 \, dx=-\frac {51 a c x^2 \sqrt {c+a^2 c x^2}}{128 \sqrt {1+a^2 x^2}}-\frac {3 a^3 c x^4 \sqrt {c+a^2 c x^2}}{128 \sqrt {1+a^2 x^2}}+\frac {45}{64} c x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)+\frac {3}{32} c x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)-\frac {27 c \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{128 a \sqrt {1+a^2 x^2}}-\frac {9 a c x^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{16 \sqrt {1+a^2 x^2}}-\frac {3 c \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{16 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3+\frac {3 c \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^4}{32 a \sqrt {1+a^2 x^2}} \]
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Time = 0.25 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {5786, 5785, 5783, 5776, 5812, 30, 5798, 14} \[ \int \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3 \, dx=-\frac {9 a c x^2 \text {arcsinh}(a x)^2 \sqrt {a^2 c x^2+c}}{16 \sqrt {a^2 x^2+1}}+\frac {1}{4} x \text {arcsinh}(a x)^3 \left (a^2 c x^2+c\right )^{3/2}+\frac {3}{8} c x \text {arcsinh}(a x)^3 \sqrt {a^2 c x^2+c}+\frac {45}{64} c x \text {arcsinh}(a x) \sqrt {a^2 c x^2+c}+\frac {3}{32} c x \left (a^2 x^2+1\right ) \text {arcsinh}(a x) \sqrt {a^2 c x^2+c}+\frac {3 c \text {arcsinh}(a x)^4 \sqrt {a^2 c x^2+c}}{32 a \sqrt {a^2 x^2+1}}-\frac {3 c \left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)^2 \sqrt {a^2 c x^2+c}}{16 a}-\frac {27 c \text {arcsinh}(a x)^2 \sqrt {a^2 c x^2+c}}{128 a \sqrt {a^2 x^2+1}}-\frac {51 a c x^2 \sqrt {a^2 c x^2+c}}{128 \sqrt {a^2 x^2+1}}-\frac {3 a^3 c x^4 \sqrt {a^2 c x^2+c}}{128 \sqrt {a^2 x^2+1}} \]
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Rule 14
Rule 30
Rule 5776
Rule 5783
Rule 5785
Rule 5786
Rule 5798
Rule 5812
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3+\frac {1}{4} (3 c) \int \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3 \, dx-\frac {\left (3 a c \sqrt {c+a^2 c x^2}\right ) \int x \left (1+a^2 x^2\right ) \text {arcsinh}(a x)^2 \, dx}{4 \sqrt {1+a^2 x^2}} \\ & = -\frac {3 c \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{16 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \int \left (1+a^2 x^2\right )^{3/2} \text {arcsinh}(a x) \, dx}{8 \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \int \frac {\text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{8 \sqrt {1+a^2 x^2}}-\frac {\left (9 a c \sqrt {c+a^2 c x^2}\right ) \int x \text {arcsinh}(a x)^2 \, dx}{8 \sqrt {1+a^2 x^2}} \\ & = \frac {3}{32} c x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)-\frac {9 a c x^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{16 \sqrt {1+a^2 x^2}}-\frac {3 c \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{16 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3+\frac {3 c \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^4}{32 a \sqrt {1+a^2 x^2}}+\frac {\left (9 c \sqrt {c+a^2 c x^2}\right ) \int \sqrt {1+a^2 x^2} \text {arcsinh}(a x) \, dx}{32 \sqrt {1+a^2 x^2}}-\frac {\left (3 a c \sqrt {c+a^2 c x^2}\right ) \int x \left (1+a^2 x^2\right ) \, dx}{32 \sqrt {1+a^2 x^2}}+\frac {\left (9 a^2 c \sqrt {c+a^2 c x^2}\right ) \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{8 \sqrt {1+a^2 x^2}} \\ & = \frac {45}{64} c x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)+\frac {3}{32} c x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)-\frac {9 a c x^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{16 \sqrt {1+a^2 x^2}}-\frac {3 c \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{16 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3+\frac {3 c \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^4}{32 a \sqrt {1+a^2 x^2}}+\frac {\left (9 c \sqrt {c+a^2 c x^2}\right ) \int \frac {\text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{64 \sqrt {1+a^2 x^2}}-\frac {\left (9 c \sqrt {c+a^2 c x^2}\right ) \int \frac {\text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{16 \sqrt {1+a^2 x^2}}-\frac {\left (3 a c \sqrt {c+a^2 c x^2}\right ) \int \left (x+a^2 x^3\right ) \, dx}{32 \sqrt {1+a^2 x^2}}-\frac {\left (9 a c \sqrt {c+a^2 c x^2}\right ) \int x \, dx}{64 \sqrt {1+a^2 x^2}}-\frac {\left (9 a c \sqrt {c+a^2 c x^2}\right ) \int x \, dx}{16 \sqrt {1+a^2 x^2}} \\ & = -\frac {51 a c x^2 \sqrt {c+a^2 c x^2}}{128 \sqrt {1+a^2 x^2}}-\frac {3 a^3 c x^4 \sqrt {c+a^2 c x^2}}{128 \sqrt {1+a^2 x^2}}+\frac {45}{64} c x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)+\frac {3}{32} c x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)-\frac {27 c \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{128 a \sqrt {1+a^2 x^2}}-\frac {9 a c x^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{16 \sqrt {1+a^2 x^2}}-\frac {3 c \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{16 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3+\frac {3 c \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^4}{32 a \sqrt {1+a^2 x^2}} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.39 \[ \int \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3 \, dx=\frac {c \sqrt {c+a^2 c x^2} \left (96 \text {arcsinh}(a x)^4-24 \text {arcsinh}(a x)^2 (16 \cosh (2 \text {arcsinh}(a x))+\cosh (4 \text {arcsinh}(a x)))-3 (64 \cosh (2 \text {arcsinh}(a x))+\cosh (4 \text {arcsinh}(a x)))+32 \text {arcsinh}(a x)^3 (8 \sinh (2 \text {arcsinh}(a x))+\sinh (4 \text {arcsinh}(a x)))+12 \text {arcsinh}(a x) (32 \sinh (2 \text {arcsinh}(a x))+\sinh (4 \text {arcsinh}(a x)))\right )}{1024 a \sqrt {1+a^2 x^2}} \]
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Time = 0.21 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.39
method | result | size |
default | \(\frac {3 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (a x \right )^{4} c}{32 \sqrt {a^{2} x^{2}+1}\, a}+\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (8 a^{5} x^{5}+8 a^{4} x^{4} \sqrt {a^{2} x^{2}+1}+12 a^{3} x^{3}+8 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+4 a x +\sqrt {a^{2} x^{2}+1}\right ) \left (32 \operatorname {arcsinh}\left (a x \right )^{3}-24 \operatorname {arcsinh}\left (a x \right )^{2}+12 \,\operatorname {arcsinh}\left (a x \right )-3\right ) c}{2048 \left (a^{2} x^{2}+1\right ) a}+\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (2 a^{3} x^{3}+2 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+2 a x +\sqrt {a^{2} x^{2}+1}\right ) \left (4 \operatorname {arcsinh}\left (a x \right )^{3}-6 \operatorname {arcsinh}\left (a x \right )^{2}+6 \,\operatorname {arcsinh}\left (a x \right )-3\right ) c}{32 \left (a^{2} x^{2}+1\right ) a}+\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (2 a^{3} x^{3}-2 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+2 a x -\sqrt {a^{2} x^{2}+1}\right ) \left (4 \operatorname {arcsinh}\left (a x \right )^{3}+6 \operatorname {arcsinh}\left (a x \right )^{2}+6 \,\operatorname {arcsinh}\left (a x \right )+3\right ) c}{32 \left (a^{2} x^{2}+1\right ) a}+\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (8 a^{5} x^{5}-8 a^{4} x^{4} \sqrt {a^{2} x^{2}+1}+12 a^{3} x^{3}-8 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+4 a x -\sqrt {a^{2} x^{2}+1}\right ) \left (32 \operatorname {arcsinh}\left (a x \right )^{3}+24 \operatorname {arcsinh}\left (a x \right )^{2}+12 \,\operatorname {arcsinh}\left (a x \right )+3\right ) c}{2048 \left (a^{2} x^{2}+1\right ) a}\) | \(484\) |
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\[ \int \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (a x\right )^{3} \,d x } \]
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\[ \int \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3 \, dx=\int \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {asinh}^{3}{\left (a x \right )}\, dx \]
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Exception generated. \[ \int \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3 \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3 \, dx=\int {\mathrm {asinh}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]
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