Optimal. Leaf size=449 \[ -\frac {27 c \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{256 a \sqrt {1+a^2 x^2}}-\frac {9 a c x^2 \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{32 \sqrt {1+a^2 x^2}}-\frac {3 c \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac {3 c \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{20 a \sqrt {1+a^2 x^2}}+\frac {3 c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {Erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a \sqrt {1+a^2 x^2}}+\frac {3 c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}}+\frac {3 c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {Erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a \sqrt {1+a^2 x^2}}+\frac {3 c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}} \]
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Rubi [A]
time = 0.39, antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps
used = 26, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {5786, 5785,
5783, 5777, 5819, 3393, 3388, 2211, 2235, 2236, 5798, 5791} \begin {gather*} \frac {3 \sqrt {\pi } c \sqrt {a^2 c x^2+c} \text {Erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a \sqrt {a^2 x^2+1}}+\frac {3 \sqrt {\frac {\pi }{2}} c \sqrt {a^2 c x^2+c} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a \sqrt {a^2 x^2+1}}+\frac {3 \sqrt {\pi } c \sqrt {a^2 c x^2+c} \text {Erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a \sqrt {a^2 x^2+1}}+\frac {3 \sqrt {\frac {\pi }{2}} c \sqrt {a^2 c x^2+c} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a \sqrt {a^2 x^2+1}}+\frac {3 c \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^{5/2}}{20 a \sqrt {a^2 x^2+1}}+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac {3}{8} c x \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}-\frac {3 c \left (a^2 x^2+1\right )^{3/2} \sqrt {a^2 c x^2+c} \sqrt {\sinh ^{-1}(a x)}}{32 a}-\frac {9 a c x^2 \sqrt {a^2 c x^2+c} \sqrt {\sinh ^{-1}(a x)}}{32 \sqrt {a^2 x^2+1}}-\frac {27 c \sqrt {a^2 c x^2+c} \sqrt {\sinh ^{-1}(a x)}}{256 a \sqrt {a^2 x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 3393
Rule 5777
Rule 5783
Rule 5785
Rule 5786
Rule 5791
Rule 5798
Rule 5819
Rubi steps
\begin {align*} \int \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2} \, dx &=\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac {1}{4} (3 c) \int \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2} \, dx-\frac {\left (3 a c \sqrt {c+a^2 c x^2}\right ) \int x \left (1+a^2 x^2\right ) \sqrt {\sinh ^{-1}(a x)} \, dx}{8 \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} \sqrt {\sinh ^{-1}(a x)}}{32 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \int \frac {\left (1+a^2 x^2\right )^{3/2}}{\sqrt {\sinh ^{-1}(a x)}} \, dx}{64 \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \int \frac {\sinh ^{-1}(a x)^{3/2}}{\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 \sqrt {\sinh ^{-1}(a x)} \, dx}{16 \sqrt {1+a^2 x^2}}\\ &=-\frac {9 a c x^2 \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{32 \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} \sqrt {\sinh ^{-1}(a x)}}{32 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac {3 c \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{20 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh ^4(x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt {1+a^2 x^2}}+\frac {\left (9 a^2 c \sqrt {c+a^2 c x^2}\right ) \int \frac {x^2}{\sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}} \, dx}{64 \sqrt {1+a^2 x^2}}\\ &=-\frac {9 a c x^2 \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{32 \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} \sqrt {\sinh ^{-1}(a x)}}{32 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac {3 c \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{20 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \left (\frac {3}{8 \sqrt {x}}+\frac {\cosh (2 x)}{2 \sqrt {x}}+\frac {\cosh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt {1+a^2 x^2}}+\frac {\left (9 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh ^2(x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt {1+a^2 x^2}}\\ &=\frac {9 c \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{256 a \sqrt {1+a^2 x^2}}-\frac {9 a c x^2 \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{32 \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} \sqrt {\sinh ^{-1}(a x)}}{32 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac {3 c \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{20 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{512 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{128 a \sqrt {1+a^2 x^2}}-\frac {\left (9 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}-\frac {\cosh (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt {1+a^2 x^2}}\\ &=-\frac {27 c \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{256 a \sqrt {1+a^2 x^2}}-\frac {9 a c x^2 \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{32 \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} \sqrt {\sinh ^{-1}(a x)}}{32 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac {3 c \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{20 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1024 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1024 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{256 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{256 a \sqrt {1+a^2 x^2}}+\frac {\left (9 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{128 a \sqrt {1+a^2 x^2}}\\ &=-\frac {27 c \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{256 a \sqrt {1+a^2 x^2}}-\frac {9 a c x^2 \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{32 \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} \sqrt {\sinh ^{-1}(a x)}}{32 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac {3 c \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{20 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{512 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{512 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{128 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{128 a \sqrt {1+a^2 x^2}}+\frac {\left (9 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{256 a \sqrt {1+a^2 x^2}}+\frac {\left (9 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{256 a \sqrt {1+a^2 x^2}}\\ &=-\frac {27 c \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{256 a \sqrt {1+a^2 x^2}}-\frac {9 a c x^2 \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{32 \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} \sqrt {\sinh ^{-1}(a x)}}{32 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac {3 c \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{20 a \sqrt {1+a^2 x^2}}+\frac {3 c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a \sqrt {1+a^2 x^2}}+\frac {3 c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{256 a \sqrt {1+a^2 x^2}}+\frac {3 c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a \sqrt {1+a^2 x^2}}+\frac {3 c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{256 a \sqrt {1+a^2 x^2}}+\frac {\left (9 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{128 a \sqrt {1+a^2 x^2}}+\frac {\left (9 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{128 a \sqrt {1+a^2 x^2}}\\ &=-\frac {27 c \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{256 a \sqrt {1+a^2 x^2}}-\frac {9 a c x^2 \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{32 \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} \sqrt {\sinh ^{-1}(a x)}}{32 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac {3 c \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{20 a \sqrt {1+a^2 x^2}}+\frac {3 c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a \sqrt {1+a^2 x^2}}+\frac {3 c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}}+\frac {3 c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a \sqrt {1+a^2 x^2}}+\frac {3 c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 186, normalized size = 0.41 \begin {gather*} \frac {c \sqrt {c+a^2 c x^2} \left (384 \sinh ^{-1}(a x)^3-480 \sinh ^{-1}(a x) \cosh \left (2 \sinh ^{-1}(a x)\right )+60 \sqrt {2 \pi } \sqrt {\sinh ^{-1}(a x)} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )+60 \sqrt {2 \pi } \sqrt {\sinh ^{-1}(a x)} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )+5 \sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {5}{2},-4 \sinh ^{-1}(a x)\right )-5 \sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {5}{2},4 \sinh ^{-1}(a x)\right )+640 \sinh ^{-1}(a x)^2 \sinh \left (2 \sinh ^{-1}(a x)\right )\right )}{2560 a \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arcsinh \left (a x \right )^{\frac {3}{2}}\, dx\]
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(-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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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {\mathrm {asinh}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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