Optimal. Leaf size=264 \[ \frac {3 c \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{4 a \sqrt {1+a^2 x^2}}+\frac {c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {Erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{32 a \sqrt {1+a^2 x^2}}+\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{4 a \sqrt {1+a^2 x^2}}+\frac {c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {Erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{32 a \sqrt {1+a^2 x^2}}+\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{4 a \sqrt {1+a^2 x^2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5791, 3393,
3388, 2211, 2235, 2236} \begin {gather*} \frac {\sqrt {\pi } c \sqrt {a^2 c x^2+c} \text {Erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{32 a \sqrt {a^2 x^2+1}}+\frac {\sqrt {\frac {\pi }{2}} c \sqrt {a^2 c x^2+c} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{4 a \sqrt {a^2 x^2+1}}+\frac {\sqrt {\pi } c \sqrt {a^2 c x^2+c} \text {Erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{32 a \sqrt {a^2 x^2+1}}+\frac {\sqrt {\frac {\pi }{2}} c \sqrt {a^2 c x^2+c} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{4 a \sqrt {a^2 x^2+1}}+\frac {3 c \sqrt {a^2 c x^2+c} \sqrt {\sinh ^{-1}(a x)}}{4 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 5791
Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\sinh ^{-1}(a x)}} \, dx &=\frac {\left (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}{\sqrt {1+a^2 x^2}}\\ &=\frac {\left (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 )}{a \sqrt {1+a^2 x^2}}\\ &=\frac {\left (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 )}{a \sqrt {1+a^2 x^2}}\\ &=\frac {3 c \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{4 a \sqrt {1+a^2 x^2}}+\frac {\left (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 )}{8 a \sqrt {1+a^2 x^2}}+\frac {\left (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 )}{2 a \sqrt {1+a^2 x^2}}\\ &=\frac {3 c \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{4 a \sqrt {1+a^2 x^2}}+\frac {\left (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 )}{16 a \sqrt {1+a^2 x^2}}+\frac {\left (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 )}{16 a \sqrt {1+a^2 x^2}}+\frac {\left (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 )}{4 a \sqrt {1+a^2 x^2}}+\frac {\left (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 )}{4 a \sqrt {1+a^2 x^2}}\\ &=\frac {3 c \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{4 a \sqrt {1+a^2 x^2}}+\frac {\left (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 )}{8 a \sqrt {1+a^2 x^2}}+\frac {\left (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 )}{8 a \sqrt {1+a^2 x^2}}+\frac {\left (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 )}{2 a \sqrt {1+a^2 x^2}}+\frac {\left (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 )}{2 a \sqrt {1+a^2 x^2}}\\ &=\frac {3 c \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{4 a \sqrt {1+a^2 x^2}}+\frac {c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{32 a \sqrt {1+a^2 x^2}}+\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{4 a \sqrt {1+a^2 x^2}}+\frac {c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{32 a \sqrt {1+a^2 x^2}}+\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{4 a \sqrt {1+a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 141, normalized size = 0.53 \begin {gather*} \frac {c \sqrt {c+a^2 c x^2} \left (\sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 \sinh ^{-1}(a x)\right )+4 \sqrt {2} \sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 \sinh ^{-1}(a x)\right )+\sqrt {\sinh ^{-1}(a x)} \left (24 \sqrt {\sinh ^{-1}(a x)}-4 \sqrt {2} \Gamma \left (\frac {1}{2},2 \sinh ^{-1}(a x)\right )-\Gamma \left (\frac {1}{2},4 \sinh ^{-1}(a x)\right )\right )\right )}{32 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 \[\int \frac {\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{\sqrt {\arcsinh \left (a x \right )}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}{\sqrt {\operatorname {asinh}{\left (a x \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 {{\left (c\,a^2\,x^2+c\right )}^{3/2}}{\sqrt {\mathrm {asinh}\left (a\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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