Optimal. Leaf size=182 \[ \frac {2 \sqrt {2 \pi } \sqrt {a^2 c x^2+c} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{3 a \sqrt {a^2 x^2+1}}+\frac {2 \sqrt {2 \pi } \sqrt {a^2 c x^2+c} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{3 a \sqrt {a^2 x^2+1}}-\frac {8 x \sqrt {a^2 c x^2+c}}{3 \sqrt {\sinh ^{-1}(a x)}}-\frac {2 \sqrt {a^2 x^2+1} \sqrt {a^2 c x^2+c}}{3 a \sinh ^{-1}(a x)^{3/2}} \]
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Rubi [A] time = 0.11, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5696, 5665, 3307, 2180, 2204, 2205} \[ \frac {2 \sqrt {2 \pi } \sqrt {a^2 c x^2+c} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{3 a \sqrt {a^2 x^2+1}}+\frac {2 \sqrt {2 \pi } \sqrt {a^2 c x^2+c} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{3 a \sqrt {a^2 x^2+1}}-\frac {8 x \sqrt {a^2 c x^2+c}}{3 \sqrt {\sinh ^{-1}(a x)}}-\frac {2 \sqrt {a^2 x^2+1} \sqrt {a^2 c x^2+c}}{3 a \sinh ^{-1}(a x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 5665
Rule 5696
Rubi steps
\begin {align*} \int \frac {\sqrt {c+a^2 c x^2}}{\sinh ^{-1}(a x)^{5/2}} \, dx &=-\frac {2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}+\frac {\left (4 a \sqrt {c+a^2 c x^2}\right ) \int \frac {x}{\sinh ^{-1}(a x)^{3/2}} \, dx}{3 \sqrt {1+a^2 x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {8 x \sqrt {c+a^2 c x^2}}{3 \sqrt {\sinh ^{-1}(a x)}}+\frac {\left (8 \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a \sqrt {1+a^2 x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {8 x \sqrt {c+a^2 c x^2}}{3 \sqrt {\sinh ^{-1}(a x)}}+\frac {\left (4 \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {\left (4 \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a \sqrt {1+a^2 x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {8 x \sqrt {c+a^2 c x^2}}{3 \sqrt {\sinh ^{-1}(a x)}}+\frac {\left (8 \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {\left (8 \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {8 x \sqrt {c+a^2 c x^2}}{3 \sqrt {\sinh ^{-1}(a x)}}+\frac {2 \sqrt {2 \pi } \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {2 \sqrt {2 \pi } \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 122, normalized size = 0.67 \[ -\frac {2 \sqrt {a^2 c x^2+c} \left (a^2 x^2+4 a x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)+\sqrt {2} \left (-\sinh ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-2 \sinh ^{-1}(a x)\right )+\sqrt {2} \sinh ^{-1}(a x)^{3/2} \Gamma \left (\frac {1}{2},2 \sinh ^{-1}(a x)\right )+1\right )}{3 a \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {\sqrt {a^{2} c x^{2} + c}}{\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a^{2} c \,x^{2}+c}}{\arcsinh \left (a x \right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {\sqrt {a^{2} c x^{2} + c}}{\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {c\,a^2\,x^2+c}}{{\mathrm {asinh}\left (a\,x\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\sqrt {c \left (a^{2} x^{2} + 1\right )}}{\operatorname {asinh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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