Optimal. Leaf size=175 \[ -\frac {2 d \left (1+c^2 x^2\right )^{3/2}}{b c x \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {d e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2}}+\frac {d e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2}}-\frac {2 d \text {Int}\left (\frac {1}{x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}},x\right )}{b c} \]
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Rubi [A]
time = 0.48, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {d+c^2 d x^2}{x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {d+c^2 d x^2}{x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac {2 d \left (1+c^2 x^2\right )^{3/2}}{b c x \sqrt {a+b \sinh ^{-1}(c x)}}-\frac {(2 d) \int \frac {\sqrt {1+c^2 x^2}}{x^2 \sqrt {a+b \sinh ^{-1}(c x)}} \, dx}{b c}+\frac {(4 c d) \int \frac {\sqrt {1+c^2 x^2}}{\sqrt {a+b \sinh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac {2 d \left (1+c^2 x^2\right )^{3/2}}{b c x \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {(4 d) \text {Subst}\left (\int \frac {\cosh ^2(x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b}-\frac {(2 d) \int \left (\frac {c^2}{\sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {1}{x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}\right ) \, dx}{b c}\\ &=-\frac {2 d \left (1+c^2 x^2\right )^{3/2}}{b c x \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {(4 d) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {a+b x}}+\frac {\cosh (2 x)}{2 \sqrt {a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b}-\frac {(2 d) \int \frac {1}{x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}} \, dx}{b c}-\frac {(2 c d) \int \frac {1}{\sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac {2 d \left (1+c^2 x^2\right )^{3/2}}{b c x \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {(2 d) \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b}-\frac {(2 d) \int \frac {1}{x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}} \, dx}{b c}\\ &=-\frac {2 d \left (1+c^2 x^2\right )^{3/2}}{b c x \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {d \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b}+\frac {d \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b}-\frac {(2 d) \int \frac {1}{x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}} \, dx}{b c}\\ &=-\frac {2 d \left (1+c^2 x^2\right )^{3/2}}{b c x \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {(2 d) \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{b^2}+\frac {(2 d) \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{b^2}-\frac {(2 d) \int \frac {1}{x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}} \, dx}{b c}\\ &=-\frac {2 d \left (1+c^2 x^2\right )^{3/2}}{b c x \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {d e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2}}+\frac {d e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2}}-\frac {(2 d) \int \frac {1}{x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}} \, dx}{b c}\\ \end {align*}
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Mathematica [A]
time = 2.74, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+c^2 d x^2}{x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {c^{2} d \,x^{2}+d}{x \left (a +b \arcsinh \left (c x \right )\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} d \left (\int \frac {c^{2} x^{2}}{a x \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b x \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx + \int \frac {1}{a x \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b x \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx\right ) \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 [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {d\,c^2\,x^2+d}{x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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