Optimal. Leaf size=63 \[ \frac{4 b \tanh ^{-1}\left (\frac{a-b \tanh \left (\frac{1}{2} \left (c+d \sqrt{x}\right )\right )}{\sqrt{a^2+b^2}}\right )}{a d \sqrt{a^2+b^2}}+\frac{2 \sqrt{x}}{a} \]
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Rubi [A] time = 0.0950436, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {5437, 3783, 2660, 618, 204} \[ \frac{4 b \tanh ^{-1}\left (\frac{a-b \tanh \left (\frac{1}{2} \left (c+d \sqrt{x}\right )\right )}{\sqrt{a^2+b^2}}\right )}{a d \sqrt{a^2+b^2}}+\frac{2 \sqrt{x}}{a} \]
Antiderivative was successfully verified.
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Rule 5437
Rule 3783
Rule 2660
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x} \left (a+b \text{csch}\left (c+d \sqrt{x}\right )\right )} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{a+b \text{csch}(c+d x)} \, dx,x,\sqrt{x}\right )\\ &=\frac{2 \sqrt{x}}{a}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{1+\frac{a \sinh (c+d x)}{b}} \, dx,x,\sqrt{x}\right )}{a}\\ &=\frac{2 \sqrt{x}}{a}+\frac{(4 i) \operatorname{Subst}\left (\int \frac{1}{1-\frac{2 i a x}{b}+x^2} \, dx,x,i \tanh \left (\frac{1}{2} \left (c+d \sqrt{x}\right )\right )\right )}{a d}\\ &=\frac{2 \sqrt{x}}{a}-\frac{(8 i) \operatorname{Subst}\left (\int \frac{1}{-4 \left (1+\frac{a^2}{b^2}\right )-x^2} \, dx,x,-\frac{2 i a}{b}+2 i \tanh \left (\frac{1}{2} \left (c+d \sqrt{x}\right )\right )\right )}{a d}\\ &=\frac{2 \sqrt{x}}{a}+\frac{4 b \tanh ^{-1}\left (\frac{b \left (\frac{a}{b}-\tanh \left (\frac{1}{2} \left (c+d \sqrt{x}\right )\right )\right )}{\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}\\ \end{align*}
Mathematica [A] time = 0.142376, size = 73, normalized size = 1.16 \[ \frac{2 \left (-\frac{2 b \tan ^{-1}\left (\frac{a-b \tanh \left (\frac{1}{2} \left (c+d \sqrt{x}\right )\right )}{\sqrt{-a^2-b^2}}\right )}{d \sqrt{-a^2-b^2}}+\frac{c}{d}+\sqrt{x}\right )}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 94, normalized size = 1.5 \begin{align*} -4\,{\frac{b}{ad\sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,b\tanh \left ( c/2+1/2\,d\sqrt{x} \right ) -2\,a}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) }-2\,{\frac{\ln \left ( \tanh \left ( c/2+1/2\,d\sqrt{x} \right ) -1 \right ) }{ad}}+2\,{\frac{\ln \left ( \tanh \left ( c/2+1/2\,d\sqrt{x} \right ) +1 \right ) }{ad}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.87965, size = 305, normalized size = 4.84 \begin{align*} \frac{2 \,{\left ({\left (a^{2} + b^{2}\right )} d \sqrt{x} + \sqrt{a^{2} + b^{2}} b \log \left (\frac{a b +{\left (a^{2} + b^{2} + \sqrt{a^{2} + b^{2}} b\right )} \cosh \left (d \sqrt{x} + c\right ) -{\left (b^{2} + \sqrt{a^{2} + b^{2}} b\right )} \sinh \left (d \sqrt{x} + c\right ) + \sqrt{a^{2} + b^{2}} a}{a \sinh \left (d \sqrt{x} + c\right ) + b}\right )\right )}}{{\left (a^{3} + a b^{2}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x} \left (a + b \operatorname{csch}{\left (c + d \sqrt{x} \right )}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22405, size = 124, normalized size = 1.97 \begin{align*} -\frac{2 \, b \log \left (\frac{{\left | 2 \, a e^{\left (d \sqrt{x} + c\right )} + 2 \, b - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{\left (d \sqrt{x} + c\right )} + 2 \, b + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{\sqrt{a^{2} + b^{2}} a d} + \frac{2 \,{\left (d \sqrt{x} + c\right )}}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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