Optimal. Leaf size=585 \[ \frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a-1} d}\right )}{d^2}+\frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{-a-1} d+\sqrt{b} c}\right )}{d^2}-\frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{1-a} d}\right )}{d^2}-\frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{1-a} d+\sqrt{b} c}\right )}{d^2}+\frac{c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{-a-1}-\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} d+\sqrt{b} c}\right )}{d^2}-\frac{c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} d+\sqrt{b} c}\right )}{d^2}+\frac{c \log \left (c+d \sqrt{x}\right ) \log \left (-\frac{d \left (\sqrt{-a-1}+\sqrt{b} \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a-1} d}\right )}{d^2}-\frac{c \log \left (c+d \sqrt{x}\right ) \log \left (-\frac{d \left (\sqrt{1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{b} c-\sqrt{1-a} d}\right )}{d^2}+\frac{c \log (-a-b x+1) \log \left (c+d \sqrt{x}\right )}{d^2}-\frac{c \log (a+b x+1) \log \left (c+d \sqrt{x}\right )}{d^2}-\frac{\sqrt{x} \log (-a-b x+1)}{d}+\frac{\sqrt{x} \log (a+b x+1)}{d}+\frac{2 \sqrt{a+1} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+1}}\right )}{\sqrt{b} d}-\frac{2 \sqrt{1-a} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{\sqrt{b} d} \]
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Rubi [A] time = 1.01346, antiderivative size = 585, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 13, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.722, Rules used = {6115, 2408, 2466, 2448, 321, 205, 2462, 260, 2416, 2394, 2393, 2391, 208} \[ \frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a-1} d}\right )}{d^2}+\frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{-a-1} d+\sqrt{b} c}\right )}{d^2}-\frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{1-a} d}\right )}{d^2}-\frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{1-a} d+\sqrt{b} c}\right )}{d^2}+\frac{c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{-a-1}-\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} d+\sqrt{b} c}\right )}{d^2}-\frac{c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} d+\sqrt{b} c}\right )}{d^2}+\frac{c \log \left (c+d \sqrt{x}\right ) \log \left (-\frac{d \left (\sqrt{-a-1}+\sqrt{b} \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a-1} d}\right )}{d^2}-\frac{c \log \left (c+d \sqrt{x}\right ) \log \left (-\frac{d \left (\sqrt{1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{b} c-\sqrt{1-a} d}\right )}{d^2}+\frac{c \log (-a-b x+1) \log \left (c+d \sqrt{x}\right )}{d^2}-\frac{c \log (a+b x+1) \log \left (c+d \sqrt{x}\right )}{d^2}-\frac{\sqrt{x} \log (-a-b x+1)}{d}+\frac{\sqrt{x} \log (a+b x+1)}{d}+\frac{2 \sqrt{a+1} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+1}}\right )}{\sqrt{b} d}-\frac{2 \sqrt{1-a} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{\sqrt{b} d} \]
Antiderivative was successfully verified.
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Rule 6115
Rule 2408
Rule 2466
Rule 2448
Rule 321
Rule 205
Rule 2462
Rule 260
Rule 2416
Rule 2394
Rule 2393
Rule 2391
Rule 208
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(a+b x)}{c+d \sqrt{x}} \, dx &=-\left (\frac{1}{2} \int \frac{\log (1-a-b x)}{c+d \sqrt{x}} \, dx\right )+\frac{1}{2} \int \frac{\log (1+a+b x)}{c+d \sqrt{x}} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{x \log \left (1-a-b x^2\right )}{c+d x} \, dx,x,\sqrt{x}\right )+\operatorname{Subst}\left (\int \frac{x \log \left (1+a+b x^2\right )}{c+d x} \, dx,x,\sqrt{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{\log \left (1-a-b x^2\right )}{d}-\frac{c \log \left (1-a-b x^2\right )}{d (c+d x)}\right ) \, dx,x,\sqrt{x}\right )+\operatorname{Subst}\left (\int \left (\frac{\log \left (1+a+b x^2\right )}{d}-\frac{c \log \left (1+a+b x^2\right )}{d (c+d x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{\operatorname{Subst}\left (\int \log \left (1-a-b x^2\right ) \, dx,x,\sqrt{x}\right )}{d}+\frac{\operatorname{Subst}\left (\int \log \left (1+a+b x^2\right ) \, dx,x,\sqrt{x}\right )}{d}+\frac{c \operatorname{Subst}\left (\int \frac{\log \left (1-a-b x^2\right )}{c+d x} \, dx,x,\sqrt{x}\right )}{d}-\frac{c \operatorname{Subst}\left (\int \frac{\log \left (1+a+b x^2\right )}{c+d x} \, dx,x,\sqrt{x}\right )}{d}\\ &=-\frac{\sqrt{x} \log (1-a-b x)}{d}+\frac{c \log \left (c+d \sqrt{x}\right ) \log (1-a-b x)}{d^2}+\frac{\sqrt{x} \log (1+a+b x)}{d}-\frac{c \log \left (c+d \sqrt{x}\right ) \log (1+a+b x)}{d^2}+\frac{(2 b c) \operatorname{Subst}\left (\int \frac{x \log (c+d x)}{1-a-b x^2} \, dx,x,\sqrt{x}\right )}{d^2}+\frac{(2 b c) \operatorname{Subst}\left (\int \frac{x \log (c+d x)}{1+a+b x^2} \, dx,x,\sqrt{x}\right )}{d^2}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{x^2}{1-a-b x^2} \, dx,x,\sqrt{x}\right )}{d}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{x^2}{1+a+b x^2} \, dx,x,\sqrt{x}\right )}{d}\\ &=-\frac{\sqrt{x} \log (1-a-b x)}{d}+\frac{c \log \left (c+d \sqrt{x}\right ) \log (1-a-b x)}{d^2}+\frac{\sqrt{x} \log (1+a+b x)}{d}-\frac{c \log \left (c+d \sqrt{x}\right ) \log (1+a+b x)}{d^2}+\frac{(2 b c) \operatorname{Subst}\left (\int \left (-\frac{\log (c+d x)}{2 \sqrt{b} \left (\sqrt{-1-a}-\sqrt{b} x\right )}+\frac{\log (c+d x)}{2 \sqrt{b} \left (\sqrt{-1-a}+\sqrt{b} x\right )}\right ) \, dx,x,\sqrt{x}\right )}{d^2}+\frac{(2 b c) \operatorname{Subst}\left (\int \left (\frac{\log (c+d x)}{2 \sqrt{b} \left (\sqrt{1-a}-\sqrt{b} x\right )}-\frac{\log (c+d x)}{2 \sqrt{b} \left (\sqrt{1-a}+\sqrt{b} x\right )}\right ) \, dx,x,\sqrt{x}\right )}{d^2}-\frac{(2 (1-a)) \operatorname{Subst}\left (\int \frac{1}{1-a-b x^2} \, dx,x,\sqrt{x}\right )}{d}+\frac{(2 (1+a)) \operatorname{Subst}\left (\int \frac{1}{1+a+b x^2} \, dx,x,\sqrt{x}\right )}{d}\\ &=\frac{2 \sqrt{1+a} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1+a}}\right )}{\sqrt{b} d}-\frac{2 \sqrt{1-a} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{\sqrt{b} d}-\frac{\sqrt{x} \log (1-a-b x)}{d}+\frac{c \log \left (c+d \sqrt{x}\right ) \log (1-a-b x)}{d^2}+\frac{\sqrt{x} \log (1+a+b x)}{d}-\frac{c \log \left (c+d \sqrt{x}\right ) \log (1+a+b x)}{d^2}-\frac{\left (\sqrt{b} c\right ) \operatorname{Subst}\left (\int \frac{\log (c+d x)}{\sqrt{-1-a}-\sqrt{b} x} \, dx,x,\sqrt{x}\right )}{d^2}+\frac{\left (\sqrt{b} c\right ) \operatorname{Subst}\left (\int \frac{\log (c+d x)}{\sqrt{1-a}-\sqrt{b} x} \, dx,x,\sqrt{x}\right )}{d^2}+\frac{\left (\sqrt{b} c\right ) \operatorname{Subst}\left (\int \frac{\log (c+d x)}{\sqrt{-1-a}+\sqrt{b} x} \, dx,x,\sqrt{x}\right )}{d^2}-\frac{\left (\sqrt{b} c\right ) \operatorname{Subst}\left (\int \frac{\log (c+d x)}{\sqrt{1-a}+\sqrt{b} x} \, dx,x,\sqrt{x}\right )}{d^2}\\ &=\frac{2 \sqrt{1+a} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1+a}}\right )}{\sqrt{b} d}-\frac{2 \sqrt{1-a} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{\sqrt{b} d}+\frac{c \log \left (\frac{d \left (\sqrt{-1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{b} c+\sqrt{-1-a} d}\right ) \log \left (c+d \sqrt{x}\right )}{d^2}-\frac{c \log \left (\frac{d \left (\sqrt{1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{b} c+\sqrt{1-a} d}\right ) \log \left (c+d \sqrt{x}\right )}{d^2}+\frac{c \log \left (-\frac{d \left (\sqrt{-1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-1-a} d}\right ) \log \left (c+d \sqrt{x}\right )}{d^2}-\frac{c \log \left (-\frac{d \left (\sqrt{1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{b} c-\sqrt{1-a} d}\right ) \log \left (c+d \sqrt{x}\right )}{d^2}-\frac{\sqrt{x} \log (1-a-b x)}{d}+\frac{c \log \left (c+d \sqrt{x}\right ) \log (1-a-b x)}{d^2}+\frac{\sqrt{x} \log (1+a+b x)}{d}-\frac{c \log \left (c+d \sqrt{x}\right ) \log (1+a+b x)}{d^2}-\frac{c \operatorname{Subst}\left (\int \frac{\log \left (\frac{d \left (\sqrt{-1-a}-\sqrt{b} x\right )}{\sqrt{b} c+\sqrt{-1-a} d}\right )}{c+d x} \, dx,x,\sqrt{x}\right )}{d}+\frac{c \operatorname{Subst}\left (\int \frac{\log \left (\frac{d \left (\sqrt{1-a}-\sqrt{b} x\right )}{\sqrt{b} c+\sqrt{1-a} d}\right )}{c+d x} \, dx,x,\sqrt{x}\right )}{d}-\frac{c \operatorname{Subst}\left (\int \frac{\log \left (\frac{d \left (\sqrt{-1-a}+\sqrt{b} x\right )}{-\sqrt{b} c+\sqrt{-1-a} d}\right )}{c+d x} \, dx,x,\sqrt{x}\right )}{d}+\frac{c \operatorname{Subst}\left (\int \frac{\log \left (\frac{d \left (\sqrt{1-a}+\sqrt{b} x\right )}{-\sqrt{b} c+\sqrt{1-a} d}\right )}{c+d x} \, dx,x,\sqrt{x}\right )}{d}\\ &=\frac{2 \sqrt{1+a} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1+a}}\right )}{\sqrt{b} d}-\frac{2 \sqrt{1-a} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{\sqrt{b} d}+\frac{c \log \left (\frac{d \left (\sqrt{-1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{b} c+\sqrt{-1-a} d}\right ) \log \left (c+d \sqrt{x}\right )}{d^2}-\frac{c \log \left (\frac{d \left (\sqrt{1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{b} c+\sqrt{1-a} d}\right ) \log \left (c+d \sqrt{x}\right )}{d^2}+\frac{c \log \left (-\frac{d \left (\sqrt{-1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-1-a} d}\right ) \log \left (c+d \sqrt{x}\right )}{d^2}-\frac{c \log \left (-\frac{d \left (\sqrt{1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{b} c-\sqrt{1-a} d}\right ) \log \left (c+d \sqrt{x}\right )}{d^2}-\frac{\sqrt{x} \log (1-a-b x)}{d}+\frac{c \log \left (c+d \sqrt{x}\right ) \log (1-a-b x)}{d^2}+\frac{\sqrt{x} \log (1+a+b x)}{d}-\frac{c \log \left (c+d \sqrt{x}\right ) \log (1+a+b x)}{d^2}-\frac{c \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{b} x}{-\sqrt{b} c+\sqrt{-1-a} d}\right )}{x} \, dx,x,c+d \sqrt{x}\right )}{d^2}-\frac{c \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{b} x}{\sqrt{b} c+\sqrt{-1-a} d}\right )}{x} \, dx,x,c+d \sqrt{x}\right )}{d^2}+\frac{c \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{b} x}{-\sqrt{b} c+\sqrt{1-a} d}\right )}{x} \, dx,x,c+d \sqrt{x}\right )}{d^2}+\frac{c \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{b} x}{\sqrt{b} c+\sqrt{1-a} d}\right )}{x} \, dx,x,c+d \sqrt{x}\right )}{d^2}\\ &=\frac{2 \sqrt{1+a} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1+a}}\right )}{\sqrt{b} d}-\frac{2 \sqrt{1-a} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{\sqrt{b} d}+\frac{c \log \left (\frac{d \left (\sqrt{-1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{b} c+\sqrt{-1-a} d}\right ) \log \left (c+d \sqrt{x}\right )}{d^2}-\frac{c \log \left (\frac{d \left (\sqrt{1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{b} c+\sqrt{1-a} d}\right ) \log \left (c+d \sqrt{x}\right )}{d^2}+\frac{c \log \left (-\frac{d \left (\sqrt{-1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-1-a} d}\right ) \log \left (c+d \sqrt{x}\right )}{d^2}-\frac{c \log \left (-\frac{d \left (\sqrt{1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{b} c-\sqrt{1-a} d}\right ) \log \left (c+d \sqrt{x}\right )}{d^2}-\frac{\sqrt{x} \log (1-a-b x)}{d}+\frac{c \log \left (c+d \sqrt{x}\right ) \log (1-a-b x)}{d^2}+\frac{\sqrt{x} \log (1+a+b x)}{d}-\frac{c \log \left (c+d \sqrt{x}\right ) \log (1+a+b x)}{d^2}+\frac{c \text{Li}_2\left (\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-1-a} d}\right )}{d^2}+\frac{c \text{Li}_2\left (\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c+\sqrt{-1-a} d}\right )}{d^2}-\frac{c \text{Li}_2\left (\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{1-a} d}\right )}{d^2}-\frac{c \text{Li}_2\left (\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c+\sqrt{1-a} d}\right )}{d^2}\\ \end{align*}
Mathematica [A] time = 0.518536, size = 549, normalized size = 0.94 \[ \frac{c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{-a-1} d}\right )+c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{-a-1} d+\sqrt{b} c}\right )-c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{b} c-\sqrt{1-a} d}\right )-c \text{PolyLog}\left (2,\frac{\sqrt{b} \left (c+d \sqrt{x}\right )}{\sqrt{1-a} d+\sqrt{b} c}\right )+c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{-a-1}-\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} d+\sqrt{b} c}\right )-c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{1-a}-\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} d+\sqrt{b} c}\right )+c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{-a-1}+\sqrt{b} \sqrt{x}\right )}{\sqrt{-a-1} d-\sqrt{b} c}\right )-c \log \left (c+d \sqrt{x}\right ) \log \left (\frac{d \left (\sqrt{1-a}+\sqrt{b} \sqrt{x}\right )}{\sqrt{1-a} d-\sqrt{b} c}\right )+c \log (-a-b x+1) \log \left (c+d \sqrt{x}\right )-c \log (a+b x+1) \log \left (c+d \sqrt{x}\right )-d \sqrt{x} \log (-a-b x+1)+d \sqrt{x} \log (a+b x+1)+\frac{2 \sqrt{a+1} d \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+1}}\right )}{\sqrt{b}}-\frac{2 \sqrt{1-a} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{1-a}}\right )}{\sqrt{b}}}{d^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.239, size = 738, normalized size = 1.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{artanh}\left (b x + a\right )}{d \sqrt{x} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{d \sqrt{x} \operatorname{artanh}\left (b x + a\right ) - c \operatorname{artanh}\left (b x + a\right )}{d^{2} x - c^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{artanh}\left (b x + a\right )}{d \sqrt{x} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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