Optimal. Leaf size=192 \[ -\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} \sqrt{b} d}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}+1\right )}{\sqrt{2} \sqrt{b} d}-\frac{\log \left (\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right )}{2 \sqrt{2} \sqrt{b} d}+\frac{\log \left (\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right )}{2 \sqrt{2} \sqrt{b} d} \]
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Rubi [A] time = 0.121426, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {3476, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} \sqrt{b} d}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}+1\right )}{\sqrt{2} \sqrt{b} d}-\frac{\log \left (\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right )}{2 \sqrt{2} \sqrt{b} d}+\frac{\log \left (\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right )}{2 \sqrt{2} \sqrt{b} d} \]
Antiderivative was successfully verified.
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Rule 3476
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{b \tan (c+d x)}} \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (b^2+x^2\right )} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{b^2+x^4} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{b-x^2}{b^2+x^4} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{d}+\frac{\operatorname{Subst}\left (\int \frac{b+x^2}{b^2+x^4} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{b-\sqrt{2} \sqrt{b} x+x^2} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{2 d}+\frac{\operatorname{Subst}\left (\int \frac{1}{b+\sqrt{2} \sqrt{b} x+x^2} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{2 d}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{b}+2 x}{-b-\sqrt{2} \sqrt{b} x-x^2} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} \sqrt{b} d}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{b}-2 x}{-b+\sqrt{2} \sqrt{b} x-x^2} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} \sqrt{b} d}\\ &=-\frac{\log \left (\sqrt{b}+\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} \sqrt{b} d}+\frac{\log \left (\sqrt{b}+\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} \sqrt{b} d}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} \sqrt{b} d}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} \sqrt{b} d}\\ &=-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} \sqrt{b} d}+\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} \sqrt{b} d}-\frac{\log \left (\sqrt{b}+\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} \sqrt{b} d}+\frac{\log \left (\sqrt{b}+\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} \sqrt{b} d}\\ \end{align*}
Mathematica [A] time = 0.0998046, size = 131, normalized size = 0.68 \[ \frac{\sqrt{\tan (c+d x)} \left (-2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )+2 \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )-\log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )+\log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )\right )}{2 \sqrt{2} d \sqrt{b \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 166, normalized size = 0.9 \begin{align*}{\frac{\sqrt{2}}{4\,bd}\sqrt [4]{{b}^{2}}\ln \left ({ \left ( b\tan \left ( dx+c \right ) +\sqrt [4]{{b}^{2}}\sqrt{b\tan \left ( dx+c \right ) }\sqrt{2}+\sqrt{{b}^{2}} \right ) \left ( b\tan \left ( dx+c \right ) -\sqrt [4]{{b}^{2}}\sqrt{b\tan \left ( dx+c \right ) }\sqrt{2}+\sqrt{{b}^{2}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}}{2\,bd}\sqrt [4]{{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{b\tan \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{b}^{2}}}}}+1 \right ) }-{\frac{\sqrt{2}}{2\,bd}\sqrt [4]{{b}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{b\tan \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{b}^{2}}}}}+1 \right ) } \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 = 1.67544, size = 1339, normalized size = 6.97 \begin{align*} -\sqrt{2} \left (\frac{1}{b^{2} d^{4}}\right )^{\frac{1}{4}} \arctan \left (-\sqrt{2} b d^{3} \sqrt{\frac{b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \left (\frac{1}{b^{2} d^{4}}\right )^{\frac{3}{4}} + \sqrt{2} b d^{3} \sqrt{\frac{b^{2} d^{2} \sqrt{\frac{1}{b^{2} d^{4}}} \cos \left (d x + c\right ) + \sqrt{2} b d \sqrt{\frac{b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \left (\frac{1}{b^{2} d^{4}}\right )^{\frac{1}{4}} \cos \left (d x + c\right ) + b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \left (\frac{1}{b^{2} d^{4}}\right )^{\frac{3}{4}} - 1\right ) - \sqrt{2} \left (\frac{1}{b^{2} d^{4}}\right )^{\frac{1}{4}} \arctan \left (-\sqrt{2} b d^{3} \sqrt{\frac{b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \left (\frac{1}{b^{2} d^{4}}\right )^{\frac{3}{4}} + \sqrt{2} b d^{3} \sqrt{\frac{b^{2} d^{2} \sqrt{\frac{1}{b^{2} d^{4}}} \cos \left (d x + c\right ) - \sqrt{2} b d \sqrt{\frac{b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \left (\frac{1}{b^{2} d^{4}}\right )^{\frac{1}{4}} \cos \left (d x + c\right ) + b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \left (\frac{1}{b^{2} d^{4}}\right )^{\frac{3}{4}} + 1\right ) + \frac{1}{4} \, \sqrt{2} \left (\frac{1}{b^{2} d^{4}}\right )^{\frac{1}{4}} \log \left (\frac{b^{2} d^{2} \sqrt{\frac{1}{b^{2} d^{4}}} \cos \left (d x + c\right ) + \sqrt{2} b d \sqrt{\frac{b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \left (\frac{1}{b^{2} d^{4}}\right )^{\frac{1}{4}} \cos \left (d x + c\right ) + b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right ) - \frac{1}{4} \, \sqrt{2} \left (\frac{1}{b^{2} d^{4}}\right )^{\frac{1}{4}} \log \left (\frac{b^{2} d^{2} \sqrt{\frac{1}{b^{2} d^{4}}} \cos \left (d x + c\right ) - \sqrt{2} b d \sqrt{\frac{b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \left (\frac{1}{b^{2} d^{4}}\right )^{\frac{1}{4}} \cos \left (d x + c\right ) + b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right ) \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{b \tan{\left (c + d x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.483, size = 251, normalized size = 1.31 \begin{align*} \frac{1}{4} \, b{\left (\frac{2 \, \sqrt{2} \sqrt{{\left | b \right |}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{{\left | b \right |}} + 2 \, \sqrt{b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt{{\left | b \right |}}}\right )}{b^{2} d} + \frac{2 \, \sqrt{2} \sqrt{{\left | b \right |}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{{\left | b \right |}} - 2 \, \sqrt{b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt{{\left | b \right |}}}\right )}{b^{2} d} + \frac{\sqrt{2} \sqrt{{\left | b \right |}} \log \left (b \tan \left (d x + c\right ) + \sqrt{2} \sqrt{b \tan \left (d x + c\right )} \sqrt{{\left | b \right |}} +{\left | b \right |}\right )}{b^{2} d} - \frac{\sqrt{2} \sqrt{{\left | b \right |}} \log \left (b \tan \left (d x + c\right ) - \sqrt{2} \sqrt{b \tan \left (d x + c\right )} \sqrt{{\left | b \right |}} +{\left | b \right |}\right )}{b^{2} d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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