Optimal. Leaf size=51 \[ -\frac{x \tan ^2(c+d x)}{\sqrt{b \tan ^4(c+d x)}}-\frac{\tan (c+d x)}{d \sqrt{b \tan ^4(c+d x)}} \]
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Rubi [A] time = 0.0217256, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3658, 3473, 8} \[ -\frac{x \tan ^2(c+d x)}{\sqrt{b \tan ^4(c+d x)}}-\frac{\tan (c+d x)}{d \sqrt{b \tan ^4(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{b \tan ^4(c+d x)}} \, dx &=\frac{\tan ^2(c+d x) \int \cot ^2(c+d x) \, dx}{\sqrt{b \tan ^4(c+d x)}}\\ &=-\frac{\tan (c+d x)}{d \sqrt{b \tan ^4(c+d x)}}-\frac{\tan ^2(c+d x) \int 1 \, dx}{\sqrt{b \tan ^4(c+d x)}}\\ &=-\frac{\tan (c+d x)}{d \sqrt{b \tan ^4(c+d x)}}-\frac{x \tan ^2(c+d x)}{\sqrt{b \tan ^4(c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.0549894, size = 43, normalized size = 0.84 \[ -\frac{\tan (c+d x) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right )}{d \sqrt{b \tan ^4(c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 40, normalized size = 0.8 \begin{align*} -{\frac{\tan \left ( dx+c \right ) \left ( \arctan \left ( \tan \left ( dx+c \right ) \right ) \tan \left ( dx+c \right ) +1 \right ) }{d}{\frac{1}{\sqrt{b \left ( \tan \left ( dx+c \right ) \right ) ^{4}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40506, size = 36, normalized size = 0.71 \begin{align*} -\frac{\frac{d x + c}{\sqrt{b}} + \frac{1}{\sqrt{b} \tan \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34884, size = 93, normalized size = 1.82 \begin{align*} -\frac{\sqrt{b \tan \left (d x + c\right )^{4}}{\left (d x \tan \left (d x + c\right ) + 1\right )}}{b d \tan \left (d x + c\right )^{3}} \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 ^{4}{\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 = 1.53974, size = 61, normalized size = 1.2 \begin{align*} -\frac{\frac{2 \,{\left (d x + c\right )}}{\sqrt{b}} - \frac{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\sqrt{b}} + \frac{1}{\sqrt{b} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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