Optimal. Leaf size=98 \[ \frac{b^2 \tan ^3(c+d x) \sqrt{b \tan ^2(c+d x)}}{4 d}-\frac{b^2 \tan (c+d x) \sqrt{b \tan ^2(c+d x)}}{2 d}-\frac{b^2 \cot (c+d x) \sqrt{b \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.0416749, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3658, 3473, 3475} \[ \frac{b^2 \tan ^3(c+d x) \sqrt{b \tan ^2(c+d x)}}{4 d}-\frac{b^2 \tan (c+d x) \sqrt{b \tan ^2(c+d x)}}{2 d}-\frac{b^2 \cot (c+d x) \sqrt{b \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3473
Rule 3475
Rubi steps
\begin{align*} \int \left (b \tan ^2(c+d x)\right )^{5/2} \, dx &=\left (b^2 \cot (c+d x) \sqrt{b \tan ^2(c+d x)}\right ) \int \tan ^5(c+d x) \, dx\\ &=\frac{b^2 \tan ^3(c+d x) \sqrt{b \tan ^2(c+d x)}}{4 d}-\left (b^2 \cot (c+d x) \sqrt{b \tan ^2(c+d x)}\right ) \int \tan ^3(c+d x) \, dx\\ &=-\frac{b^2 \tan (c+d x) \sqrt{b \tan ^2(c+d x)}}{2 d}+\frac{b^2 \tan ^3(c+d x) \sqrt{b \tan ^2(c+d x)}}{4 d}+\left (b^2 \cot (c+d x) \sqrt{b \tan ^2(c+d x)}\right ) \int \tan (c+d x) \, dx\\ &=-\frac{b^2 \cot (c+d x) \log (\cos (c+d x)) \sqrt{b \tan ^2(c+d x)}}{d}-\frac{b^2 \tan (c+d x) \sqrt{b \tan ^2(c+d x)}}{2 d}+\frac{b^2 \tan ^3(c+d x) \sqrt{b \tan ^2(c+d x)}}{4 d}\\ \end{align*}
Mathematica [A] time = 0.376226, size = 56, normalized size = 0.57 \[ -\frac{\cot (c+d x) \left (b \tan ^2(c+d x)\right )^{5/2} \left (2 \cot ^2(c+d x)+4 \cot ^4(c+d x) \log (\cos (c+d x))-1\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 58, normalized size = 0.6 \begin{align*}{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{4}-2\, \left ( \tan \left ( dx+c \right ) \right ) ^{2}+2\,\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{4\,d \left ( \tan \left ( dx+c \right ) \right ) ^{5}} \left ( b \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56432, size = 63, normalized size = 0.64 \begin{align*} \frac{b^{\frac{5}{2}} \tan \left (d x + c\right )^{4} - 2 \, b^{\frac{5}{2}} \tan \left (d x + c\right )^{2} + 2 \, b^{\frac{5}{2}} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29161, size = 180, normalized size = 1.84 \begin{align*} \frac{{\left (b^{2} \tan \left (d x + c\right )^{4} - 2 \, b^{2} \tan \left (d x + c\right )^{2} - 2 \, b^{2} \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 3 \, b^{2}\right )} \sqrt{b \tan \left (d x + c\right )^{2}}}{4 \, d \tan \left (d x + c\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 \left (b \tan ^{2}{\left (c + d x \right )}\right )^{\frac{5}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.58912, size = 74, normalized size = 0.76 \begin{align*} \frac{1}{4} \, b^{\frac{5}{2}}{\left (\frac{2 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{d} + \frac{d \tan \left (d x + c\right )^{4} - 2 \, d \tan \left (d x + c\right )^{2}}{d^{2}}\right )} \mathrm{sgn}\left (\tan \left (d x + c\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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