Optimal. Leaf size=50 \[ \frac {\cot (c+d x) \sqrt {b \tan ^4(c+d x)}}{d}-x \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)} \]
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Rubi [A] time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, 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} \[ \frac {\cot (c+d x) \sqrt {b \tan ^4(c+d x)}}{d}-x \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rule 3658
Rubi steps
\begin {align*} \int \sqrt {b \tan ^4(c+d x)} \, dx &=\left (\cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int \tan ^2(c+d x) \, dx\\ &=\frac {\cot (c+d x) \sqrt {b \tan ^4(c+d x)}}{d}-\left (\cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int 1 \, dx\\ &=\frac {\cot (c+d x) \sqrt {b \tan ^4(c+d x)}}{d}-x \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 41, normalized size = 0.82 \[ -\frac {\cot (c+d x) \sqrt {b \tan ^4(c+d x)} \left (\tan ^{-1}(\tan (c+d x)) \cot (c+d x)-1\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 37, normalized size = 0.74 \[ -\frac {\sqrt {b \tan \left (d x + c\right )^{4}} {\left (d x - \tan \left (d x + c\right )\right )}}{d \tan \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 42, normalized size = 0.84 \[ -\frac {\sqrt {b \left (\tan ^{4}\left (d x +c \right )\right )}\, \left (-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d \tan \left (d x +c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 26, normalized size = 0.52 \[ -\frac {{\left (d x + c\right )} \sqrt {b} - \sqrt {b} \tan \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {b\,{\mathrm {tan}\left (c+d\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \tan ^{4}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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