Optimal. Leaf size=62 \[ \frac {2 \tan (c+d x) \, _2F_1\left (1,\frac {2-p}{4};\frac {6-p}{4};-\tan ^2(c+d x)\right )}{d (2-p) \sqrt {b \tan ^p(c+d x)}} \]
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Rubi [A] time = 0.05, antiderivative size = 62, 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 = {3659, 3476, 364} \[ \frac {2 \tan (c+d x) \, _2F_1\left (1,\frac {2-p}{4};\frac {6-p}{4};-\tan ^2(c+d x)\right )}{d (2-p) \sqrt {b \tan ^p(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 3476
Rule 3659
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {b \tan ^p(c+d x)}} \, dx &=\frac {\tan ^{\frac {p}{2}}(c+d x) \int \tan ^{-\frac {p}{2}}(c+d x) \, dx}{\sqrt {b \tan ^p(c+d x)}}\\ &=\frac {\tan ^{\frac {p}{2}}(c+d x) \operatorname {Subst}\left (\int \frac {x^{-p/2}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d \sqrt {b \tan ^p(c+d x)}}\\ &=\frac {2 \, _2F_1\left (1,\frac {2-p}{4};\frac {6-p}{4};-\tan ^2(c+d x)\right ) \tan (c+d x)}{d (2-p) \sqrt {b \tan ^p(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 60, normalized size = 0.97 \[ -\frac {2 \tan (c+d x) \, _2F_1\left (1,\frac {2-p}{4};\frac {6-p}{4};-\tan ^2(c+d x)\right )}{d (p-2) \sqrt {b \tan ^p(c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \tan \left (d x + c\right )^{p}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.60, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \left (\tan ^{p}\left (d x +c \right )\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \tan \left (d x + c\right )^{p}}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {b\,{\mathrm {tan}\left (c+d\,x\right )}^p}} \,d x \]
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 \[ \int \frac {1}{\sqrt {b \tan ^{p}{\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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