Optimal. Leaf size=71 \[ \frac {2 \tan ^{1-2 p}(c+d x) \, _2F_1\left (1,\frac {1}{4} (2-5 p);\frac {1}{4} (6-5 p);-\tan ^2(c+d x)\right )}{b^2 d (2-5 p) \sqrt {b \tan ^p(c+d x)}} \]
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Rubi [A] time = 0.05, antiderivative size = 71, 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} \[ \frac {2 \tan ^{1-2 p}(c+d x) \, _2F_1\left (1,\frac {1}{4} (2-5 p);\frac {1}{4} (6-5 p);-\tan ^2(c+d x)\right )}{b^2 d (2-5 p) \sqrt {b \tan ^p(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 3476
Rule 3659
Rubi steps
\begin {align*} \int \frac {1}{\left (b \tan ^p(c+d x)\right )^{5/2}} \, dx &=\frac {\tan ^{\frac {p}{2}}(c+d x) \int \tan ^{-\frac {5 p}{2}}(c+d x) \, dx}{b^2 \sqrt {b \tan ^p(c+d x)}}\\ &=\frac {\tan ^{\frac {p}{2}}(c+d x) \operatorname {Subst}\left (\int \frac {x^{-5 p/2}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{b^2 d \sqrt {b \tan ^p(c+d x)}}\\ &=\frac {2 \, _2F_1\left (1,\frac {1}{4} (2-5 p);\frac {1}{4} (6-5 p);-\tan ^2(c+d x)\right ) \tan ^{1-2 p}(c+d x)}{b^2 d (2-5 p) \sqrt {b \tan ^p(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 62, normalized size = 0.87 \[ -\frac {2 \tan (c+d x) \, _2F_1\left (1,\frac {1}{4} (2-5 p);\frac {1}{4} (6-5 p);-\tan ^2(c+d x)\right )}{d (5 p-2) \left (b \tan ^p(c+d x)\right )^{5/2}} \]
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 \[ \int \frac {1}{\left (b \tan \left (d x + c\right )^{p}\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.77, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \left (\tan ^{p}\left (d x +c \right )\right )\right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {1}{\left (b \tan \left (d x + c\right )^{p}\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^p\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (b \tan ^{p}{\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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