Optimal. Leaf size=71 \[ \frac {2 \, _2F_1\left (1,\frac {1}{4} (2-5 p);\frac {1}{4} (6-5 p);-\tan ^2(c+d x)\right ) \tan ^{1-2 p}(c+d x)}{b^2 d (2-5 p) \sqrt {b \tan ^p(c+d x)}} \]
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Rubi [A]
time = 0.03, 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 = {3740, 3557,
371} \begin {gather*} \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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 3557
Rule 3740
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) \text {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 = 0.08, size = 62, normalized size = 0.87 \begin {gather*} -\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 (c+d x)}{d (-2+5 p) \left (b \tan ^p(c+d x)\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.24, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (b \tan ^{p}{\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^p\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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