Optimal. Leaf size=78 \[ \frac {\sec ^5(a+b x) \cos ^2(a+b x)^{\frac {n+6}{2}} (d \tan (a+b x))^{n+1} \, _2F_1\left (\frac {n+1}{2},\frac {n+6}{2};\frac {n+3}{2};\sin ^2(a+b x)\right )}{b d (n+1)} \]
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Rubi [A] time = 0.04, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2617} \[ \frac {\sec ^5(a+b x) \cos ^2(a+b x)^{\frac {n+6}{2}} (d \tan (a+b x))^{n+1} \, _2F_1\left (\frac {n+1}{2},\frac {n+6}{2};\frac {n+3}{2};\sin ^2(a+b x)\right )}{b d (n+1)} \]
Antiderivative was successfully verified.
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Rule 2617
Rubi steps
\begin {align*} \int \sec ^5(a+b x) (d \tan (a+b x))^n \, dx &=\frac {\cos ^2(a+b x)^{\frac {6+n}{2}} \, _2F_1\left (\frac {1+n}{2},\frac {6+n}{2};\frac {3+n}{2};\sin ^2(a+b x)\right ) \sec ^5(a+b x) (d \tan (a+b x))^{1+n}}{b d (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 72, normalized size = 0.92 \[ \frac {d \sec ^5(a+b x) \left (-\tan ^2(a+b x)\right )^{\frac {1-n}{2}} (d \tan (a+b x))^{n-1} \, _2F_1\left (\frac {5}{2},\frac {1-n}{2};\frac {7}{2};\sec ^2(a+b x)\right )}{5 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d \tan \left (b x + a\right )\right )^{n} \sec \left (b x + a\right )^{5}, x\right ) \]
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 \[ \int \left (d \tan \left (b x + a\right )\right )^{n} \sec \left (b x + a\right )^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.46, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{5}\left (b x +a \right )\right ) \left (d \tan \left (b x +a \right )\right )^{n}\, dx \]
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 \[ \int \left (d \tan \left (b x + a\right )\right )^{n} \sec \left (b x + a\right )^{5}\,{d x} \]
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 \[ \int \frac {{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^n}{{\cos \left (a+b\,x\right )}^5} \,d x \]
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 \[ \int \left (d \tan {\left (a + b x \right )}\right )^{n} \sec ^{5}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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