Optimal. Leaf size=76 \[ \frac {\sec (a+b x) \cos ^2(a+b x)^{\frac {n+2}{2}} (d \tan (a+b x))^{n+1} \, _2F_1\left (\frac {n+1}{2},\frac {n+2}{2};\frac {n+3}{2};\sin ^2(a+b x)\right )}{b d (n+1)} \]
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Rubi [A] time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2617} \[ \frac {\sec (a+b x) \cos ^2(a+b x)^{\frac {n+2}{2}} (d \tan (a+b x))^{n+1} \, _2F_1\left (\frac {n+1}{2},\frac {n+2}{2};\frac {n+3}{2};\sin ^2(a+b x)\right )}{b d (n+1)} \]
Antiderivative was successfully verified.
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Rule 2617
Rubi steps
\begin {align*} \int \sec (a+b x) (d \tan (a+b x))^n \, dx &=\frac {\cos ^2(a+b x)^{\frac {2+n}{2}} \, _2F_1\left (\frac {1+n}{2},\frac {2+n}{2};\frac {3+n}{2};\sin ^2(a+b x)\right ) \sec (a+b x) (d \tan (a+b x))^{1+n}}{b d (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 64, normalized size = 0.84 \[ \frac {\csc (a+b x) \left (-\tan ^2(a+b x)\right )^{\frac {1-n}{2}} (d \tan (a+b x))^n \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3}{2};\sec ^2(a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d \tan \left (b x + a\right )\right )^{n} \sec \left (b x + a\right ), x\right ) \]
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 \[ \int \left (d \tan \left (b x + a\right )\right )^{n} \sec \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.69, size = 0, normalized size = 0.00 \[ \int \sec \left (b x +a \right ) \left (d \tan \left (b x +a \right )\right )^{n}\, dx \]
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 \[ \int \left (d \tan \left (b x + a\right )\right )^{n} \sec \left (b x + a\right )\,{d x} \]
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 \[ \int \frac {{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^n}{\cos \left (a+b\,x\right )} \,d x \]
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 \[ \int \left (d \tan {\left (a + b x \right )}\right )^{n} \sec {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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