Optimal. Leaf size=78 \[ \frac {\cos ^2(a+b x)^{\frac {4+n}{2}} \, _2F_1\left (\frac {1+n}{2},\frac {4+n}{2};\frac {3+n}{2};\sin ^2(a+b x)\right ) \sec ^3(a+b x) (d \tan (a+b x))^{1+n}}{b d (1+n)} \]
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Rubi [A]
time = 0.03, 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 = {2697}
\begin {gather*} \frac {\sec ^3(a+b x) \cos ^2(a+b x)^{\frac {n+4}{2}} (d \tan (a+b x))^{n+1} \, _2F_1\left (\frac {n+1}{2},\frac {n+4}{2};\frac {n+3}{2};\sin ^2(a+b x)\right )}{b d (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2697
Rubi steps
\begin {align*} \int \sec ^3(a+b x) (d \tan (a+b x))^n \, dx &=\frac {\cos ^2(a+b x)^{\frac {4+n}{2}} \, _2F_1\left (\frac {1+n}{2},\frac {4+n}{2};\frac {3+n}{2};\sin ^2(a+b x)\right ) \sec ^3(a+b x) (d \tan (a+b x))^{1+n}}{b d (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 72, normalized size = 0.92 \begin {gather*} \frac {d \, _2F_1\left (\frac {3}{2},\frac {1-n}{2};\frac {5}{2};\sec ^2(a+b x)\right ) \sec ^3(a+b x) (d \tan (a+b x))^{-1+n} \left (-\tan ^2(a+b x)\right )^{\frac {1-n}{2}}}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \left (\sec ^{3}\left (b x +a \right )\right ) \left (d \tan \left (b x +a \right )\right )^{n}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d \tan {\left (a + b x \right )}\right )^{n} \sec ^{3}{\left (a + b x \right )}\, 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 {{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^n}{{\cos \left (a+b\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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