Optimal. Leaf size=73 \[ -\frac {c \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(a+b x)\right ) (c \sec (a+b x))^{-1+n} \sin (a+b x)}{b (1-n) \sqrt {\sin ^2(a+b x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3857, 2722}
\begin {gather*} -\frac {c \sin (a+b x) (c \sec (a+b x))^{n-1} \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(a+b x)\right )}{b (1-n) \sqrt {\sin ^2(a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2722
Rule 3857
Rubi steps
\begin {align*} \int (c \sec (a+b x))^n \, dx &=\left (\frac {\cos (a+b x)}{c}\right )^n (c \sec (a+b x))^n \int \left (\frac {\cos (a+b x)}{c}\right )^{-n} \, dx\\ &=-\frac {\cos (a+b x) \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(a+b x)\right ) (c \sec (a+b x))^n \sin (a+b x)}{b (1-n) \sqrt {\sin ^2(a+b x)}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 61, normalized size = 0.84 \begin {gather*} \frac {\cot (a+b x) \, _2F_1\left (\frac {1}{2},\frac {n}{2};\frac {2+n}{2};\sec ^2(a+b x)\right ) (c \sec (a+b x))^n \sqrt {-\tan ^2(a+b x)}}{b n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \left (c \sec \left (b x +a \right )\right )^{n}\, dx\]
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 (c \sec {\left (a + b x \right )}\right )^{n}\, 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 {\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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