Optimal. Leaf size=80 \[ -\frac {2 \, _2F_1\left (\frac {1}{2},\frac {1}{4} (1-2 n);\frac {1}{4} (5-2 n);\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) \sqrt {\sec (c+d x)} \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A]
time = 0.03, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {20, 3857, 2722}
\begin {gather*} -\frac {2 \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (1-2 n);\frac {1}{4} (5-2 n);\cos ^2(c+d x)\right )}{d (1-2 n) \sqrt {\sin ^2(c+d x)} \sqrt {\sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 20
Rule 2722
Rule 3857
Rubi steps
\begin {align*} \int \sqrt {\sec (c+d x)} (b \sec (c+d x))^n \, dx &=\left (\sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{\frac {1}{2}+n}(c+d x) \, dx\\ &=\left (\cos ^{\frac {1}{2}+n}(c+d x) \sqrt {\sec (c+d x)} (b \sec (c+d x))^n\right ) \int \cos ^{-\frac {1}{2}-n}(c+d x) \, dx\\ &=-\frac {2 \, _2F_1\left (\frac {1}{2},\frac {1}{4} (1-2 n);\frac {1}{4} (5-2 n);\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) \sqrt {\sec (c+d x)} \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 81, normalized size = 1.01 \begin {gather*} \frac {\csc (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} \left (\frac {1}{2}+n\right );\frac {1}{2} \left (\frac {5}{2}+n\right );\sec ^2(c+d x)\right ) (b \sec (c+d x))^n \sqrt {-\tan ^2(c+d x)}}{d \left (\frac {1}{2}+n\right ) \sqrt {\sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.35, size = 0, normalized size = 0.00 \[\int \left (b \sec \left (d x +c \right )\right )^{n} \left (\sqrt {\sec }\left (d x +c \right )\right )\, dx\]
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 (b \sec {\left (c + d x \right )}\right )^{n} \sqrt {\sec {\left (c + d 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 {\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^n\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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