Optimal. Leaf size=81 \[ -\frac {\sin (c+d x) \cos (c+d x) \left ((b \sec (c+d x))^p\right )^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (1-n p);\frac {1}{2} (3-n p);\cos ^2(c+d x)\right )}{d (1-n p) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.05, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4123, 3772, 2643} \[ -\frac {\sin (c+d x) \cos (c+d x) \left ((b \sec (c+d x))^p\right )^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (1-n p);\frac {1}{2} (3-n p);\cos ^2(c+d x)\right )}{d (1-n p) \sqrt {\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 3772
Rule 4123
Rubi steps
\begin {align*} \int \left ((b \sec (c+d x))^p\right )^n \, dx &=\left ((b \sec (c+d x))^{-n p} \left ((b \sec (c+d x))^p\right )^n\right ) \int (b \sec (c+d x))^{n p} \, dx\\ &=\left (\left (\frac {\cos (c+d x)}{b}\right )^{n p} \left ((b \sec (c+d x))^p\right )^n\right ) \int \left (\frac {\cos (c+d x)}{b}\right )^{-n p} \, dx\\ &=-\frac {\cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (1-n p);\frac {1}{2} (3-n p);\cos ^2(c+d x)\right ) \left ((b \sec (c+d x))^p\right )^n \sin (c+d x)}{d (1-n p) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 69, normalized size = 0.85 \[ \frac {\sqrt {-\tan ^2(c+d x)} \cot (c+d x) \left ((b \sec (c+d x))^p\right )^n \, _2F_1\left (\frac {1}{2},\frac {n p}{2};\frac {1}{2} (n p+2);\sec ^2(c+d x)\right )}{d n p} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (\left (b \sec \left (d x + c\right )\right )^{p}\right )}^{n}, x\right ) \]
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 \[ \int {\left (\left (b \sec \left (d x + c\right )\right )^{p}\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.83, size = 0, normalized size = 0.00 \[ \int \left (\left (b \sec \left (d x +c \right )\right )^{p}\right )^{n}\, dx \]
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 \[ \int {\left (\left (b \sec \left (d x + c\right )\right )^{p}\right )}^{n}\,{d x} \]
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 \[ \int {\left ({\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^p\right )}^n \,d x \]
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 \[ \int \left (\left (b \sec {\left (c + d x \right )}\right )^{p}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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