Optimal. Leaf size=172 \[ \frac {B \sin (c+d x) \sec ^m(c+d x) (b \sec (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-m-n);\frac {1}{2} (-m-n+2);\cos ^2(c+d x)\right )}{d (m+n) \sqrt {\sin ^2(c+d x)}}-\frac {A \sin (c+d x) \sec ^{m-1}(c+d x) (b \sec (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-m-n+1);\frac {1}{2} (-m-n+3);\cos ^2(c+d x)\right )}{d (-m-n+1) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.11, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {20, 3787, 3772, 2643} \[ \frac {B \sin (c+d x) \sec ^m(c+d x) (b \sec (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-m-n);\frac {1}{2} (-m-n+2);\cos ^2(c+d x)\right )}{d (m+n) \sqrt {\sin ^2(c+d x)}}-\frac {A \sin (c+d x) \sec ^{m-1}(c+d x) (b \sec (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-m-n+1);\frac {1}{2} (-m-n+3);\cos ^2(c+d x)\right )}{d (-m-n+1) \sqrt {\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 2643
Rule 3772
Rule 3787
Rubi steps
\begin {align*} \int \sec ^m(c+d x) (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx &=\left (\sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{m+n}(c+d x) (A+B \sec (c+d x)) \, dx\\ &=\left (A \sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{m+n}(c+d x) \, dx+\left (B \sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{1+m+n}(c+d x) \, dx\\ &=\left (A \cos ^{m+n}(c+d x) \sec ^m(c+d x) (b \sec (c+d x))^n\right ) \int \cos ^{-m-n}(c+d x) \, dx+\left (B \cos ^{m+n}(c+d x) \sec ^m(c+d x) (b \sec (c+d x))^n\right ) \int \cos ^{-1-m-n}(c+d x) \, dx\\ &=-\frac {A \, _2F_1\left (\frac {1}{2},\frac {1}{2} (1-m-n);\frac {1}{2} (3-m-n);\cos ^2(c+d x)\right ) \sec ^{-1+m}(c+d x) (b \sec (c+d x))^n \sin (c+d x)}{d (1-m-n) \sqrt {\sin ^2(c+d x)}}+\frac {B \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-m-n);\frac {1}{2} (2-m-n);\cos ^2(c+d x)\right ) \sec ^m(c+d x) (b \sec (c+d x))^n \sin (c+d x)}{d (m+n) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 126, normalized size = 0.73 \[ \frac {\sqrt {-\tan ^2(c+d x)} \csc (c+d x) \sec ^m(c+d x) (b \sec (c+d x))^n \left (A (m+n+1) \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {m+n}{2};\frac {1}{2} (m+n+2);\sec ^2(c+d x)\right )+B (m+n) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (m+n+1);\frac {1}{2} (m+n+3);\sec ^2(c+d x)\right )\right )}{d (m+n) (m+n+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{m}, x\right ) \]
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 \[ \int {\left (B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.11, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{m}\left (d x +c \right )\right ) \left (b \sec \left (d x +c \right )\right )^{n} \left (A +B \sec \left (d x +c \right )\right )\, dx \]
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 \[ \int {\left (B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{m}\,{d x} \]
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 \[ \int \left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^n\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^m \,d x \]
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 \[ \int \left (b \sec {\left (c + d x \right )}\right )^{n} \left (A + B \sec {\left (c + d x \right )}\right ) \sec ^{m}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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