Optimal. Leaf size=145 \[ \frac{C \sin (c+d x) \sec ^{m+1}(c+d x) (b \sec (c+d x))^n}{d (m+n+1)}-\frac{(A (m+n+1)+C (m+n)) \sin (c+d x) \sec ^{m-1}(c+d x) (b \sec (c+d x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2} (-m-n+1),\frac{1}{2} (-m-n+3),\cos ^2(c+d x)\right )}{d (-m-n+1) (m+n+1) \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.110575, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {20, 4046, 3772, 2643} \[ \frac{C \sin (c+d x) \sec ^{m+1}(c+d x) (b \sec (c+d x))^n}{d (m+n+1)}-\frac{(A (m+n+1)+C (m+n)) \sin (c+d x) \sec ^{m-1}(c+d x) (b \sec (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{2} (-m-n+1);\frac{1}{2} (-m-n+3);\cos ^2(c+d x)\right )}{d (-m-n+1) (m+n+1) \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 4046
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \sec ^m(c+d x) (b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right ) \, dx &=\left (\sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{m+n}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx\\ &=\frac{C \sec ^{1+m}(c+d x) (b \sec (c+d x))^n \sin (c+d x)}{d (1+m+n)}+\left (\left (A+\frac{C (m+n)}{1+m+n}\right ) \sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{m+n}(c+d x) \, dx\\ &=\frac{C \sec ^{1+m}(c+d x) (b \sec (c+d x))^n \sin (c+d x)}{d (1+m+n)}+\left (\left (A+\frac{C (m+n)}{1+m+n}\right ) \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\\ &=\frac{C \sec ^{1+m}(c+d x) (b \sec (c+d x))^n \sin (c+d x)}{d (1+m+n)}-\frac{\left (A+\frac{C (m+n)}{1+m+n}\right ) \, _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)}}\\ \end{align*}
Mathematica [C] time = 7.79207, size = 289, normalized size = 1.99 \[ -\frac{i 2^{m+n+1} e^{-i (m+n+1) (c+d x)} \left (\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{m+n+1} \sec ^{-n-2}(c+d x) \left (A+C \sec ^2(c+d x)\right ) (b \sec (c+d x))^n \left (\frac{2 (A+2 C) e^{i (m+n+2) (c+d x)} \text{Hypergeometric2F1}\left (1,\frac{1}{2} (-m-n),\frac{1}{2} (m+n+4),-e^{2 i (c+d x)}\right )}{m+n+2}+\frac{A e^{i (m+n) (c+d x)} \text{Hypergeometric2F1}\left (1,\frac{1}{2} (-m-n-2),\frac{1}{2} (m+n+2),-e^{2 i (c+d x)}\right )}{m+n}+\frac{A e^{i (m+n+4) (c+d x)} \text{Hypergeometric2F1}\left (1,\frac{1}{2} (-m-n+2),\frac{1}{2} (m+n+6),-e^{2 i (c+d x)}\right )}{m+n+4}\right )}{d (A \cos (2 c+2 d x)+A+2 C)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.011, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( dx+c \right ) \right ) ^{m} \left ( b\sec \left ( dx+c \right ) \right ) ^{n} \left ( A+C \left ( \sec \left ( dx+c \right ) \right ) ^{2} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec{\left (c + d x \right )}\right )^{n} \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{m}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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