Optimal. Leaf size=223 \[ -\frac{2 (A (3-2 n)+C (5-2 n)) \sin (c+d x) (b \sec (c+d x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{4} (7-2 n),\frac{1}{4} (11-2 n),\cos ^2(c+d x)\right )}{d (3-2 n) (7-2 n) \sqrt{\sin ^2(c+d x)} \sec ^{\frac{7}{2}}(c+d x)}-\frac{2 B \sin (c+d x) (b \sec (c+d x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{4} (5-2 n),\frac{1}{4} (9-2 n),\cos ^2(c+d x)\right )}{d (5-2 n) \sqrt{\sin ^2(c+d x)} \sec ^{\frac{5}{2}}(c+d x)}-\frac{2 C \sin (c+d x) (b \sec (c+d x))^n}{d (3-2 n) \sec ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 0.201044, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.122, Rules used = {20, 4047, 3772, 2643, 4046} \[ -\frac{2 (A (3-2 n)+C (5-2 n)) \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{4} (7-2 n);\frac{1}{4} (11-2 n);\cos ^2(c+d x)\right )}{d (3-2 n) (7-2 n) \sqrt{\sin ^2(c+d x)} \sec ^{\frac{7}{2}}(c+d x)}-\frac{2 B \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{4} (5-2 n);\frac{1}{4} (9-2 n);\cos ^2(c+d x)\right )}{d (5-2 n) \sqrt{\sin ^2(c+d x)} \sec ^{\frac{5}{2}}(c+d x)}-\frac{2 C \sin (c+d x) (b \sec (c+d x))^n}{d (3-2 n) \sec ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 20
Rule 4047
Rule 3772
Rule 2643
Rule 4046
Rubi steps
\begin{align*} \int \frac{(b \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac{5}{2}}(c+d x)} \, dx &=\left (\sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{-\frac{5}{2}+n}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\\ &=\left (\sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{-\frac{5}{2}+n}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx+\left (B \sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{-\frac{3}{2}+n}(c+d x) \, dx\\ &=-\frac{2 C (b \sec (c+d x))^n \sin (c+d x)}{d (3-2 n) \sec ^{\frac{3}{2}}(c+d x)}+\left (B \cos ^{\frac{1}{2}+n}(c+d x) \sqrt{\sec (c+d x)} (b \sec (c+d x))^n\right ) \int \cos ^{\frac{3}{2}-n}(c+d x) \, dx+\frac{\left (\left (C \left (-\frac{5}{2}+n\right )+A \left (-\frac{3}{2}+n\right )\right ) \sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{-\frac{5}{2}+n}(c+d x) \, dx}{-\frac{3}{2}+n}\\ &=-\frac{2 C (b \sec (c+d x))^n \sin (c+d x)}{d (3-2 n) \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 B \, _2F_1\left (\frac{1}{2},\frac{1}{4} (5-2 n);\frac{1}{4} (9-2 n);\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d (5-2 n) \sec ^{\frac{5}{2}}(c+d x) \sqrt{\sin ^2(c+d x)}}+\frac{\left (\left (C \left (-\frac{5}{2}+n\right )+A \left (-\frac{3}{2}+n\right )\right ) \cos ^{\frac{1}{2}+n}(c+d x) \sqrt{\sec (c+d x)} (b \sec (c+d x))^n\right ) \int \cos ^{\frac{5}{2}-n}(c+d x) \, dx}{-\frac{3}{2}+n}\\ &=-\frac{2 C (b \sec (c+d x))^n \sin (c+d x)}{d (3-2 n) \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 (A (3-2 n)+C (5-2 n)) \, _2F_1\left (\frac{1}{2},\frac{1}{4} (7-2 n);\frac{1}{4} (11-2 n);\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d (3-2 n) (7-2 n) \sec ^{\frac{7}{2}}(c+d x) \sqrt{\sin ^2(c+d x)}}-\frac{2 B \, _2F_1\left (\frac{1}{2},\frac{1}{4} (5-2 n);\frac{1}{4} (9-2 n);\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d (5-2 n) \sec ^{\frac{5}{2}}(c+d x) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [F] time = 180.001, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
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Maple [F] time = 0.24, size = 0, normalized size = 0. \begin{align*} \int{ \left ( b\sec \left ( dx+c \right ) \right ) ^{n} \left ( A+B\sec \left ( dx+c \right ) +C \left ( \sec \left ( dx+c \right ) \right ) ^{2} \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}}\, 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 \frac{{\left (C \sec \left (d x + c\right )^{2} + 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 )^{\frac{5}{2}}}\,{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 (\frac{{\left (C \sec \left (d x + c\right )^{2} + 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 )^{\frac{5}{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \frac{{\left (C \sec \left (d x + c\right )^{2} + 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 )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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