Optimal. Leaf size=142 \[ \frac{2 (A (2 n+5)+C (2 n+3)) \sin (c+d x) \sqrt{\sec (c+d x)} (b \sec (c+d x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{4} (-2 n-1),\frac{1}{4} (3-2 n),\cos ^2(c+d x)\right )}{d (2 n+1) (2 n+5) \sqrt{\sin ^2(c+d x)}}+\frac{2 C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (b \sec (c+d x))^n}{d (2 n+5)} \]
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Rubi [A] time = 0.126935, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {20, 4046, 3772, 2643} \[ \frac{2 (A (2 n+5)+C (2 n+3)) \sin (c+d x) \sqrt{\sec (c+d x)} (b \sec (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{4} (-2 n-1);\frac{1}{4} (3-2 n);\cos ^2(c+d x)\right )}{d (2 n+1) (2 n+5) \sqrt{\sin ^2(c+d x)}}+\frac{2 C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (b \sec (c+d x))^n}{d (2 n+5)} \]
Antiderivative was successfully verified.
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Rule 20
Rule 4046
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \sec ^{\frac{3}{2}}(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 ^{\frac{3}{2}+n}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx\\ &=\frac{2 C \sec ^{\frac{5}{2}}(c+d x) (b \sec (c+d x))^n \sin (c+d x)}{d (5+2 n)}+\frac{\left (\left (C \left (\frac{3}{2}+n\right )+A \left (\frac{5}{2}+n\right )\right ) \sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{\frac{3}{2}+n}(c+d x) \, dx}{\frac{5}{2}+n}\\ &=\frac{2 C \sec ^{\frac{5}{2}}(c+d x) (b \sec (c+d x))^n \sin (c+d x)}{d (5+2 n)}+\frac{\left (\left (C \left (\frac{3}{2}+n\right )+A \left (\frac{5}{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{3}{2}-n}(c+d x) \, dx}{\frac{5}{2}+n}\\ &=\frac{2 C \sec ^{\frac{5}{2}}(c+d x) (b \sec (c+d x))^n \sin (c+d x)}{d (5+2 n)}+\frac{2 (C (3+2 n)+A (5+2 n)) \, _2F_1\left (\frac{1}{2},\frac{1}{4} (-1-2 n);\frac{1}{4} (3-2 n);\cos ^2(c+d x)\right ) \sqrt{\sec (c+d x)} (b \sec (c+d x))^n \sin (c+d x)}{d (1+2 n) (5+2 n) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [C] time = 2.2733, size = 303, normalized size = 2.13 \[ -\frac{i 2^{n+\frac{7}{2}} e^{-\frac{1}{2} i (2 n+5) (c+d x)} \left (\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{n+\frac{5}{2}} \sec ^{-n-2}(c+d x) \left (A+C \sec ^2(c+d x)\right ) (b \sec (c+d x))^n \left (\frac{2 (A+2 C) e^{\frac{1}{2} i (2 n+7) (c+d x)} \text{Hypergeometric2F1}\left (1,\frac{1}{4} (-2 n-3),\frac{1}{4} (2 n+11),-e^{2 i (c+d x)}\right )}{2 n+7}+\frac{A e^{\frac{1}{2} i (2 n+3) (c+d x)} \text{Hypergeometric2F1}\left (1,\frac{1}{4} (-2 n-7),\frac{1}{4} (2 n+7),-e^{2 i (c+d x)}\right )}{2 n+3}+\frac{A e^{\frac{1}{2} i (2 n+11) (c+d x)} \text{Hypergeometric2F1}\left (1,\frac{1}{4} (1-2 n),\frac{1}{4} (2 n+15),-e^{2 i (c+d x)}\right )}{2 n+11}\right )}{d (A \cos (2 c+2 d x)+A+2 C)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.234, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}} \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(-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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C \sec \left (d x + c\right )^{3} + A \sec \left (d x + c\right )\right )} \left (b \sec \left (d x + c\right )\right )^{n} \sqrt{\sec \left (d x + c\right )}, 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{\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 )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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