Optimal. Leaf size=140 \[ -\frac {2 (-2 A n+A+C (3-2 n)) \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (5-2 n);\frac {1}{4} (9-2 n);\cos ^2(c+d x)\right )}{d (1-2 n) (5-2 n) \sqrt {\sin ^2(c+d x)} \sec ^{\frac {5}{2}}(c+d x)}-\frac {2 C \sin (c+d x) (b \sec (c+d x))^n}{d (1-2 n) \sqrt {\sec (c+d x)}} \]
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Rubi [A] time = 0.13, antiderivative size = 140, normalized size of antiderivative = 1.00, 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} \[ -\frac {2 (-2 A n+A+C (3-2 n)) \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (5-2 n);\frac {1}{4} (9-2 n);\cos ^2(c+d x)\right )}{d (1-2 n) (5-2 n) \sqrt {\sin ^2(c+d x)} \sec ^{\frac {5}{2}}(c+d x)}-\frac {2 C \sin (c+d x) (b \sec (c+d x))^n}{d (1-2 n) \sqrt {\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 2643
Rule 3772
Rule 4046
Rubi steps
\begin {align*} \int \frac {(b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, 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 (b \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) \sqrt {\sec (c+d x)}}+\frac {\left (\left (C \left (-\frac {3}{2}+n\right )+A \left (-\frac {1}{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 {1}{2}+n}\\ &=-\frac {2 C (b \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) \sqrt {\sec (c+d x)}}+\frac {\left (\left (C \left (-\frac {3}{2}+n\right )+A \left (-\frac {1}{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 {1}{2}+n}\\ &=-\frac {2 C (b \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) \sqrt {\sec (c+d x)}}-\frac {2 (A (1-2 n)+C (3-2 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 ) (b \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) (5-2 n) \sec ^{\frac {5}{2}}(c+d x) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [C] time = 3.07, size = 343, normalized size = 2.45 \[ -\frac {i 2^{n+\frac {1}{2}} e^{-\frac {1}{2} i (4 c+d (2 n+1) x)} \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{n+\frac {1}{2}} \left (1+e^{2 i (c+d x)}\right )^{n+\frac {1}{2}} \sec ^{-n-2}(c+d x) \left (A+C \sec ^2(c+d x)\right ) (b \sec (c+d x))^n \left (\frac {e^{\frac {1}{2} i (4 c+d (2 n+1) x)} \left (2 (2 n+5) (A+2 C) \, _2F_1\left (n+\frac {1}{2},\frac {1}{4} (2 n+1);\frac {1}{4} (2 n+5);-e^{2 i (c+d x)}\right )+A (2 n+1) e^{2 i (c+d x)} \, _2F_1\left (n+\frac {1}{2},\frac {1}{4} (2 n+5);\frac {1}{4} (2 n+9);-e^{2 i (c+d x)}\right )\right )}{d (2 n+1) (2 n+5)}+\frac {A e^{\frac {1}{2} i d (2 n-3) x} \, _2F_1\left (n+\frac {1}{2},\frac {1}{4} (2 n-3);\frac {1}{4} (2 n+1);-e^{2 i (c+d x)}\right )}{d (2 n-3)}\right )}{A \cos (2 c+2 d x)+A+2 C} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n}}{\sec \left (d x + c\right )^{\frac {3}{2}}}, x\right ) \]
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 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n}}{\sec \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.88, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \sec \left (d x +c \right )\right )^{n} \left (A +C \left (\sec ^{2}\left (d x +c \right )\right )\right )}{\sec \left (d x +c \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n}}{\sec \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^n}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (b \sec {\left (c + d x \right )}\right )^{n} \left (A + C \sec ^{2}{\left (c + d x \right )}\right )}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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