Optimal. Leaf size=142 \[ \frac {2 (A (2 n+5)+C (2 n+3)) \sin (c+d x) \sqrt {\sec (c+d x)} (b \sec (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (-2 n-1);\frac {1}{4} (3-2 n);\cos ^2(c+d x)\right )}{d (2 n+1) (2 n+5) \sqrt {\sin ^2(c+d x)}}+\frac {2 C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (b \sec (c+d x))^n}{d (2 n+5)} \]
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Rubi [A] time = 0.13, antiderivative size = 142, 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} \[ \frac {2 (A (2 n+5)+C (2 n+3)) \sin (c+d x) \sqrt {\sec (c+d x)} (b \sec (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (-2 n-1);\frac {1}{4} (3-2 n);\cos ^2(c+d x)\right )}{d (2 n+1) (2 n+5) \sqrt {\sin ^2(c+d x)}}+\frac {2 C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (b \sec (c+d x))^n}{d (2 n+5)} \]
Antiderivative was successfully verified.
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Rule 20
Rule 2643
Rule 3772
Rule 4046
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*}
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Mathematica [C] time = 2.37, size = 303, normalized size = 2.13 \[ -\frac {i 2^{n+\frac {7}{2}} e^{-\frac {1}{2} i (2 n+5) (c+d x)} \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{n+\frac {5}{2}} \sec ^{-n-2}(c+d x) \left (A+C \sec ^2(c+d x)\right ) (b \sec (c+d x))^n \left (\frac {2 (A+2 C) e^{\frac {1}{2} i (2 n+7) (c+d x)} \, _2F_1\left (1,\frac {1}{4} (-2 n-3);\frac {1}{4} (2 n+11);-e^{2 i (c+d x)}\right )}{2 n+7}+\frac {A e^{\frac {1}{2} i (2 n+3) (c+d x)} \, _2F_1\left (1,\frac {1}{4} (-2 n-7);\frac {1}{4} (2 n+7);-e^{2 i (c+d x)}\right )}{2 n+3}+\frac {A e^{\frac {1}{2} i (2 n+11) (c+d x)} \, _2F_1\left (1,\frac {1}{4} (1-2 n);\frac {1}{4} (2 n+15);-e^{2 i (c+d x)}\right )}{2 n+11}\right )}{d (A \cos (2 c+2 d x)+A+2 C)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C \sec \left (d x + c\right )^{3} + A \sec \left (d x + c\right )\right )} \left (b \sec \left (d x + c\right )\right )^{n} \sqrt {\sec \left (d x + c\right )}, x\right ) \]
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 \[ \int {\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.52, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{\frac {3}{2}}\left (d x +c \right )\right ) \left (b \sec \left (d x +c \right )\right )^{n} \left (A +C \left (\sec ^{2}\left (d x +c \right )\right )\right )\, dx \]
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 \[ \int {\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int \left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^n\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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