Optimal. Leaf size=223 \[ -\frac {2 (A (3-2 n)+C (5-2 n)) \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (7-2 n);\frac {1}{4} (11-2 n);\cos ^2(c+d x)\right )}{d (3-2 n) (7-2 n) \sqrt {\sin ^2(c+d x)} \sec ^{\frac {7}{2}}(c+d x)}-\frac {2 B \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (5-2 n);\frac {1}{4} (9-2 n);\cos ^2(c+d x)\right )}{d (5-2 n) \sqrt {\sin ^2(c+d x)} \sec ^{\frac {5}{2}}(c+d x)}-\frac {2 C \sin (c+d x) (b \sec (c+d x))^n}{d (3-2 n) \sec ^{\frac {3}{2}}(c+d x)} \]
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Rubi [A] time = 0.20, antiderivative size = 223, normalized size of antiderivative = 1.00, 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 2643
Rule 3772
Rule 4046
Rule 4047
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*}
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Mathematica [F] time = 180.00, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]
Verification is Not applicable to the result.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{n}}{\sec \left (d x + c\right )^{\frac {5}{2}}}, x\right ) \]
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 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{n}}{\sec \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.76, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \sec \left (d x +c \right )\right )^{n} \left (A +B \sec \left (d x +c \right )+C \left (\sec ^{2}\left (d x +c \right )\right )\right )}{\sec \left (d x +c \right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{n}}{\sec \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^n\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\left (b \sec {\left (c + d x \right )}\right )^{n} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right )}{\sec ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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