Optimal. Leaf size=457 \[ -\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{315 a^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{315 a^3 d \sqrt {\sec (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (-75 a^3 B+6 a^2 b (6 A+7 C)-24 a b^2 B+16 A b^3\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{315 a^4 d \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (-21 a^4 (7 A+9 C)-57 a^3 b B+6 a^2 b^2 (4 A+7 C)-24 a b^3 B+16 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{315 a^4 d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{63 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)} \]
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Rubi [A] time = 1.64, antiderivative size = 457, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661} \[ -\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{315 a^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (a^2 b (13 A+21 C)+75 a^3 B-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{315 a^3 d \sqrt {\sec (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (6 a^2 b (6 A+7 C)-75 a^3 B-24 a b^2 B+16 A b^3\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{315 a^4 d \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)-57 a^3 b B-24 a b^3 B+16 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{315 a^4 d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{63 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 3856
Rule 3858
Rule 4035
Rule 4094
Rule 4104
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx &=\frac {2 A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2}{9} \int \frac {\frac {1}{2} (A b+9 a B)+\frac {1}{2} (7 a A+9 b B+9 a C) \sec (c+d x)+\frac {3}{2} b (2 A+3 C) \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx\\ &=\frac {2 A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (A b+9 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {4 \int \frac {\frac {1}{4} \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right )-\frac {1}{4} a (47 A b+45 a B+63 b C) \sec (c+d x)-b (A b+9 a B) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx}{63 a}\\ &=\frac {2 A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (A b+9 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {8 \int \frac {\frac {3}{8} \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right )+\frac {1}{8} a \left (2 A b^2+207 a b B+21 a^2 (7 A+9 C)\right ) \sec (c+d x)-\frac {1}{4} b \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx}{315 a^2}\\ &=\frac {2 A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (A b+9 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^3 d \sqrt {\sec (c+d x)}}-\frac {16 \int \frac {\frac {3}{16} \left (16 A b^4-57 a^3 b B-24 a b^3 B+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\right )+\frac {3}{16} a \left (4 A b^3-75 a^3 B-6 a b^2 B-3 a^2 b (37 A+49 C)\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{945 a^3}\\ &=\frac {2 A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (A b+9 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^3 d \sqrt {\sec (c+d x)}}-\frac {\left (\left (a^2-b^2\right ) \left (16 A b^3-75 a^3 B-24 a b^2 B+6 a^2 b (6 A+7 C)\right )\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx}{315 a^4}-\frac {\left (16 A b^4-57 a^3 b B-24 a b^3 B+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{315 a^4}\\ &=\frac {2 A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (A b+9 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^3 d \sqrt {\sec (c+d x)}}-\frac {\left (\left (a^2-b^2\right ) \left (16 A b^3-75 a^3 B-24 a b^2 B+6 a^2 b (6 A+7 C)\right ) \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{315 a^4 \sqrt {a+b \sec (c+d x)}}-\frac {\left (\left (16 A b^4-57 a^3 b B-24 a b^3 B+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{315 a^4 \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}}\\ &=\frac {2 A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (A b+9 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^3 d \sqrt {\sec (c+d x)}}-\frac {\left (\left (a^2-b^2\right ) \left (16 A b^3-75 a^3 B-24 a b^2 B+6 a^2 b (6 A+7 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{315 a^4 \sqrt {a+b \sec (c+d x)}}-\frac {\left (\left (16 A b^4-57 a^3 b B-24 a b^3 B+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}} \, dx}{315 a^4 \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}\\ &=-\frac {2 \left (a^2-b^2\right ) \left (16 A b^3-75 a^3 B-24 a b^2 B+6 a^2 b (6 A+7 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{315 a^4 d \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (16 A b^4-57 a^3 b B-24 a b^3 B+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{315 a^4 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (A b+9 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^3 d \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 7.04, size = 5993, normalized size = 13.11 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sqrt {b \sec \left (d x + c\right ) + a}}{\sec \left (d x + c\right )^{\frac {9}{2}}}, x\right ) \]
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 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sqrt {b \sec \left (d x + c\right ) + a}}{\sec \left (d x + c\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.12, size = 6552, normalized size = 14.34 \[ \text {output too large to display} \]
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 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sqrt {b \sec \left (d x + c\right ) + a}}{\sec \left (d x + c\right )^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x \]
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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