Optimal. Leaf size=232 \[ \frac {2 a^2 (52 A+72 B+63 C) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {4 a^2 (136 A+156 B+189 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{315 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a^2 (136 A+156 B+189 C) \sin (c+d x)}{315 d \sqrt {\sec (c+d x)} \sqrt {a \sec (c+d x)+a}}+\frac {2 a (A+3 B) \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{21 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)} \]
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Rubi [A] time = 0.65, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4086, 4017, 4015, 3805, 3804} \[ \frac {2 a^2 (52 A+72 B+63 C) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {4 a^2 (136 A+156 B+189 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{315 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a^2 (136 A+156 B+189 C) \sin (c+d x)}{315 d \sqrt {\sec (c+d x)} \sqrt {a \sec (c+d x)+a}}+\frac {2 a (A+3 B) \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{21 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 3804
Rule 3805
Rule 4015
Rule 4017
Rule 4086
Rubi steps
\begin {align*} \int \frac {(a+a \sec (c+d x))^{3/2} \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 (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \int \frac {(a+a \sec (c+d x))^{3/2} \left (\frac {3}{2} a (A+3 B)+\frac {1}{2} a (4 A+9 C) \sec (c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx}{9 a}\\ &=\frac {2 a (A+3 B) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{21 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {4 \int \frac {\sqrt {a+a \sec (c+d x)} \left (\frac {1}{4} a^2 (52 A+72 B+63 C)+\frac {1}{4} a^2 (40 A+36 B+63 C) \sec (c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx}{63 a}\\ &=\frac {2 a^2 (52 A+72 B+63 C) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {2 a (A+3 B) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{21 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {1}{105} (a (136 A+156 B+189 C)) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 (52 A+72 B+63 C) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (136 A+156 B+189 C) \sin (c+d x)}{315 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {2 a (A+3 B) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{21 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {1}{315} (2 a (136 A+156 B+189 C)) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx\\ &=\frac {2 a^2 (52 A+72 B+63 C) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (136 A+156 B+189 C) \sin (c+d x)}{315 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {4 a^2 (136 A+156 B+189 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{315 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (A+3 B) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{21 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A] time = 1.55, size = 123, normalized size = 0.53 \[ \frac {a \tan \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\sec (c+d x)+1)} (2 (799 A+759 B+756 C) \cos (c+d x)+4 (137 A+117 B+63 C) \cos (2 (c+d x))+170 A \cos (3 (c+d x))+35 A \cos (4 (c+d x))+2689 A+90 B \cos (3 (c+d x))+2964 B+3276 C)}{1260 d \sqrt {\sec (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 140, normalized size = 0.60 \[ \frac {2 \, {\left (35 \, A a \cos \left (d x + c\right )^{5} + 5 \, {\left (17 \, A + 9 \, B\right )} a \cos \left (d x + c\right )^{4} + 3 \, {\left (34 \, A + 39 \, B + 21 \, C\right )} a \cos \left (d x + c\right )^{3} + {\left (136 \, A + 156 \, B + 189 \, C\right )} a \cos \left (d x + c\right )^{2} + 2 \, {\left (136 \, A + 156 \, B + 189 \, C\right )} a \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right ) + d\right )} \sqrt {\cos \left (d x + c\right )}} \]
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 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\sec \left (d x + c\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.68, size = 164, normalized size = 0.71 \[ -\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (35 A \left (\cos ^{4}\left (d x +c \right )\right )+85 A \left (\cos ^{3}\left (d x +c \right )\right )+45 B \left (\cos ^{3}\left (d x +c \right )\right )+102 A \left (\cos ^{2}\left (d x +c \right )\right )+117 B \left (\cos ^{2}\left (d x +c \right )\right )+63 C \left (\cos ^{2}\left (d x +c \right )\right )+136 A \cos \left (d x +c \right )+156 B \cos \left (d x +c \right )+189 C \cos \left (d x +c \right )+272 A +312 B +378 C \right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (\cos ^{5}\left (d x +c \right )\right ) \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {9}{2}} a}{315 d \sin \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.84, size = 908, normalized size = 3.91 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.03, size = 188, normalized size = 0.81 \[ \frac {a\,\cos \left (c+d\,x\right )\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,\sqrt {\frac {a\,\left (\cos \left (c+d\,x\right )+1\right )}{\cos \left (c+d\,x\right )}}\,\left (4830\,A\,\sin \left (c+d\,x\right )+5460\,B\,\sin \left (c+d\,x\right )+6300\,C\,\sin \left (c+d\,x\right )+1428\,A\,\sin \left (2\,c+2\,d\,x\right )+513\,A\,\sin \left (3\,c+3\,d\,x\right )+170\,A\,\sin \left (4\,c+4\,d\,x\right )+35\,A\,\sin \left (5\,c+5\,d\,x\right )+1428\,B\,\sin \left (2\,c+2\,d\,x\right )+468\,B\,\sin \left (3\,c+3\,d\,x\right )+90\,B\,\sin \left (4\,c+4\,d\,x\right )+1512\,C\,\sin \left (2\,c+2\,d\,x\right )+252\,C\,\sin \left (3\,c+3\,d\,x\right )\right )}{2520\,d\,\left (\cos \left (c+d\,x\right )+1\right )} \]
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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