Optimal. Leaf size=427 \[ \frac {2 \left (49 a^2 A+72 a b B+3 A b^2\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{315 a d}+\frac {2 \left (75 a^3 B+88 a^2 A b+9 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}{315 a^2 d}+\frac {2 \left (a^2-b^2\right ) \left (75 a^3 B+39 a^2 A b-18 a b^2 B+8 A b^3\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{315 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4 A+246 a^3 b B+33 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{315 a^3 d \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {2 (9 a B+10 A b) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{63 d}+\frac {2 a A \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d} \]
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Rubi [A] time = 1.71, antiderivative size = 427, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2955, 4025, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661} \[ \frac {2 \left (49 a^2 A+72 a b B+3 A b^2\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{315 a d}+\frac {2 \left (88 a^2 A b+75 a^3 B+9 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}{315 a^2 d}+\frac {2 \left (a^2-b^2\right ) \left (39 a^2 A b+75 a^3 B-18 a b^2 B+8 A b^3\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{315 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (33 a^2 A b^2+147 a^4 A+246 a^3 b B-18 a b^3 B+8 A b^4\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{315 a^3 d \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {2 (9 a B+10 A b) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{63 d}+\frac {2 a A \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2955
Rule 3856
Rule 3858
Rule 4025
Rule 4035
Rule 4104
Rubi steps
\begin {align*} \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\\ &=\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d}-\frac {1}{9} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{2} a (10 A b+9 a B)-\frac {1}{2} \left (7 a^2 A+9 A b^2+18 a b B\right ) \sec (c+d x)-\frac {3}{2} b (2 a A+3 b B) \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx\\ &=\frac {2 (10 A b+9 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d}+\frac {\left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{4} a \left (49 a^2 A+3 A b^2+72 a b B\right )+\frac {1}{4} a \left (92 a A b+45 a^2 B+63 b^2 B\right ) \sec (c+d x)+a b (10 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 \left (49 a^2 A+3 A b^2+72 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac {2 (10 A b+9 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d}-\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {3}{8} a \left (88 a^2 A b-4 A b^3+75 a^3 B+9 a b^2 B\right )-\frac {1}{8} a^2 \left (147 a^2 A+209 A b^2+396 a b B\right ) \sec (c+d x)-\frac {1}{4} a b \left (49 a^2 A+3 A b^2+72 a b B\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 \left (88 a^2 A b-4 A b^3+75 a^3 B+9 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d}+\frac {2 \left (49 a^2 A+3 A b^2+72 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac {2 (10 A b+9 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d}+\frac {\left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {3}{16} a \left (147 a^4 A+33 a^2 A b^2+8 A b^4+246 a^3 b B-18 a b^3 B\right )+\frac {3}{16} a^2 \left (186 a^2 A b+2 A b^3+75 a^3 B+153 a b^2 B\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{945 a^3}\\ &=\frac {2 \left (88 a^2 A b-4 A b^3+75 a^3 B+9 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d}+\frac {2 \left (49 a^2 A+3 A b^2+72 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac {2 (10 A b+9 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d}+\frac {\left (\left (a^2-b^2\right ) \left (39 a^2 A b+8 A b^3+75 a^3 B-18 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx}{315 a^3}+\frac {\left (\left (147 a^4 A+33 a^2 A b^2+8 A b^4+246 a^3 b B-18 a b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{315 a^3}\\ &=\frac {2 \left (88 a^2 A b-4 A b^3+75 a^3 B+9 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d}+\frac {2 \left (49 a^2 A+3 A b^2+72 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac {2 (10 A b+9 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d}+\frac {\left (\left (a^2-b^2\right ) \left (39 a^2 A b+8 A b^3+75 a^3 B-18 a b^2 B\right ) \sqrt {b+a \cos (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{315 a^3 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (147 a^4 A+33 a^2 A b^2+8 A b^4+246 a^3 b B-18 a b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{315 a^3 \sqrt {b+a \cos (c+d x)}}\\ &=\frac {2 \left (88 a^2 A b-4 A b^3+75 a^3 B+9 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d}+\frac {2 \left (49 a^2 A+3 A b^2+72 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac {2 (10 A b+9 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d}+\frac {\left (\left (a^2-b^2\right ) \left (39 a^2 A b+8 A b^3+75 a^3 B-18 a b^2 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{315 a^3 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (147 a^4 A+33 a^2 A b^2+8 A b^4+246 a^3 b B-18 a b^3 B\right ) \sqrt {\cos (c+d x)} \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^3 \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}\\ &=\frac {2 \left (a^2-b^2\right ) \left (39 a^2 A b+8 A b^3+75 a^3 B-18 a b^2 B\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 )}{315 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4 A+33 a^2 A b^2+8 A b^4+246 a^3 b B-18 a b^3 B\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{315 a^3 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {2 \left (88 a^2 A b-4 A b^3+75 a^3 B+9 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d}+\frac {2 \left (49 a^2 A+3 A b^2+72 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac {2 (10 A b+9 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d}\\ \end {align*}
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Mathematica [C] time = 18.65, size = 540, normalized size = 1.26 \[ \frac {\cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left (\frac {\left (133 a^2 A+144 a b B+6 A b^2\right ) \sin (2 (c+d x))}{630 a}+\frac {\left (345 a^3 B+402 a^2 A b+36 a b^2 B-16 A b^3\right ) \sin (c+d x)}{630 a^2}+\frac {1}{126} (9 a B+10 A b) \sin (3 (c+d x))+\frac {1}{36} a A \sin (4 (c+d x))\right )}{d (a \cos (c+d x)+b)}-\frac {2 \cos ^{\frac {3}{2}}(c+d x) \left (\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)\right )^{3/2} (a+b \sec (c+d x))^{3/2} \left (i a (a+b) \left (3 a^3 (49 A+25 B)+3 a^2 b (13 A+57 B)-6 a b^2 (A+3 B)+8 A b^3\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{a+b}} F\left (i \sinh ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {b-a}{a+b}\right )-\left (147 a^4 A+246 a^3 b B+33 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )^{3/2} (a \cos (c+d x)+b)-i (a+b) \left (147 a^4 A+246 a^3 b B+33 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{a+b}} E\left (i \sinh ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {b-a}{a+b}\right )\right )}{315 a^3 d \sec ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B b \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{2} + A a \cos \left (d x + c\right )^{4} + {\left (B a + A b\right )} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )\right )} \sqrt {b \sec \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}, x\right ) \]
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 \[ \int {\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{\frac {9}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.38, size = 3069, normalized size = 7.19 \[ \text {output too large to display} \]
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 \[ \int {\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{\frac {9}{2}}\,{d x} \]
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 \[ \int {\cos \left (c+d\,x\right )}^{9/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
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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