Optimal. Leaf size=140 \[ \frac {2 \left (a^2 B+2 a A b+3 b^2 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}+\frac {2 \left (3 a^2 A+5 b (2 a B+A b)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a (5 a B+7 A b) \sin (c+d x) \sqrt {\cos (c+d x)}}{15 d}+\frac {2 a A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)}{5 d} \]
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Rubi [A] time = 0.33, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2954, 2990, 3023, 2748, 2641, 2639} \[ \frac {2 \left (a^2 B+2 a A b+3 b^2 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}+\frac {2 \left (3 a^2 A+5 b (2 a B+A b)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a (5 a B+7 A b) \sin (c+d x) \sqrt {\cos (c+d x)}}{15 d}+\frac {2 a A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)}{5 d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2748
Rule 2954
Rule 2990
Rule 3023
Rubi steps
\begin {align*} \int \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx &=\int \frac {(b+a \cos (c+d x))^2 (B+A \cos (c+d x))}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 a A \sqrt {\cos (c+d x)} (b+a \cos (c+d x)) \sin (c+d x)}{5 d}+\frac {2}{5} \int \frac {\frac {1}{2} b (a A+5 b B)+\frac {1}{2} \left (3 a^2 A+5 b (A b+2 a B)\right ) \cos (c+d x)+\frac {1}{2} a (7 A b+5 a B) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 a (7 A b+5 a B) \sqrt {\cos (c+d x)} \sin (c+d x)}{15 d}+\frac {2 a A \sqrt {\cos (c+d x)} (b+a \cos (c+d x)) \sin (c+d x)}{5 d}+\frac {4}{15} \int \frac {\frac {5}{4} \left (2 a A b+a^2 B+3 b^2 B\right )+\frac {3}{4} \left (3 a^2 A+5 b (A b+2 a B)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 a (7 A b+5 a B) \sqrt {\cos (c+d x)} \sin (c+d x)}{15 d}+\frac {2 a A \sqrt {\cos (c+d x)} (b+a \cos (c+d x)) \sin (c+d x)}{5 d}+\frac {1}{3} \left (2 a A b+a^2 B+3 b^2 B\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\frac {1}{5} \left (3 a^2 A+5 b (A b+2 a B)\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {2 \left (3 a^2 A+5 b (A b+2 a B)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (2 a A b+a^2 B+3 b^2 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}+\frac {2 a (7 A b+5 a B) \sqrt {\cos (c+d x)} \sin (c+d x)}{15 d}+\frac {2 a A \sqrt {\cos (c+d x)} (b+a \cos (c+d x)) \sin (c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.67, size = 106, normalized size = 0.76 \[ \frac {2 \left (5 \left (a^2 B+2 a A b+3 b^2 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+3 \left (3 a^2 A+10 a b B+5 A b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+a \sin (c+d x) \sqrt {\cos (c+d x)} (3 a A \cos (c+d x)+5 a B+10 A b)\right )}{15 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B b^{2} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{3} + A a^{2} \cos \left (d x + c\right )^{2} + {\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{2} + {\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]
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 \[ \int {\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 5.32, size = 487, normalized size = 3.48 \[ -\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-24 A \,a^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (24 a^{2} A +40 A a b +20 a^{2} B \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-6 a^{2} A -20 A a b -10 a^{2} B \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+10 A a b \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-9 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2}-15 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{2}+5 a^{2} B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+15 b^{2} B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-30 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a b \right )}{15 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
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 \[ \int {\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.04, size = 177, normalized size = 1.26 \[ \frac {B\,a^2\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}+\frac {2\,A\,b^2\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,B\,b^2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,A\,a\,b\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}+\frac {4\,B\,a\,b\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}-\frac {2\,A\,a^2\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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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