Optimal. Leaf size=234 \[ \frac {(3 A+4 B+17 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{42 a^4 d}+\frac {(15 A-B-83 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{210 a^4 d (\cos (c+d x)+1)^2}+\frac {(B+8 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^4 d}-\frac {(B+8 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{10 a^4 d (\cos (c+d x)+1)}+\frac {(5 A+2 B-9 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{35 a d (a \cos (c+d x)+a)^3}-\frac {(A-B+C) \sin (c+d x) \sqrt {\cos (c+d x)}}{7 d (a \cos (c+d x)+a)^4} \]
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Rubi [A] time = 0.76, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {4112, 3041, 2978, 2748, 2641, 2639} \[ \frac {(3 A+4 B+17 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{42 a^4 d}+\frac {(15 A-B-83 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{210 a^4 d (\cos (c+d x)+1)^2}+\frac {(B+8 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^4 d}-\frac {(B+8 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{10 a^4 d (\cos (c+d x)+1)}+\frac {(5 A+2 B-9 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{35 a d (a \cos (c+d x)+a)^3}-\frac {(A-B+C) \sin (c+d x) \sqrt {\cos (c+d x)}}{7 d (a \cos (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2748
Rule 2978
Rule 3041
Rule 4112
Rubi steps
\begin {align*} \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^4} \, dx &=\int \frac {C+B \cos (c+d x)+A \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} (a+a \cos (c+d x))^4} \, dx\\ &=-\frac {(A-B+C) \sqrt {\cos (c+d x)} \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {\int \frac {-\frac {1}{2} a (A-B-13 C)+\frac {1}{2} a (9 A+5 B-5 C) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac {(A-B+C) \sqrt {\cos (c+d x)} \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(5 A+2 B-9 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {\frac {7}{2} a^2 (B+8 C)+\frac {3}{2} a^2 (5 A+2 B-9 C) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=\frac {(15 A-B-83 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{210 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \sqrt {\cos (c+d x)} \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(5 A+2 B-9 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {\frac {1}{4} a^3 (15 A+41 B+253 C)+\frac {1}{4} a^3 (15 A-B-83 C) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+a \cos (c+d x))} \, dx}{105 a^6}\\ &=\frac {(15 A-B-83 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{210 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \sqrt {\cos (c+d x)} \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(5 A+2 B-9 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {(B+8 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{10 d \left (a^4+a^4 \cos (c+d x)\right )}+\frac {\int \frac {\frac {5}{4} a^4 (3 A+4 B+17 C)+\frac {21}{4} a^4 (B+8 C) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx}{105 a^8}\\ &=\frac {(15 A-B-83 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{210 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \sqrt {\cos (c+d x)} \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(5 A+2 B-9 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {(B+8 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{10 d \left (a^4+a^4 \cos (c+d x)\right )}+\frac {(B+8 C) \int \sqrt {\cos (c+d x)} \, dx}{20 a^4}+\frac {(3 A+4 B+17 C) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{84 a^4}\\ &=\frac {(B+8 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^4 d}+\frac {(3 A+4 B+17 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{42 a^4 d}+\frac {(15 A-B-83 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{210 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \sqrt {\cos (c+d x)} \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(5 A+2 B-9 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {(B+8 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{10 d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end {align*}
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Mathematica [C] time = 7.11, size = 1862, normalized size = 7.96 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sqrt {\cos \left (d x + c\right )}}{a^{4} \cos \left (d x + c\right )^{3} \sec \left (d x + c\right )^{4} + 4 \, a^{4} \cos \left (d x + c\right )^{3} \sec \left (d x + c\right )^{3} + 6 \, a^{4} \cos \left (d x + c\right )^{3} \sec \left (d x + c\right )^{2} + 4 \, a^{4} \cos \left (d x + c\right )^{3} \sec \left (d x + c\right ) + a^{4} \cos \left (d x + c\right )^{3}}, x\right ) \]
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 \[ \int \frac {C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{{\left (a \sec \left (d x + c\right ) + a\right )}^{4} \cos \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 6.65, size = 595, normalized size = 2.54 \[ -\frac {\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 (60 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{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 ) \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-168 B \left (\cos ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+80 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{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 ) \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-84 B \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{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 )-1344 C \left (\cos ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+340 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{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 ) \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-672 C \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{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 )+60 A \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+248 B \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1684 C \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-30 A \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-54 B \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-282 C \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-90 A \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 B \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-34 C \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+75 A \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-33 B \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-9 C \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 A +15 B -15 C \right )}{840 a^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} \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(-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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{{\cos \left (c+d\,x\right )}^{5/2}\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^4} \,d x \]
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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