Optimal. Leaf size=344 \[ -\frac {2 A b \sin (c+d x)}{7 a^2 d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2 b^3 \left (a^2 C+A b^2\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a^5 d (a+b)}-\frac {2 b \left (a^2 (5 A+7 C)+7 A b^2\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 a^4 d}-\frac {2 b \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{21 a^4 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (a^2 (7 A+9 C)+9 A b^2\right ) \sin (c+d x)}{45 a^3 d \cos ^{\frac {5}{2}}(c+d x)}-\frac {2 \left (a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 a^5 d}+\frac {2 \left (a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right ) \sin (c+d x)}{15 a^5 d \sqrt {\cos (c+d x)}}+\frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)} \]
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Rubi [A] time = 1.94, antiderivative size = 344, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3056, 3055, 3059, 2639, 3002, 2641, 2805} \[ -\frac {2 b \left (a^2 (5 A+7 C)+7 A b^2\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 a^4 d}-\frac {2 \left (3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+15 A b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 a^5 d}-\frac {2 b^3 \left (a^2 C+A b^2\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a^5 d (a+b)}-\frac {2 b \left (a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{21 a^4 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (a^2 (7 A+9 C)+9 A b^2\right ) \sin (c+d x)}{45 a^3 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+15 A b^4\right ) \sin (c+d x)}{15 a^5 d \sqrt {\cos (c+d x)}}-\frac {2 A b \sin (c+d x)}{7 a^2 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2805
Rule 3002
Rule 3055
Rule 3056
Rule 3059
Rubi steps
\begin {align*} \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {11}{2}}(c+d x) (a+b \cos (c+d x))} \, dx &=\frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \int \frac {-\frac {9 A b}{2}+\frac {1}{2} a (7 A+9 C) \cos (c+d x)+\frac {7}{2} A b \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{9 a}\\ &=\frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {2 A b \sin (c+d x)}{7 a^2 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4 \int \frac {\frac {7}{4} \left (9 A b^2+a^2 (7 A+9 C)\right )+a A b \cos (c+d x)-\frac {45}{4} A b^2 \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{63 a^2}\\ &=\frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {2 A b \sin (c+d x)}{7 a^2 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (9 A b^2+a^2 (7 A+9 C)\right ) \sin (c+d x)}{45 a^3 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {8 \int \frac {-\frac {45}{8} b \left (7 A b^2+a^2 (5 A+7 C)\right )-\frac {3}{8} a \left (12 A b^2-7 a^2 (7 A+9 C)\right ) \cos (c+d x)+\frac {21}{8} b \left (9 A b^2+a^2 (7 A+9 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{315 a^3}\\ &=\frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {2 A b \sin (c+d x)}{7 a^2 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (9 A b^2+a^2 (7 A+9 C)\right ) \sin (c+d x)}{45 a^3 d \cos ^{\frac {5}{2}}(c+d x)}-\frac {2 b \left (7 A b^2+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^4 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {16 \int \frac {\frac {63}{16} \left (15 A b^4+3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right )+\frac {9}{4} a b \left (7 A b^2+a^2 (6 A+7 C)\right ) \cos (c+d x)-\frac {45}{16} b^2 \left (7 A b^2+a^2 (5 A+7 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{945 a^4}\\ &=\frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {2 A b \sin (c+d x)}{7 a^2 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (9 A b^2+a^2 (7 A+9 C)\right ) \sin (c+d x)}{45 a^3 d \cos ^{\frac {5}{2}}(c+d x)}-\frac {2 b \left (7 A b^2+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^4 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (15 A b^4+3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) \sin (c+d x)}{15 a^5 d \sqrt {\cos (c+d x)}}+\frac {32 \int \frac {-\frac {45}{32} b \left (21 A b^4+7 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right )-\frac {9}{32} a \left (140 A b^4+7 a^4 (7 A+9 C)+4 a^2 b^2 (22 A+35 C)\right ) \cos (c+d x)-\frac {63}{32} b \left (15 A b^4+3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{945 a^5}\\ &=\frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {2 A b \sin (c+d x)}{7 a^2 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (9 A b^2+a^2 (7 A+9 C)\right ) \sin (c+d x)}{45 a^3 d \cos ^{\frac {5}{2}}(c+d x)}-\frac {2 b \left (7 A b^2+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^4 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (15 A b^4+3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) \sin (c+d x)}{15 a^5 d \sqrt {\cos (c+d x)}}-\frac {32 \int \frac {\frac {45}{32} b^2 \left (21 A b^4+7 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right )+\frac {45}{32} a b^3 \left (7 A b^2+a^2 (5 A+7 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{945 a^5 b}-\frac {\left (15 A b^4+3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) \int \sqrt {\cos (c+d x)} \, dx}{15 a^5}\\ &=-\frac {2 \left (15 A b^4+3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 a^5 d}+\frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {2 A b \sin (c+d x)}{7 a^2 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (9 A b^2+a^2 (7 A+9 C)\right ) \sin (c+d x)}{45 a^3 d \cos ^{\frac {5}{2}}(c+d x)}-\frac {2 b \left (7 A b^2+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^4 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (15 A b^4+3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) \sin (c+d x)}{15 a^5 d \sqrt {\cos (c+d x)}}-\frac {\left (b^3 \left (A b^2+a^2 C\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{a^5}-\frac {\left (b \left (7 A b^2+a^2 (5 A+7 C)\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 a^4}\\ &=-\frac {2 \left (15 A b^4+3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 a^5 d}-\frac {2 b \left (7 A b^2+a^2 (5 A+7 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 a^4 d}-\frac {2 b^3 \left (A b^2+a^2 C\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a^5 (a+b) d}+\frac {2 A \sin (c+d x)}{9 a d \cos ^{\frac {9}{2}}(c+d x)}-\frac {2 A b \sin (c+d x)}{7 a^2 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (9 A b^2+a^2 (7 A+9 C)\right ) \sin (c+d x)}{45 a^3 d \cos ^{\frac {5}{2}}(c+d x)}-\frac {2 b \left (7 A b^2+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^4 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (15 A b^4+3 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) \sin (c+d x)}{15 a^5 d \sqrt {\cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 4.29, size = 427, normalized size = 1.24 \[ \frac {\frac {2 \left (70 a^4 A \tan (c+d x)+7 \left (a^4 (7 A+9 C)+9 a^2 A b^2\right ) \sin (2 (c+d x))+6 \sin (c+d x) \left (-15 a^3 A b-5 a b \left (a^2 (5 A+7 C)+7 A b^2\right ) \cos ^2(c+d x)+7 \left (a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right ) \cos ^3(c+d x)\right )\right )}{\cos ^{\frac {7}{2}}(c+d x)}-3 \left (\frac {4 \left (7 a^5 (7 A+9 C)+4 a^3 b^2 (22 A+35 C)+140 a A b^4\right ) \left ((a+b) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-a \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{b (a+b)}+\frac {2 \left (a^4 b (99 A+133 C)+7 a^2 b^3 (19 A+45 C)+315 A b^5\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a+b}+\frac {14 \left (a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right ) \sin (c+d x) \left (\left (b^2-2 a^2\right ) \Pi \left (-\frac {b}{a};\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) F\left (\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )-2 a b E\left (\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )\right )}{a b \sqrt {\sin ^2(c+d x)}}\right )}{630 a^5 d} \]
Antiderivative was successfully verified.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {C \cos \left (d x + c\right )^{2} + A}{{\left (b \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 12.62, size = 1320, normalized size = 3.84 \[ \text {result too large to display} \]
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 \[ \int \frac {C \cos \left (d x + c\right )^{2} + A}{{\left (b \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {11}{2}}}\,{d x} \]
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 \[ \int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{{\cos \left (c+d\,x\right )}^{11/2}\,\left (a+b\,\cos \left (c+d\,x\right )\right )} \,d x \]
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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