Optimal. Leaf size=299 \[ -\frac {2 a \left (7 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 b^4 d}+\frac {2 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{45 b^3 d}+\frac {2 a^4 \left (a^2 C+A b^2\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{b^6 d (a+b)}-\frac {2 a \left (21 a^4 C+7 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 b^6 d}+\frac {2 \left (15 a^4 C+3 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 b^5 d}-\frac {2 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b^2 d}+\frac {2 C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 b d} \]
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Rubi [A] time = 1.50, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3050, 3049, 3059, 2639, 3002, 2641, 2805} \[ -\frac {2 a \left (7 a^2 b^2 (3 A+C)+21 a^4 C+b^4 (7 A+5 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 b^6 d}+\frac {2 \left (3 a^2 b^2 (5 A+3 C)+15 a^4 C+b^4 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 b^5 d}+\frac {2 a^4 \left (a^2 C+A b^2\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{b^6 d (a+b)}+\frac {2 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{45 b^3 d}-\frac {2 a \left (7 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 b^4 d}-\frac {2 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b^2 d}+\frac {2 C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 b d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2805
Rule 3002
Rule 3049
Rule 3050
Rule 3059
Rubi steps
\begin {align*} \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx &=\frac {2 C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 b d}+\frac {2 \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (\frac {7 a C}{2}+\frac {1}{2} b (9 A+7 C) \cos (c+d x)-\frac {9}{2} a C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{9 b}\\ &=-\frac {2 a C \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 b^2 d}+\frac {2 C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 b d}+\frac {4 \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (-\frac {45 a^2 C}{4}+a b C \cos (c+d x)+\frac {7}{4} \left (9 a^2 C+b^2 (9 A+7 C)\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{63 b^2}\\ &=\frac {2 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 b^3 d}-\frac {2 a C \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 b^2 d}+\frac {2 C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 b d}+\frac {8 \int \frac {\sqrt {\cos (c+d x)} \left (\frac {21}{8} a \left (9 a^2 C+b^2 (9 A+7 C)\right )+\frac {3}{8} b \left (63 A b^2-12 a^2 C+49 b^2 C\right ) \cos (c+d x)-\frac {45}{8} a \left (7 A b^2+7 a^2 C+5 b^2 C\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{315 b^3}\\ &=-\frac {2 a \left (7 A b^2+7 a^2 C+5 b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 b^4 d}+\frac {2 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 b^3 d}-\frac {2 a C \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 b^2 d}+\frac {2 C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 b d}+\frac {16 \int \frac {-\frac {45}{16} a^2 \left (7 A b^2+7 a^2 C+5 b^2 C\right )+\frac {9}{4} a b \left (7 A b^2+7 a^2 C+6 b^2 C\right ) \cos (c+d x)+\frac {63}{16} \left (15 a^4 C+3 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{945 b^4}\\ &=-\frac {2 a \left (7 A b^2+7 a^2 C+5 b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 b^4 d}+\frac {2 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 b^3 d}-\frac {2 a C \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 b^2 d}+\frac {2 C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 b d}-\frac {16 \int \frac {\frac {45}{16} a^2 b \left (7 A b^2+7 a^2 C+5 b^2 C\right )+\frac {45}{16} a \left (21 a^4 C+7 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{945 b^5}+\frac {\left (15 a^4 C+3 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) \int \sqrt {\cos (c+d x)} \, dx}{15 b^5}\\ &=\frac {2 \left (15 a^4 C+3 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 b^5 d}-\frac {2 a \left (7 A b^2+7 a^2 C+5 b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 b^4 d}+\frac {2 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 b^3 d}-\frac {2 a C \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 b^2 d}+\frac {2 C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 b d}+\frac {\left (a^4 \left (A b^2+a^2 C\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{b^6}-\frac {\left (a \left (21 a^4 C+7 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 b^6}\\ &=\frac {2 \left (15 a^4 C+3 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 b^5 d}-\frac {2 a \left (21 a^4 C+7 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 b^6 d}+\frac {2 a^4 \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 )}{b^6 (a+b) d}-\frac {2 a \left (7 A b^2+7 a^2 C+5 b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 b^4 d}+\frac {2 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 b^3 d}-\frac {2 a C \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 b^2 d}+\frac {2 C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 b d}\\ \end {align*}
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Mathematica [A] time = 2.65, size = 360, normalized size = 1.20 \[ \frac {6 \left (\frac {8 a \left (7 a^2 C+7 A b^2+6 b^2 C\right ) \left ((a+b) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-a \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{a+b}+\frac {\left (35 a^4 C+a^2 b^2 (35 A+13 C)+7 b^4 (9 A+7 C)\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a+b}+\frac {7 \left (15 a^4 C+3 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) \sin (c+d x) \left (\left (b^2-2 a^2\right ) \Pi \left (-\frac {b}{a};\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) F\left (\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )-2 a b E\left (\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )\right )}{a b^2 \sqrt {\sin ^2(c+d x)}}\right )+\sin (c+d x) \sqrt {\cos (c+d x)} \left (7 b \left (36 a^2 C+36 A b^2+43 b^2 C\right ) \cos (c+d x)-5 \left (84 a^3 C+84 a A b^2+18 a b^2 C \cos (2 (c+d x))+78 a b^2 C-7 b^3 C \cos (3 (c+d x))\right )\right )}{630 b^4 d} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{\frac {7}{2}}}{b \cos \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.91, size = 1554, normalized size = 5.20 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{\frac {7}{2}}}{b \cos \left (d x + c\right ) + a}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^{7/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )}{a+b\,\cos \left (c+d\,x\right )} \,d x \]
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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