Optimal. Leaf size=243 \[ \frac {2 a \left (a^2 (5 A+7 C)+21 b^2 (A+3 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {2 b \left (3 a^2 (3 A+5 C)+5 b^2 (A-C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a \left (5 a^2 (5 A+7 C)+24 A b^2\right ) \sin (c+d x)}{105 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {6 b \left (7 a^2 (3 A+5 C)+8 A b^2\right ) \sin (c+d x)}{35 d \sqrt {\cos (c+d x)}}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {12 A b \sin (c+d x) (a+b \cos (c+d x))^2}{35 d \cos ^{\frac {5}{2}}(c+d x)} \]
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Rubi [A] time = 0.72, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3048, 3047, 3031, 3021, 2748, 2641, 2639} \[ \frac {2 a \left (a^2 (5 A+7 C)+21 b^2 (A+3 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {2 b \left (3 a^2 (3 A+5 C)+5 b^2 (A-C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a \left (5 a^2 (5 A+7 C)+24 A b^2\right ) \sin (c+d x)}{105 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {6 b \left (7 a^2 (3 A+5 C)+8 A b^2\right ) \sin (c+d x)}{35 d \sqrt {\cos (c+d x)}}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {12 A b \sin (c+d x) (a+b \cos (c+d x))^2}{35 d \cos ^{\frac {5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2748
Rule 3021
Rule 3031
Rule 3047
Rule 3048
Rubi steps
\begin {align*} \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx &=\frac {2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2}{7} \int \frac {(a+b \cos (c+d x))^2 \left (3 A b+\frac {1}{2} a (5 A+7 C) \cos (c+d x)-\frac {1}{2} b (A-7 C) \cos ^2(c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {12 A b (a+b \cos (c+d x))^2 \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4}{35} \int \frac {(a+b \cos (c+d x)) \left (\frac {1}{4} \left (24 A b^2+5 a^2 (5 A+7 C)\right )+\frac {1}{2} a b (19 A+35 C) \cos (c+d x)-\frac {1}{4} b^2 (11 A-35 C) \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 a \left (24 A b^2+5 a^2 (5 A+7 C)\right ) \sin (c+d x)}{105 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {12 A b (a+b \cos (c+d x))^2 \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}-\frac {8}{105} \int \frac {-\frac {9}{8} b \left (8 A b^2+7 a^2 (3 A+5 C)\right )-\frac {5}{8} a \left (21 b^2 (A+3 C)+a^2 (5 A+7 C)\right ) \cos (c+d x)+\frac {3}{8} b^3 (11 A-35 C) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a \left (24 A b^2+5 a^2 (5 A+7 C)\right ) \sin (c+d x)}{105 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {6 b \left (8 A b^2+7 a^2 (3 A+5 C)\right ) \sin (c+d x)}{35 d \sqrt {\cos (c+d x)}}+\frac {12 A b (a+b \cos (c+d x))^2 \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}-\frac {16}{105} \int \frac {-\frac {5}{16} a \left (21 b^2 (A+3 C)+a^2 (5 A+7 C)\right )+\frac {21}{16} b \left (5 b^2 (A-C)+3 a^2 (3 A+5 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 a \left (24 A b^2+5 a^2 (5 A+7 C)\right ) \sin (c+d x)}{105 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {6 b \left (8 A b^2+7 a^2 (3 A+5 C)\right ) \sin (c+d x)}{35 d \sqrt {\cos (c+d x)}}+\frac {12 A b (a+b \cos (c+d x))^2 \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}-\frac {1}{5} \left (b \left (5 b^2 (A-C)+3 a^2 (3 A+5 C)\right )\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} \left (a \left (21 b^2 (A+3 C)+a^2 (5 A+7 C)\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 b \left (5 b^2 (A-C)+3 a^2 (3 A+5 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a \left (21 b^2 (A+3 C)+a^2 (5 A+7 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 a \left (24 A b^2+5 a^2 (5 A+7 C)\right ) \sin (c+d x)}{105 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {6 b \left (8 A b^2+7 a^2 (3 A+5 C)\right ) \sin (c+d x)}{35 d \sqrt {\cos (c+d x)}}+\frac {12 A b (a+b \cos (c+d x))^2 \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A] time = 1.58, size = 241, normalized size = 0.99 \[ \frac {25 a^3 A \sin (2 (c+d x))+30 a^3 A \tan (c+d x)+35 a^3 C \sin (2 (c+d x))+10 a \left (a^2 (5 A+7 C)+21 b^2 (A+3 C)\right ) \cos ^{\frac {5}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-42 b \left (3 a^2 (3 A+5 C)+5 b^2 (A-C)\right ) \cos ^{\frac {5}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+126 a^2 A b \sin (c+d x)+378 a^2 A b \sin (c+d x) \cos ^2(c+d x)+630 a^2 b C \sin (c+d x) \cos ^2(c+d x)+105 a A b^2 \sin (2 (c+d x))+210 A b^3 \sin (c+d x) \cos ^2(c+d x)}{105 d \cos ^{\frac {5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C b^{3} \cos \left (d x + c\right )^{5} + 3 \, C a b^{2} \cos \left (d x + c\right )^{4} + 3 \, A a^{2} b \cos \left (d x + c\right ) + A a^{3} + {\left (3 \, C a^{2} b + A b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{3} + 3 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2}}{\cos \left (d x + c\right )^{\frac {9}{2}}}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 8.09, size = 1113, normalized size = 4.58 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.11, size = 283, normalized size = 1.16 \[ \frac {2\,\left (C\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )\,b^3+3\,C\,a\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )\,b^2\right )}{d}+\frac {\frac {2\,A\,a^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {7}{4},\frac {1}{2};\ -\frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7}+2\,A\,b^3\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )+2\,A\,a\,b^2\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )+\frac {6\,A\,a^2\,b\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{4},\frac {1}{2};\ -\frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{5}}{d\,{\cos \left (c+d\,x\right )}^{7/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {2\,C\,a^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {6\,C\,a^2\,b\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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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