Optimal. Leaf size=244 \[ \frac {2 b \left (21 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {2 a \left (5 a^2 (A-C)-3 b^2 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}-\frac {2 b \left (6 a^2 (7 A-3 C)-b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d}-\frac {2 a b^2 (35 A-11 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{35 d}-\frac {2 b (7 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}} \]
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Rubi [A] time = 0.74, antiderivative size = 244, 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, 3049, 3033, 3023, 2748, 2641, 2639} \[ \frac {2 b \left (21 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {2 a \left (5 a^2 (A-C)-3 b^2 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}-\frac {2 b \left (6 a^2 (7 A-3 C)-b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d}-\frac {2 a b^2 (35 A-11 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{35 d}-\frac {2 b (7 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2748
Rule 3023
Rule 3033
Rule 3048
Rule 3049
Rubi steps
\begin {align*} \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx &=\frac {2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+2 \int \frac {(a+b \cos (c+d x))^2 \left (3 A b-\frac {1}{2} a (A-C) \cos (c+d x)-\frac {1}{2} b (7 A-C) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 b (7 A-C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {4}{7} \int \frac {(a+b \cos (c+d x)) \left (\frac {1}{4} a b (35 A+C)-\frac {1}{4} \left (7 a^2 (A-C)-b^2 (7 A+5 C)\right ) \cos (c+d x)-\frac {1}{4} a b (35 A-11 C) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 a b^2 (35 A-11 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{35 d}-\frac {2 b (7 A-C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {8}{35} \int \frac {\frac {5}{8} a^2 b (35 A+C)-\frac {7}{8} a \left (5 a^2 (A-C)-3 b^2 (5 A+3 C)\right ) \cos (c+d x)-\frac {5}{8} b \left (6 a^2 (7 A-3 C)-b^2 (7 A+5 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 b \left (6 a^2 (7 A-3 C)-b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}-\frac {2 a b^2 (35 A-11 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{35 d}-\frac {2 b (7 A-C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {16}{105} \int \frac {\frac {5}{16} b \left (21 a^2 (3 A+C)+b^2 (7 A+5 C)\right )-\frac {21}{16} a \left (5 a^2 (A-C)-3 b^2 (5 A+3 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 b \left (6 a^2 (7 A-3 C)-b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}-\frac {2 a b^2 (35 A-11 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{35 d}-\frac {2 b (7 A-C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {1}{5} \left (a \left (5 a^2 (A-C)-3 b^2 (5 A+3 C)\right )\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} \left (b \left (21 a^2 (3 A+C)+b^2 (7 A+5 C)\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 a \left (5 a^2 (A-C)-3 b^2 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 b \left (21 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {2 b \left (6 a^2 (7 A-3 C)-b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}-\frac {2 a b^2 (35 A-11 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{35 d}-\frac {2 b (7 A-C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 1.80, size = 172, normalized size = 0.70 \[ \frac {-84 \left (5 a^3 (A-C)-3 a b^2 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+20 b \left (21 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\frac {\sin (c+d x) \left (3 \left (140 a^3 A+42 a b^2 C \cos (2 (c+d x))+42 a b^2 C+5 b^3 C \cos (3 (c+d x))\right )+5 b \left (84 a^2 C+28 A b^2+29 b^2 C\right ) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}}}{210 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C b^{3} \cos \left (d x + c\right )^{5} + 3 \, C a b^{2} \cos \left (d x + c\right )^{4} + 3 \, A a^{2} b \cos \left (d x + c\right ) + A a^{3} + {\left (3 \, C a^{2} b + A b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{3} + 3 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2}}{\cos \left (d x + c\right )^{\frac {3}{2}}}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.42, size = 943, normalized size = 3.86 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.82, size = 274, normalized size = 1.12 \[ \frac {2\,\left (C\,a^3\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+C\,a^2\,b\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+C\,a^2\,b\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{d}+\frac {A\,b^3\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}+\frac {6\,A\,a\,b^2\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {6\,A\,a^2\,b\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,A\,a^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,b^3\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,C\,a\,b^2\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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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