Optimal. Leaf size=192 \[ \frac {2 a^2 (3 a B+7 A b) \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}+\frac {2 \left (a^3 A+9 a^2 b B+9 a A b^2+b^3 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}-\frac {2 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 b^2 (a A-b B) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)} \]
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Rubi [A] time = 0.47, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2989, 3031, 3023, 2748, 2641, 2639} \[ \frac {2 \left (a^3 A+9 a^2 b B+9 a A b^2+b^3 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}-\frac {2 \left (3 a^2 A b+a^3 B-3 a b^2 B-A b^3\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {2 a^2 (3 a B+7 A b) \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}-\frac {2 b^2 (a A-b B) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2748
Rule 2989
Rule 3023
Rule 3031
Rubi steps
\begin {align*} \int \frac {(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\cos ^{\frac {5}{2}}(c+d x)} \, dx &=\frac {2 a A (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2}{3} \int \frac {(a+b \cos (c+d x)) \left (\frac {1}{2} a (7 A b+3 a B)+\frac {1}{2} \left (a^2 A+3 A b^2+6 a b B\right ) \cos (c+d x)-\frac {3}{2} b (a A-b B) \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 (7 A b+3 a B) \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}+\frac {2 a A (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {4}{3} \int \frac {-\frac {1}{4} a \left (a^2 A+10 A b^2+9 a b B\right )+\frac {3}{4} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \cos (c+d x)+\frac {3}{4} b^2 (a A-b B) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 a^2 (7 A b+3 a B) \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}-\frac {2 b^2 (a A-b B) \sqrt {\cos (c+d x)} \sin (c+d x)}{3 d}+\frac {2 a A (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {8}{9} \int \frac {-\frac {3}{8} \left (a^3 A+9 a A b^2+9 a^2 b B+b^3 B\right )+\frac {9}{8} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 a^2 (7 A b+3 a B) \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}-\frac {2 b^2 (a A-b B) \sqrt {\cos (c+d x)} \sin (c+d x)}{3 d}+\frac {2 a A (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}-\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \int \sqrt {\cos (c+d x)} \, dx-\frac {1}{3} \left (-a^3 A-9 a A b^2-9 a^2 b B-b^3 B\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {2 \left (a^3 A+9 a A b^2+9 a^2 b B+b^3 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}+\frac {2 a^2 (7 A b+3 a B) \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}-\frac {2 b^2 (a A-b B) \sqrt {\cos (c+d x)} \sin (c+d x)}{3 d}+\frac {2 a A (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A] time = 1.11, size = 165, normalized size = 0.86 \[ \frac {2 a^3 A \tan (c+d x)+6 a^3 B \sin (c+d x)+18 a^2 A b \sin (c+d x)+2 \left (a^3 A+9 a^2 b B+9 a A b^2+b^3 B\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-6 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+b^3 B \sin (2 (c+d x))}{3 d \sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B b^{3} \cos \left (d x + c\right )^{4} + A a^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{\frac {5}{2}}}, x\right ) \]
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 \[ \int \frac {{\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.80, size = 1212, normalized size = 6.31 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.34, size = 255, normalized size = 1.33 \[ \frac {2\,\left (A\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )\,b^3+3\,A\,a\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )\,b^2\right )}{d}+\frac {B\,b^3\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}+\frac {6\,B\,a\,b^2\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {6\,B\,a^2\,b\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,A\,a^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {2\,B\,a^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {6\,A\,a^2\,b\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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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