Optimal. Leaf size=473 \[ -\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (8 a^2 C+7 A b^2-b^2 C\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b^2 d \left (a^2-b^2\right )}-\frac {2 a \left (48 a^2 C+35 A b^2-13 b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{35 b^3 d \left (a^2-b^2\right )}+\frac {2 \left (192 a^4 C+2 a^2 b^2 (70 A-31 C)-5 b^4 (7 A+5 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{105 b^4 d \left (a^2-b^2\right )}+\frac {2 \left (384 a^4 C+4 a^2 b^2 (70 A+29 C)+5 b^4 (7 A+5 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^5 d \sqrt {a+b \cos (c+d x)}}-\frac {2 a \left (384 a^4 C+4 a^2 b^2 (70 A-43 C)-b^4 (175 A+107 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^5 d \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}} \]
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Rubi [A] time = 1.13, antiderivative size = 473, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {3048, 3049, 3023, 2752, 2663, 2661, 2655, 2653} \[ -\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (8 a^2 C+7 A b^2-b^2 C\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b^2 d \left (a^2-b^2\right )}-\frac {2 a \left (48 a^2 C+35 A b^2-13 b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{35 b^3 d \left (a^2-b^2\right )}+\frac {2 \left (2 a^2 b^2 (70 A-31 C)+192 a^4 C-5 b^4 (7 A+5 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{105 b^4 d \left (a^2-b^2\right )}+\frac {2 \left (4 a^2 b^2 (70 A+29 C)+384 a^4 C+5 b^4 (7 A+5 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^5 d \sqrt {a+b \cos (c+d x)}}-\frac {2 a \left (4 a^2 b^2 (70 A-43 C)+384 a^4 C-b^4 (175 A+107 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^5 d \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 3023
Rule 3048
Rule 3049
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx &=-\frac {2 \left (A b^2+a^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}-\frac {2 \int \frac {\cos ^2(c+d x) \left (3 \left (A b^2+a^2 C\right )-\frac {1}{2} a b (A+C) \cos (c+d x)-\frac {1}{2} \left (7 A b^2+8 a^2 C-b^2 C\right ) \cos ^2(c+d x)\right )}{\sqrt {a+b \cos (c+d x)}} \, dx}{b \left (a^2-b^2\right )}\\ &=-\frac {2 \left (A b^2+a^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (7 A b^2+8 a^2 C-b^2 C\right ) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{7 b^2 \left (a^2-b^2\right ) d}-\frac {4 \int \frac {\cos (c+d x) \left (-a \left (7 A b^2+\left (8 a^2-b^2\right ) C\right )+\frac {1}{4} b \left (7 A b^2+2 a^2 C+5 b^2 C\right ) \cos (c+d x)+\frac {1}{4} a \left (35 A b^2+48 a^2 C-13 b^2 C\right ) \cos ^2(c+d x)\right )}{\sqrt {a+b \cos (c+d x)}} \, dx}{7 b^2 \left (a^2-b^2\right )}\\ &=-\frac {2 \left (A b^2+a^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}-\frac {2 a \left (35 A b^2+48 a^2 C-13 b^2 C\right ) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{35 b^3 \left (a^2-b^2\right ) d}+\frac {2 \left (7 A b^2+8 a^2 C-b^2 C\right ) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{7 b^2 \left (a^2-b^2\right ) d}-\frac {8 \int \frac {\frac {1}{4} a^2 \left (35 A b^2+48 a^2 C-13 b^2 C\right )-\frac {1}{8} a b \left (35 A b^2+16 a^2 C+19 b^2 C\right ) \cos (c+d x)-\frac {1}{8} \left (2 a^2 b^2 (70 A-31 C)+192 a^4 C-5 b^4 (7 A+5 C)\right ) \cos ^2(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{35 b^3 \left (a^2-b^2\right )}\\ &=-\frac {2 \left (A b^2+a^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (2 a^2 b^2 (70 A-31 C)+192 a^4 C-5 b^4 (7 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b^4 \left (a^2-b^2\right ) d}-\frac {2 a \left (35 A b^2+48 a^2 C-13 b^2 C\right ) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{35 b^3 \left (a^2-b^2\right ) d}+\frac {2 \left (7 A b^2+8 a^2 C-b^2 C\right ) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{7 b^2 \left (a^2-b^2\right ) d}-\frac {16 \int \frac {\frac {1}{16} b \left (2 a^2 b^2 (35 A-8 C)+96 a^4 C+5 b^4 (7 A+5 C)\right )+\frac {1}{16} a \left (4 a^2 b^2 (70 A-43 C)+384 a^4 C-b^4 (175 A+107 C)\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{105 b^4 \left (a^2-b^2\right )}\\ &=-\frac {2 \left (A b^2+a^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (2 a^2 b^2 (70 A-31 C)+192 a^4 C-5 b^4 (7 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b^4 \left (a^2-b^2\right ) d}-\frac {2 a \left (35 A b^2+48 a^2 C-13 b^2 C\right ) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{35 b^3 \left (a^2-b^2\right ) d}+\frac {2 \left (7 A b^2+8 a^2 C-b^2 C\right ) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{7 b^2 \left (a^2-b^2\right ) d}+\frac {\left (384 a^4 C+5 b^4 (7 A+5 C)+4 a^2 b^2 (70 A+29 C)\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{105 b^5}-\frac {\left (a \left (4 a^2 b^2 (70 A-43 C)+384 a^4 C-b^4 (175 A+107 C)\right )\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{105 b^5 \left (a^2-b^2\right )}\\ &=-\frac {2 \left (A b^2+a^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (2 a^2 b^2 (70 A-31 C)+192 a^4 C-5 b^4 (7 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b^4 \left (a^2-b^2\right ) d}-\frac {2 a \left (35 A b^2+48 a^2 C-13 b^2 C\right ) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{35 b^3 \left (a^2-b^2\right ) d}+\frac {2 \left (7 A b^2+8 a^2 C-b^2 C\right ) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{7 b^2 \left (a^2-b^2\right ) d}-\frac {\left (a \left (4 a^2 b^2 (70 A-43 C)+384 a^4 C-b^4 (175 A+107 C)\right ) \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{105 b^5 \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (\left (384 a^4 C+5 b^4 (7 A+5 C)+4 a^2 b^2 (70 A+29 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{105 b^5 \sqrt {a+b \cos (c+d x)}}\\ &=-\frac {2 a \left (4 a^2 b^2 (70 A-43 C)+384 a^4 C-b^4 (175 A+107 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^5 \left (a^2-b^2\right ) d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (384 a^4 C+5 b^4 (7 A+5 C)+4 a^2 b^2 (70 A+29 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^5 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (A b^2+a^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (2 a^2 b^2 (70 A-31 C)+192 a^4 C-5 b^4 (7 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b^4 \left (a^2-b^2\right ) d}-\frac {2 a \left (35 A b^2+48 a^2 C-13 b^2 C\right ) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{35 b^3 \left (a^2-b^2\right ) d}+\frac {2 \left (7 A b^2+8 a^2 C-b^2 C\right ) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{7 b^2 \left (a^2-b^2\right ) d}\\ \end {align*}
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Mathematica [A] time = 1.68, size = 358, normalized size = 0.76 \[ \frac {b (a-b) (a+b) \left (\left (a^2-b^2\right ) \left (348 a^2 C+140 A b^2+115 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))-78 a b C \left (a^2-b^2\right ) \sin (2 (c+d x)) (a+b \cos (c+d x))+15 b^2 C \left (a^2-b^2\right ) \sin (3 (c+d x)) (a+b \cos (c+d x))+420 a^3 \left (a^2 C+A b^2\right ) \sin (c+d x)\right )-4 \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b \left (96 a^4 b C+2 a^2 b^3 (35 A-8 C)+5 b^5 (7 A+5 C)\right ) F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+a \left (384 a^4 C+4 a^2 b^2 (70 A-43 C)-b^4 (175 A+107 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )\right )\right )}{210 b^5 d (a-b) (a+b) \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.06, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{5} + A \cos \left (d x + c\right )^{3}\right )} \sqrt {b \cos \left (d x + c\right ) + a}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{3}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 9.98, size = 1788, normalized size = 3.78 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{3}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^3\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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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