Optimal. Leaf size=523 \[ \frac {\left (24 a^4 A b-65 a^2 A b^3+35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (8 a^4 A-61 a^2 A b^2+35 A b^4+33 a^3 b B-15 a b^3 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {b \left (63 a^4 A b-86 a^2 A b^3+35 A b^5-35 a^5 B+38 a^3 b^2 B-15 a b^4 B\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{4 a^4 (a-b)^2 (a+b)^3 d}+\frac {\left (8 a^4 A-61 a^2 A b^2+35 A b^4+33 a^3 b B-15 a b^3 B\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (24 a^4 A b-65 a^2 A b^3+35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B\right ) \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}}+\frac {b (A b-a B) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac {b \left (13 a^2 A b-7 A b^3-9 a^3 B+3 a b^2 B\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))} \]
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Rubi [A]
time = 1.26, antiderivative size = 523, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {3079, 3134,
3138, 2719, 3081, 2720, 2884} \begin {gather*} \frac {b (A b-a B) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac {b \left (-9 a^3 B+13 a^2 A b+3 a b^2 B-7 A b^3\right ) \sin (c+d x)}{4 a^2 d \left (a^2-b^2\right )^2 \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac {\left (8 a^4 A+33 a^3 b B-61 a^2 A b^2-15 a b^3 B+35 A b^4\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{12 a^3 d \left (a^2-b^2\right )^2}+\frac {\left (8 a^4 A+33 a^3 b B-61 a^2 A b^2-15 a b^3 B+35 A b^4\right ) \sin (c+d x)}{12 a^3 d \left (a^2-b^2\right )^2 \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (-8 a^5 B+24 a^4 A b+29 a^3 b^2 B-65 a^2 A b^3-15 a b^4 B+35 A b^5\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 a^4 d \left (a^2-b^2\right )^2}+\frac {b \left (-35 a^5 B+63 a^4 A b+38 a^3 b^2 B-86 a^2 A b^3-15 a b^4 B+35 A b^5\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{4 a^4 d (a-b)^2 (a+b)^3}-\frac {\left (-8 a^5 B+24 a^4 A b+29 a^3 b^2 B-65 a^2 A b^3-15 a b^4 B+35 A b^5\right ) \sin (c+d x)}{4 a^4 d \left (a^2-b^2\right )^2 \sqrt {\cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2720
Rule 2884
Rule 3079
Rule 3081
Rule 3134
Rule 3138
Rubi steps
\begin {align*} \int \frac {A+B \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx &=\frac {b (A b-a B) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac {\int \frac {\frac {1}{2} \left (4 a^2 A-7 A b^2+3 a b B\right )-2 a (A b-a B) \cos (c+d x)+\frac {5}{2} b (A b-a B) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=\frac {b (A b-a B) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac {b \left (13 a^2 A b-7 A b^3-9 a^3 B+3 a b^2 B\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac {\int \frac {\frac {1}{4} \left (8 a^4 A-61 a^2 A b^2+35 A b^4+33 a^3 b B-15 a b^3 B\right )-a \left (4 a^2 A b-A b^3-2 a^3 B-a b^2 B\right ) \cos (c+d x)+\frac {3}{4} b \left (13 a^2 A b-7 A b^3-9 a^3 B+3 a b^2 B\right ) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {\left (8 a^4 A-61 a^2 A b^2+35 A b^4+33 a^3 b B-15 a b^3 B\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {b (A b-a B) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac {b \left (13 a^2 A b-7 A b^3-9 a^3 B+3 a b^2 B\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac {\int \frac {-\frac {3}{8} \left (24 a^4 A b-65 a^2 A b^3+35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B\right )+\frac {1}{2} a \left (2 a^4 A+14 a^2 A b^2-7 A b^4-12 a^3 b B+3 a b^3 B\right ) \cos (c+d x)+\frac {1}{8} b \left (8 a^4 A-61 a^2 A b^2+35 A b^4+33 a^3 b B-15 a b^3 B\right ) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{3 a^3 \left (a^2-b^2\right )^2}\\ &=\frac {\left (8 a^4 A-61 a^2 A b^2+35 A b^4+33 a^3 b B-15 a b^3 B\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (24 a^4 A b-65 a^2 A b^3+35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B\right ) \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}}+\frac {b (A b-a B) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac {b \left (13 a^2 A b-7 A b^3-9 a^3 B+3 a b^2 B\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac {2 \int \frac {\frac {1}{16} \left (8 a^6 A+128 a^4 A b^2-223 a^2 A b^4+105 A b^6-72 a^5 b B+99 a^3 b^3 B-45 a b^5 B\right )+\frac {1}{4} a \left (20 a^4 A b-64 a^2 A b^3+35 A b^5-6 a^5 B+30 a^3 b^2 B-15 a b^4 B\right ) \cos (c+d x)+\frac {3}{16} b \left (24 a^4 A b-65 a^2 A b^3+35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{3 a^4 \left (a^2-b^2\right )^2}\\ &=\frac {\left (8 a^4 A-61 a^2 A b^2+35 A b^4+33 a^3 b B-15 a b^3 B\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (24 a^4 A b-65 a^2 A b^3+35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B\right ) \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}}+\frac {b (A b-a B) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac {b \left (13 a^2 A b-7 A b^3-9 a^3 B+3 a b^2 B\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {2 \int \frac {-\frac {1}{16} b \left (8 a^6 A+128 a^4 A b^2-223 a^2 A b^4+105 A b^6-72 a^5 b B+99 a^3 b^3 B-45 a b^5 B\right )-\frac {1}{16} a b^2 \left (8 a^4 A-61 a^2 A b^2+35 A b^4+33 a^3 b B-15 a b^3 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{3 a^4 b \left (a^2-b^2\right )^2}+\frac {\left (24 a^4 A b-65 a^2 A b^3+35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B\right ) \int \sqrt {\cos (c+d x)} \, dx}{8 a^4 \left (a^2-b^2\right )^2}\\ &=\frac {\left (24 a^4 A b-65 a^2 A b^3+35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (8 a^4 A-61 a^2 A b^2+35 A b^4+33 a^3 b B-15 a b^3 B\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (24 a^4 A b-65 a^2 A b^3+35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B\right ) \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}}+\frac {b (A b-a B) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac {b \left (13 a^2 A b-7 A b^3-9 a^3 B+3 a b^2 B\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac {\left (8 a^4 A-61 a^2 A b^2+35 A b^4+33 a^3 b B-15 a b^3 B\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{24 a^3 \left (a^2-b^2\right )^2}+\frac {\left (b \left (63 a^4 A b-86 a^2 A b^3+35 A b^5-35 a^5 B+38 a^3 b^2 B-15 a b^4 B\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{8 a^4 \left (a^2-b^2\right )^2}\\ &=\frac {\left (24 a^4 A b-65 a^2 A b^3+35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (8 a^4 A-61 a^2 A b^2+35 A b^4+33 a^3 b B-15 a b^3 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {b \left (63 a^4 A b-86 a^2 A b^3+35 A b^5-35 a^5 B+38 a^3 b^2 B-15 a b^4 B\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{4 a^4 (a-b)^2 (a+b)^3 d}+\frac {\left (8 a^4 A-61 a^2 A b^2+35 A b^4+33 a^3 b B-15 a b^3 B\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (24 a^4 A b-65 a^2 A b^3+35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B\right ) \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}}+\frac {b (A b-a B) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac {b \left (13 a^2 A b-7 A b^3-9 a^3 B+3 a b^2 B\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 17.33, size = 570, normalized size = 1.09 \begin {gather*} \frac {\frac {2 \left (16 a^6 A+328 a^4 A b^2-641 a^2 A b^4+315 A b^6-168 a^5 b B+285 a^3 b^3 B-135 a b^5 B\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a+b}+\frac {\left (160 a^5 A b-512 a^3 A b^3+280 a A b^5-48 a^6 B+240 a^4 b^2 B-120 a^2 b^4 B\right ) \left (2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-\frac {2 a \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a+b}\right )}{b}+\frac {2 \left (72 a^4 A b^2-195 a^2 A b^4+105 A b^6-24 a^5 b B+87 a^3 b^3 B-45 a b^5 B\right ) \cos (2 (c+d x)) \left (-2 a b E\left (\left .\text {ArcSin}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) F\left (\left .\text {ArcSin}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+\left (-2 a^2+b^2\right ) \Pi \left (-\frac {b}{a};\left .\text {ArcSin}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )\right ) \sin (c+d x)}{a b^2 \sqrt {1-\cos ^2(c+d x)} \left (-1+2 \cos ^2(c+d x)\right )}}{48 a^4 (a-b)^2 (a+b)^2 d}+\frac {\sqrt {\cos (c+d x)} \left (\frac {2 \sec (c+d x) (-3 A b \sin (c+d x)+a B \sin (c+d x))}{a^4}+\frac {A b^4 \sin (c+d x)-a b^3 B \sin (c+d x)}{2 a^3 \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac {17 a^2 A b^4 \sin (c+d x)-11 A b^6 \sin (c+d x)-13 a^3 b^3 B \sin (c+d x)+7 a b^5 B \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac {2 A \sec (c+d x) \tan (c+d x)}{3 a^3}\right )}{d} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2130\) vs.
\(2(579)=1158\).
time = 2.87, size = 2131, normalized size = 4.07
method | result | size |
default | \(\text {Expression too large to display}\) | \(2131\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {A+B\,\cos \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{5/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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