Optimal. Leaf size=664 \[ -\frac {(a-b) \sqrt {a+b} \left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right ) \cot (c+d x) E\left (\text {ArcSin}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{192 a b d}+\frac {\sqrt {a+b} \left (15 a^3 B+8 b^3 (16 A+9 B)+2 a^2 b (132 A+59 B)+4 a b^2 (52 A+71 B)\right ) \cot (c+d x) F\left (\text {ArcSin}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{192 b d}-\frac {\sqrt {a+b} \left (40 a^3 A b+160 a A b^3-5 a^4 B+120 a^2 b^2 B+48 b^4 B\right ) \cot (c+d x) \Pi \left (\frac {a+b}{b};\text {ArcSin}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{64 b^2 d}+\frac {\left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{192 b d \sqrt {\cos (c+d x)}}+\frac {\left (24 a A b+5 a^2 B+12 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 d}+\frac {(8 A b+11 a B) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {b B \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{4 d} \]
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Rubi [A]
time = 1.43, antiderivative size = 664, 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 = {3069, 3128,
3140, 3132, 2888, 3077, 2895, 3073} \begin {gather*} \frac {\left (5 a^2 B+24 a A b+12 b^2 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}{32 d}+\frac {\sqrt {a+b} \left (15 a^3 B+2 a^2 b (132 A+59 B)+4 a b^2 (52 A+71 B)+8 b^3 (16 A+9 B)\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} F\left (\text {ArcSin}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{192 b d}-\frac {(a-b) \sqrt {a+b} \left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\text {ArcSin}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{192 a b d}+\frac {\left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{192 b d \sqrt {\cos (c+d x)}}-\frac {\sqrt {a+b} \left (-5 a^4 B+40 a^3 A b+120 a^2 b^2 B+160 a A b^3+48 b^4 B\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac {a+b}{b};\text {ArcSin}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{64 b^2 d}+\frac {(11 a B+8 A b) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{24 d}+\frac {b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2888
Rule 2895
Rule 3069
Rule 3073
Rule 3077
Rule 3128
Rule 3132
Rule 3140
Rubi steps
\begin {align*} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx &=\frac {b B \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac {1}{4} \int \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \left (\frac {1}{2} a (8 a A+3 b B)+\left (8 a A b+4 a^2 B+3 b^2 B\right ) \cos (c+d x)+\frac {1}{2} b (8 A b+11 a B) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {(8 A b+11 a B) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {b B \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac {\int \frac {\sqrt {a+b \cos (c+d x)} \left (\frac {1}{4} a b (8 A b+11 a B)+\frac {1}{2} b \left (24 a^2 A+16 A b^2+31 a b B\right ) \cos (c+d x)+\frac {3}{4} b \left (24 a A b+5 a^2 B+12 b^2 B\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{12 b}\\ &=\frac {\left (24 a A b+5 a^2 B+12 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 d}+\frac {(8 A b+11 a B) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {b B \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac {\int \frac {\frac {1}{8} a b \left (104 a A b+59 a^2 B+36 b^2 B\right )+\frac {1}{4} b \left (96 a^3 A+152 a A b^2+161 a^2 b B+36 b^3 B\right ) \cos (c+d x)+\frac {1}{8} b \left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{24 b}\\ &=\frac {\left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{192 b d \sqrt {\cos (c+d x)}}+\frac {\left (24 a A b+5 a^2 B+12 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 d}+\frac {(8 A b+11 a B) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {b B \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac {\int \frac {-\frac {1}{8} a b \left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right )+\frac {1}{4} a b^2 \left (104 a A b+59 a^2 B+36 b^2 B\right ) \cos (c+d x)+\frac {3}{8} b \left (40 a^3 A b+160 a A b^3-5 a^4 B+120 a^2 b^2 B+48 b^4 B\right ) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{48 b^2}\\ &=\frac {\left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{192 b d \sqrt {\cos (c+d x)}}+\frac {\left (24 a A b+5 a^2 B+12 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 d}+\frac {(8 A b+11 a B) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {b B \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac {\int \frac {-\frac {1}{8} a b \left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right )+\frac {1}{4} a b^2 \left (104 a A b+59 a^2 B+36 b^2 B\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{48 b^2}+\frac {\left (40 a^3 A b+160 a A b^3-5 a^4 B+120 a^2 b^2 B+48 b^4 B\right ) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}} \, dx}{128 b}\\ &=-\frac {\sqrt {a+b} \left (40 a^3 A b+160 a A b^3-5 a^4 B+120 a^2 b^2 B+48 b^4 B\right ) \cot (c+d x) \Pi \left (\frac {a+b}{b};\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{64 b^2 d}+\frac {\left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{192 b d \sqrt {\cos (c+d x)}}+\frac {\left (24 a A b+5 a^2 B+12 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 d}+\frac {(8 A b+11 a B) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {b B \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{4 d}-\frac {\left (a \left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right )\right ) \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{384 b}+\frac {\left (a \left (15 a^3 B+8 b^3 (16 A+9 B)+2 a^2 b (132 A+59 B)+4 a b^2 (52 A+71 B)\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{384 b}\\ &=-\frac {(a-b) \sqrt {a+b} \left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{192 a b d}+\frac {\sqrt {a+b} \left (15 a^3 B+8 b^3 (16 A+9 B)+2 a^2 b (132 A+59 B)+4 a b^2 (52 A+71 B)\right ) \cot (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{192 b d}-\frac {\sqrt {a+b} \left (40 a^3 A b+160 a A b^3-5 a^4 B+120 a^2 b^2 B+48 b^4 B\right ) \cot (c+d x) \Pi \left (\frac {a+b}{b};\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{64 b^2 d}+\frac {\left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{192 b d \sqrt {\cos (c+d x)}}+\frac {\left (24 a A b+5 a^2 B+12 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 d}+\frac {(8 A b+11 a B) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {b B \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{4 d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 6.44, size = 1287, normalized size = 1.94 \begin {gather*} \frac {-\frac {4 a \left (472 a^2 A b+128 A b^3+133 a^3 B+356 a b^2 B\right ) \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right )|-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}-4 a \left (384 a^3 A+608 a A b^2+644 a^2 b B+144 b^3 B\right ) \left (\frac {\sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right )|-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}-\frac {\sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \Pi \left (-\frac {a}{b};\text {ArcSin}\left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right )|-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{b \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}\right )+2 \left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right ) \left (\frac {i \cos \left (\frac {1}{2} (c+d x)\right ) \sqrt {a+b \cos (c+d x)} E\left (i \sinh ^{-1}\left (\frac {\sin \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\cos (c+d x)}}\right )|-\frac {2 a}{-a-b}\right ) \sec (c+d x)}{b \sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)} \sqrt {\frac {(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac {2 a \left (\frac {a \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right )|-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}-\frac {a \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \Pi \left (-\frac {a}{b};\text {ArcSin}\left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right )|-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{b \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}\right )}{b}+\frac {\sqrt {a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt {\cos (c+d x)}}\right )}{384 d}+\frac {\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \left (\frac {1}{96} \left (104 a A b+59 a^2 B+42 b^2 B\right ) \sin (c+d x)+\frac {1}{48} b (8 A b+17 a B) \sin (2 (c+d x))+\frac {1}{16} b^2 B \sin (3 (c+d x))\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. \(4237\) vs.
\(2(610)=1220\).
time = 0.61, size = 4238, normalized size = 6.38
method | result | size |
default | \(\text {Expression too large to display}\) | \(4238\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 \sqrt {\cos \left (c+d\,x\right )}\,\left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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