Optimal. Leaf size=506 \[ -\frac {(a-b) \sqrt {a+b} \left (33 a^2+16 b^2\right ) \cot (c+d x) E\left (\text {ArcSin}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{24 a d}+\frac {\sqrt {a+b} \left (33 a^2+26 a b+16 b^2\right ) \cot (c+d x) F\left (\text {ArcSin}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{24 d}-\frac {5 a \sqrt {a+b} \left (a^2+4 b^2\right ) \cot (c+d x) \Pi \left (\frac {a+b}{b};\text {ArcSin}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{8 b d}+\frac {\left (33 a^2+16 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{24 d \sqrt {\cos (c+d x)}}+\frac {13 a b \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{12 d}+\frac {b^2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 d} \]
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Rubi [A]
time = 0.87, antiderivative size = 506, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2872, 3128,
3140, 3132, 2888, 3077, 2895, 3073} \begin {gather*} \frac {\sqrt {a+b} \left (33 a^2+26 a b+16 b^2\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 )}{24 d}-\frac {(a-b) \sqrt {a+b} \left (33 a^2+16 b^2\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 )}{24 a d}-\frac {5 a \sqrt {a+b} \left (a^2+4 b^2\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 )}{8 b d}+\frac {\left (33 a^2+16 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{24 d \sqrt {\cos (c+d x)}}+\frac {b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {13 a b \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}{12 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2872
Rule 2888
Rule 2895
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} \, dx &=\frac {b^2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 d}+\frac {1}{3} \int \frac {\sqrt {\cos (c+d x)} \left (\frac {3}{2} a \left (2 a^2+b^2\right )+b \left (9 a^2+2 b^2\right ) \cos (c+d x)+\frac {13}{2} a b^2 \cos ^2(c+d x)\right )}{\sqrt {a+b \cos (c+d x)}} \, dx\\ &=\frac {13 a b \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{12 d}+\frac {b^2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 d}+\frac {\int \frac {\frac {13 a^2 b^2}{4}+\frac {1}{2} a b \left (12 a^2+19 b^2\right ) \cos (c+d x)+\frac {1}{4} b^2 \left (33 a^2+16 b^2\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{6 b}\\ &=\frac {\left (33 a^2+16 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{24 d \sqrt {\cos (c+d x)}}+\frac {13 a b \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{12 d}+\frac {b^2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 d}+\frac {\int \frac {-\frac {1}{4} a b^2 \left (33 a^2+16 b^2\right )+\frac {13}{2} a^2 b^3 \cos (c+d x)+\frac {15}{4} a b^2 \left (a^2+4 b^2\right ) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{12 b^2}\\ &=\frac {\left (33 a^2+16 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{24 d \sqrt {\cos (c+d x)}}+\frac {13 a b \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{12 d}+\frac {b^2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 d}+\frac {\int \frac {-\frac {1}{4} a b^2 \left (33 a^2+16 b^2\right )+\frac {13}{2} a^2 b^3 \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{12 b^2}+\frac {1}{16} \left (5 a \left (a^2+4 b^2\right )\right ) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}} \, dx\\ &=-\frac {5 a \sqrt {a+b} \left (a^2+4 b^2\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}}}{8 b d}+\frac {\left (33 a^2+16 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{24 d \sqrt {\cos (c+d x)}}+\frac {13 a b \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{12 d}+\frac {b^2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 d}-\frac {1}{48} \left (a \left (33 a^2+16 b^2\right )\right ) \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx+\frac {1}{48} \left (a \left (33 a^2+26 a b+16 b^2\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx\\ &=-\frac {(a-b) \sqrt {a+b} \left (33 a^2+16 b^2\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}}}{24 a d}+\frac {\sqrt {a+b} \left (33 a^2+26 a b+16 b^2\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}}}{24 d}-\frac {5 a \sqrt {a+b} \left (a^2+4 b^2\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}}}{8 b d}+\frac {\left (33 a^2+16 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{24 d \sqrt {\cos (c+d x)}}+\frac {13 a b \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{12 d}+\frac {b^2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 19.04, size = 1203, normalized size = 2.38 \begin {gather*} \frac {-\frac {4 a \left (59 a^2 b+16 b^3\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 (48 a^3+76 a b^2\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 (33 a^2 b+16 b^3\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 )}{48 d}+\frac {\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \left (\frac {13}{12} a b \sin (c+d x)+\frac {1}{6} b^2 \sin (2 (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. \(1865\) vs.
\(2(458)=916\).
time = 0.29, size = 1866, normalized size = 3.69
method | result | size |
default | \(\text {Expression too large to display}\) | \(1866\) |
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 )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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