Optimal. Leaf size=522 \[ \frac {2 (a-b) b \sqrt {a+b} \left (741 a^4+51 a^2 b^2+8 b^4\right ) \cot (c+d x) E\left (\text {ArcSin}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{693 a^4 d}+\frac {2 (a-b) \sqrt {a+b} \left (135 a^4-606 a^3 b+57 a^2 b^2+6 a b^3+8 b^4\right ) \cot (c+d x) F\left (\text {ArcSin}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{693 a^3 d}+\frac {2 a^2 \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {46 a b \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2+113 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 b \left (229 a^2+3 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (135 a^4+205 a^2 b^2-4 b^4\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 a^2 d \cos ^{\frac {3}{2}}(c+d x)} \]
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Rubi [A]
time = 1.14, antiderivative size = 522, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2871, 3134,
3077, 2895, 3073} \begin {gather*} \frac {2 \left (81 a^2+113 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 b \left (229 a^2+3 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{693 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 a^2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 b (a-b) \sqrt {a+b} \left (741 a^4+51 a^2 b^2+8 b^4\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 )}{693 a^4 d}+\frac {2 \left (135 a^4+205 a^2 b^2-4 b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{693 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (a-b) \sqrt {a+b} \left (135 a^4-606 a^3 b+57 a^2 b^2+6 a b^3+8 b^4\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 )}{693 a^3 d}+\frac {46 a b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{99 d \cos ^{\frac {9}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2871
Rule 2895
Rule 3073
Rule 3077
Rule 3134
Rubi steps
\begin {align*} \int \frac {(a+b \cos (c+d x))^{5/2}}{\cos ^{\frac {13}{2}}(c+d x)} \, dx &=\frac {2 a^2 \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2}{11} \int \frac {\frac {23 a^2 b}{2}+\frac {3}{2} a \left (3 a^2+11 b^2\right ) \cos (c+d x)+\frac {1}{2} b \left (8 a^2+11 b^2\right ) \cos ^2(c+d x)}{\cos ^{\frac {11}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx\\ &=\frac {2 a^2 \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {46 a b \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {4 \int \frac {\frac {1}{4} a^2 \left (81 a^2+113 b^2\right )+\frac {1}{4} a b \left (233 a^2+99 b^2\right ) \cos (c+d x)+\frac {69}{2} a^2 b^2 \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{99 a}\\ &=\frac {2 a^2 \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {46 a b \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2+113 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {8 \int \frac {\frac {5}{8} a^2 b \left (229 a^2+3 b^2\right )+\frac {1}{8} a^3 \left (405 a^2+1531 b^2\right ) \cos (c+d x)+\frac {1}{2} a^2 b \left (81 a^2+113 b^2\right ) \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{693 a^2}\\ &=\frac {2 a^2 \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {46 a b \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2+113 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 b \left (229 a^2+3 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {16 \int \frac {\frac {15}{16} a^2 \left (135 a^4+205 a^2 b^2-4 b^4\right )+\frac {5}{16} a^3 b \left (1011 a^2+461 b^2\right ) \cos (c+d x)+\frac {5}{8} a^2 b^2 \left (229 a^2+3 b^2\right ) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{3465 a^3}\\ &=\frac {2 a^2 \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {46 a b \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2+113 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 b \left (229 a^2+3 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (135 a^4+205 a^2 b^2-4 b^4\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {32 \int \frac {\frac {15}{32} a^2 b \left (741 a^4+51 a^2 b^2+8 b^4\right )+\frac {15}{32} a^3 \left (135 a^4+663 a^2 b^2+2 b^4\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{10395 a^4}\\ &=\frac {2 a^2 \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {46 a b \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2+113 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 b \left (229 a^2+3 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (135 a^4+205 a^2 b^2-4 b^4\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (b \left (741 a^4+51 a^2 b^2+8 b^4\right )\right ) \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{693 a^2}+\frac {\left ((a-b) \left (135 a^4-606 a^3 b+57 a^2 b^2+6 a b^3+8 b^4\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{693 a^2}\\ &=\frac {2 (a-b) b \sqrt {a+b} \left (741 a^4+51 a^2 b^2+8 b^4\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}}}{693 a^4 d}+\frac {2 (a-b) \sqrt {a+b} \left (135 a^4-606 a^3 b+57 a^2 b^2+6 a b^3+8 b^4\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}}}{693 a^3 d}+\frac {2 a^2 \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {46 a b \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2+113 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 b \left (229 a^2+3 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (135 a^4+205 a^2 b^2-4 b^4\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 a^2 d \cos ^{\frac {3}{2}}(c+d x)}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 6.34, size = 1431, normalized size = 2.74 \begin {gather*} \frac {-\frac {4 a \left (135 a^6-78 a^4 b^2-49 a^2 b^4-8 b^6\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 (-741 a^5 b-51 a^3 b^3-8 a b^5\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 (-741 a^4 b^2-51 a^2 b^4-8 b^6\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 )}{693 a^3 d}+\frac {\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \left (\frac {2}{693} \sec ^4(c+d x) \left (81 a^2 \sin (c+d x)+113 b^2 \sin (c+d x)\right )+\frac {2 \sec ^3(c+d x) \left (229 a^2 b \sin (c+d x)+3 b^3 \sin (c+d x)\right )}{693 a}+\frac {2 \sec ^2(c+d x) \left (135 a^4 \sin (c+d x)+205 a^2 b^2 \sin (c+d x)-4 b^4 \sin (c+d x)\right )}{693 a^2}+\frac {2 \sec (c+d x) \left (741 a^4 b \sin (c+d x)+51 a^2 b^3 \sin (c+d x)+8 b^5 \sin (c+d x)\right )}{693 a^3}+\frac {46}{99} a b \sec ^4(c+d x) \tan (c+d x)+\frac {2}{11} a^2 \sec ^5(c+d x) \tan (c+d x)\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2788\) vs.
\(2(472)=944\).
time = 0.57, size = 2789, normalized size = 5.34
method | result | size |
default | \(\text {Expression too large to display}\) | \(2789\) |
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(-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 {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^{13/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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