Integrand size = 44, antiderivative size = 522 \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {2 (a-b) \sqrt {a+b} \left (147 a^4 B+279 a^2 b^2 B-10 b^4 B+435 a^3 b C+45 a b^3 C\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{315 a^3 d}-\frac {2 (a-b) \sqrt {a+b} \left (10 b^3 B-6 a^2 b (19 B-60 C)+3 a^3 (49 B-25 C)+15 a b^2 (11 B-3 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{315 a^2 d}+\frac {2 a (4 b B+3 a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{21 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (49 a^2 B+75 b^2 B+135 a b C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (163 a^2 b B+5 b^3 B+75 a^3 C+135 a b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 a d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)} \]
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Time = 2.15 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.159, Rules used = {3108, 3068, 3126, 3134, 3077, 2895, 3073} \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {2 \left (49 a^2 B+135 a b C+75 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{315 d \cos ^{\frac {5}{2}}(c+d x)}-\frac {2 (a-b) \sqrt {a+b} \left (3 a^3 (49 B-25 C)-6 a^2 b (19 B-60 C)+15 a b^2 (11 B-3 C)+10 b^3 B\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{315 a^2 d}+\frac {2 \left (75 a^3 C+163 a^2 b B+135 a b^2 C+5 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{315 a d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (a-b) \sqrt {a+b} \left (147 a^4 B+435 a^3 b C+279 a^2 b^2 B+45 a b^3 C-10 b^4 B\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{315 a^3 d}+\frac {2 a (3 a C+4 b B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{21 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)} \]
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Rule 2895
Rule 3068
Rule 3073
Rule 3077
Rule 3108
Rule 3126
Rule 3134
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b \cos (c+d x))^{5/2} (B+C \cos (c+d x))}{\cos ^{\frac {11}{2}}(c+d x)} \, dx \\ & = \frac {2 a B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2}{9} \int \frac {\sqrt {a+b \cos (c+d x)} \left (\frac {3}{2} a (4 b B+3 a C)+\frac {1}{2} \left (7 a^2 B+9 b^2 B+18 a b C\right ) \cos (c+d x)+\frac {1}{2} b (4 a B+9 b C) \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx \\ & = \frac {2 a (4 b B+3 a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{21 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {4}{63} \int \frac {\frac {1}{4} a \left (49 a^2 B+75 b^2 B+135 a b C\right )+\frac {1}{4} \left (137 a^2 b B+63 b^3 B+45 a^3 C+189 a b^2 C\right ) \cos (c+d x)+\frac {1}{4} b \left (76 a b B+36 a^2 C+63 b^2 C\right ) \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx \\ & = \frac {2 a (4 b B+3 a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{21 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (49 a^2 B+75 b^2 B+135 a b C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 a B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {8 \int \frac {\frac {3}{8} a \left (163 a^2 b B+5 b^3 B+75 a^3 C+135 a b^2 C\right )+\frac {1}{8} a \left (147 a^3 B+605 a b^2 B+585 a^2 b C+315 b^3 C\right ) \cos (c+d x)+\frac {1}{4} a b \left (49 a^2 B+75 b^2 B+135 a b C\right ) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{315 a} \\ & = \frac {2 a (4 b B+3 a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{21 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (49 a^2 B+75 b^2 B+135 a b C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (163 a^2 b B+5 b^3 B+75 a^3 C+135 a b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 a d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {16 \int \frac {\frac {3}{16} a \left (147 a^4 B+279 a^2 b^2 B-10 b^4 B+435 a^3 b C+45 a b^3 C\right )+\frac {3}{16} a^2 \left (261 a^2 b B+155 b^3 B+75 a^3 C+405 a b^2 C\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{945 a^2} \\ & = \frac {2 a (4 b B+3 a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{21 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (49 a^2 B+75 b^2 B+135 a b C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (163 a^2 b B+5 b^3 B+75 a^3 C+135 a b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 a d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}-\frac {\left ((a-b) \left (10 b^3 B-6 a^2 b (19 B-60 C)+3 a^3 (49 B-25 C)+15 a b^2 (11 B-3 C)\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{315 a}+\frac {\left (147 a^4 B+279 a^2 b^2 B-10 b^4 B+435 a^3 b C+45 a b^3 C\right ) \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{315 a} \\ & = \frac {2 (a-b) \sqrt {a+b} \left (147 a^4 B+279 a^2 b^2 B-10 b^4 B+435 a^3 b C+45 a b^3 C\right ) \cot (c+d x) E\left (\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}}}{315 a^3 d}-\frac {2 (a-b) \sqrt {a+b} \left (10 b^3 B-6 a^2 b (19 B-60 C)+3 a^3 (49 B-25 C)+15 a b^2 (11 B-3 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\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}}}{315 a^2 d}+\frac {2 a (4 b B+3 a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{21 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (49 a^2 B+75 b^2 B+135 a b C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (163 a^2 b B+5 b^3 B+75 a^3 C+135 a b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 a d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)} \\ \end{align*}
Result contains complex when optimal does not.
Time = 7.44 (sec) , antiderivative size = 1517, normalized size of antiderivative = 2.91 \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=-\frac {-\frac {4 a \left (-114 a^4 b B+124 a^2 b^3 B-10 b^5 B-75 a^5 C+30 a^3 b^2 C+45 a b^4 C\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) \operatorname {EllipticF}\left (\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 (147 a^5 B+279 a^3 b^2 B-10 a b^4 B+435 a^4 b C+45 a^2 b^3 C\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) \operatorname {EllipticF}\left (\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) \operatorname {EllipticPi}\left (-\frac {a}{b},\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 (147 a^4 b B+279 a^2 b^3 B-10 b^5 B+435 a^3 b^2 C+45 a b^4 C\right ) \left (\frac {i \cos \left (\frac {1}{2} (c+d x)\right ) \sqrt {a+b \cos (c+d x)} E\left (i \text {arcsinh}\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) \operatorname {EllipticF}\left (\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) \operatorname {EllipticPi}\left (-\frac {a}{b},\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 )}{315 a^2 d}+\frac {\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \left (\frac {2}{63} \sec ^4(c+d x) \left (19 a b B \sin (c+d x)+9 a^2 C \sin (c+d x)\right )+\frac {2}{315} \sec ^3(c+d x) \left (49 a^2 B \sin (c+d x)+75 b^2 B \sin (c+d x)+135 a b C \sin (c+d x)\right )+\frac {2 \sec ^2(c+d x) \left (163 a^2 b B \sin (c+d x)+5 b^3 B \sin (c+d x)+75 a^3 C \sin (c+d x)+135 a b^2 C \sin (c+d x)\right )}{315 a}+\frac {2 \sec (c+d x) \left (147 a^4 B \sin (c+d x)+279 a^2 b^2 B \sin (c+d x)-10 b^4 B \sin (c+d x)+435 a^3 b C \sin (c+d x)+45 a b^3 C \sin (c+d x)\right )}{315 a^2}+\frac {2}{9} a^2 B \sec ^4(c+d x) \tan (c+d x)\right )}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(5954\) vs. \(2(478)=956\).
Time = 48.85 (sec) , antiderivative size = 5955, normalized size of antiderivative = 11.41
method | result | size |
parts | \(\text {Expression too large to display}\) | \(5955\) |
default | \(\text {Expression too large to display}\) | \(6034\) |
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\[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cos \left (d x + c\right )^{\frac {13}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cos \left (d x + c\right )^{\frac {13}{2}}} \,d x } \]
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\[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cos \left (d x + c\right )^{\frac {13}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^{13/2}} \,d x \]
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