Integrand size = 21, antiderivative size = 128 \[ \int \frac {\cos ^7(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\frac {30 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{77 b d \sqrt {b \cos (c+d x)}}+\frac {30 \sqrt {b \cos (c+d x)} \sin (c+d x)}{77 b^2 d}+\frac {18 (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 b^4 d}+\frac {2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^6 d} \]
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Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {16, 2715, 2721, 2720} \[ \int \frac {\cos ^7(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\frac {2 \sin (c+d x) (b \cos (c+d x))^{9/2}}{11 b^6 d}+\frac {18 \sin (c+d x) (b \cos (c+d x))^{5/2}}{77 b^4 d}+\frac {30 \sin (c+d x) \sqrt {b \cos (c+d x)}}{77 b^2 d}+\frac {30 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{77 b d \sqrt {b \cos (c+d x)}} \]
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Rule 16
Rule 2715
Rule 2720
Rule 2721
Rubi steps \begin{align*} \text {integral}& = \frac {\int (b \cos (c+d x))^{11/2} \, dx}{b^7} \\ & = \frac {2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^6 d}+\frac {9 \int (b \cos (c+d x))^{7/2} \, dx}{11 b^5} \\ & = \frac {18 (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 b^4 d}+\frac {2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^6 d}+\frac {45 \int (b \cos (c+d x))^{3/2} \, dx}{77 b^3} \\ & = \frac {30 \sqrt {b \cos (c+d x)} \sin (c+d x)}{77 b^2 d}+\frac {18 (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 b^4 d}+\frac {2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^6 d}+\frac {15 \int \frac {1}{\sqrt {b \cos (c+d x)}} \, dx}{77 b} \\ & = \frac {30 \sqrt {b \cos (c+d x)} \sin (c+d x)}{77 b^2 d}+\frac {18 (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 b^4 d}+\frac {2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^6 d}+\frac {\left (15 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{77 b \sqrt {b \cos (c+d x)}} \\ & = \frac {30 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{77 b d \sqrt {b \cos (c+d x)}}+\frac {30 \sqrt {b \cos (c+d x)} \sin (c+d x)}{77 b^2 d}+\frac {18 (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 b^4 d}+\frac {2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^6 d} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.59 \[ \int \frac {\cos ^7(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\frac {480 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+347 \sin (2 (c+d x))+64 \sin (4 (c+d x))+7 \sin (6 (c+d x))}{1232 b d \sqrt {b \cos (c+d x)}} \]
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Time = 5.53 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.84
method | result | size |
default | \(-\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) b \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (448 \left (\cos ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1568 \left (\cos ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2384 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2040 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1084 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-370 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+62 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{77 b \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) b}\, d}\) | \(236\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.79 \[ \int \frac {\cos ^7(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\frac {2 \, {\left (7 \, \cos \left (d x + c\right )^{4} + 9 \, \cos \left (d x + c\right )^{2} + 15\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right ) - 15 i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )}{77 \, b^{2} d} \]
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Timed out. \[ \int \frac {\cos ^7(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\cos ^7(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{7}}{\left (b \cos \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\cos ^7(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{7}}{\left (b \cos \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^7(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^7}{{\left (b\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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