Integrand size = 25, antiderivative size = 169 \[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{13/2}} \, dx=-\frac {2 a^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{77 d e^6 \sqrt {e \cos (c+d x)}}+\frac {4 a^7 \sqrt {e \cos (c+d x)}}{11 d e^7 (a-a \sin (c+d x))^3}-\frac {2 a^8 \sqrt {e \cos (c+d x)}}{77 d e^7 \left (a^2-a^2 \sin (c+d x)\right )^2}-\frac {2 a^8 \sqrt {e \cos (c+d x)}}{77 d e^7 \left (a^4-a^4 \sin (c+d x)\right )} \]
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Time = 0.18 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2749, 2759, 2760, 2762, 2721, 2720} \[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{13/2}} \, dx=\frac {4 a^7 \sqrt {e \cos (c+d x)}}{11 d e^7 (a-a \sin (c+d x))^3}-\frac {2 a^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{77 d e^6 \sqrt {e \cos (c+d x)}}-\frac {2 a^8 \sqrt {e \cos (c+d x)}}{77 d e^7 \left (a^4-a^4 \sin (c+d x)\right )}-\frac {2 a^8 \sqrt {e \cos (c+d x)}}{77 d e^7 \left (a^2-a^2 \sin (c+d x)\right )^2} \]
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Rule 2720
Rule 2721
Rule 2749
Rule 2759
Rule 2760
Rule 2762
Rubi steps \begin{align*} \text {integral}& = \frac {a^8 \int \frac {(e \cos (c+d x))^{3/2}}{(a-a \sin (c+d x))^4} \, dx}{e^8} \\ & = \frac {4 a^7 \sqrt {e \cos (c+d x)}}{11 d e^7 (a-a \sin (c+d x))^3}-\frac {a^6 \int \frac {1}{\sqrt {e \cos (c+d x)} (a-a \sin (c+d x))^2} \, dx}{11 e^6} \\ & = \frac {4 a^7 \sqrt {e \cos (c+d x)}}{11 d e^7 (a-a \sin (c+d x))^3}-\frac {2 a^6 \sqrt {e \cos (c+d x)}}{77 d e^7 (a-a \sin (c+d x))^2}-\frac {\left (3 a^5\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} (a-a \sin (c+d x))} \, dx}{77 e^6} \\ & = \frac {4 a^7 \sqrt {e \cos (c+d x)}}{11 d e^7 (a-a \sin (c+d x))^3}-\frac {2 a^6 \sqrt {e \cos (c+d x)}}{77 d e^7 (a-a \sin (c+d x))^2}-\frac {2 a^5 \sqrt {e \cos (c+d x)}}{77 d e^7 (a-a \sin (c+d x))}-\frac {a^4 \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{77 e^6} \\ & = \frac {4 a^7 \sqrt {e \cos (c+d x)}}{11 d e^7 (a-a \sin (c+d x))^3}-\frac {2 a^6 \sqrt {e \cos (c+d x)}}{77 d e^7 (a-a \sin (c+d x))^2}-\frac {2 a^5 \sqrt {e \cos (c+d x)}}{77 d e^7 (a-a \sin (c+d x))}-\frac {\left (a^4 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{77 e^6 \sqrt {e \cos (c+d x)}} \\ & = -\frac {2 a^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{77 d e^6 \sqrt {e \cos (c+d x)}}+\frac {4 a^7 \sqrt {e \cos (c+d x)}}{11 d e^7 (a-a \sin (c+d x))^3}-\frac {2 a^6 \sqrt {e \cos (c+d x)}}{77 d e^7 (a-a \sin (c+d x))^2}-\frac {2 a^5 \sqrt {e \cos (c+d x)}}{77 d e^7 (a-a \sin (c+d x))} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.14 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.39 \[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{13/2}} \, dx=\frac {4 \sqrt [4]{2} a^4 \operatorname {Hypergeometric2F1}\left (-\frac {11}{4},-\frac {1}{4},-\frac {7}{4},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{11/4}}{11 d e (e \cos (c+d x))^{11/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(582\) vs. \(2(177)=354\).
Time = 15.88 (sec) , antiderivative size = 583, normalized size of antiderivative = 3.45
method | result | size |
default | \(\frac {2 \left (32 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+32 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-80 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-64 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+80 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+176 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-40 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-144 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+10 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+176 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-78 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{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 )-176 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{77 \left (32 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-80 \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+80 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-40 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, e^{6} d}\) | \(583\) |
parts | \(\text {Expression too large to display}\) | \(1058\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.50 \[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{13/2}} \, dx=-\frac {{\left (-3 i \, \sqrt {2} a^{4} \cos \left (d x + c\right )^{2} + 4 i \, \sqrt {2} a^{4} + {\left (i \, \sqrt {2} a^{4} \cos \left (d x + c\right )^{2} - 4 i \, \sqrt {2} a^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + {\left (3 i \, \sqrt {2} a^{4} \cos \left (d x + c\right )^{2} - 4 i \, \sqrt {2} a^{4} + {\left (-i \, \sqrt {2} a^{4} \cos \left (d x + c\right )^{2} + 4 i \, \sqrt {2} a^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (a^{4} \cos \left (d x + c\right )^{2} + 3 \, a^{4} \sin \left (d x + c\right ) + 11 \, a^{4}\right )} \sqrt {e \cos \left (d x + c\right )}}{77 \, {\left (3 \, d e^{7} \cos \left (d x + c\right )^{2} - 4 \, d e^{7} - {\left (d e^{7} \cos \left (d x + c\right )^{2} - 4 \, d e^{7}\right )} \sin \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{13/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{13/2}} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}{\left (e \cos \left (d x + c\right )\right )^{\frac {13}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{13/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{13/2}} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^4}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{13/2}} \,d x \]
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