Integrand size = 25, antiderivative size = 160 \[ \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{7/2}} \, dx=\frac {2 a b}{5 d e^3 \sqrt {e \cos (c+d x)}}-\frac {2 \left (3 a^2-2 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d e^4 \sqrt {\cos (c+d x)}}+\frac {2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{5 d e^3 \sqrt {e \cos (c+d x)}}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{5 d e (e \cos (c+d x))^{5/2}} \]
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Time = 0.12 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2770, 2748, 2716, 2721, 2719} \[ \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{7/2}} \, dx=-\frac {2 \left (3 a^2-2 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d e^4 \sqrt {\cos (c+d x)}}+\frac {2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{5 d e^3 \sqrt {e \cos (c+d x)}}+\frac {2 a b}{5 d e^3 \sqrt {e \cos (c+d x)}}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{5 d e (e \cos (c+d x))^{5/2}} \]
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Rule 2716
Rule 2719
Rule 2721
Rule 2748
Rule 2770
Rubi steps \begin{align*} \text {integral}& = \frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{5 d e (e \cos (c+d x))^{5/2}}-\frac {2 \int \frac {-\frac {3 a^2}{2}+b^2-\frac {1}{2} a b \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx}{5 e^2} \\ & = \frac {2 a b}{5 d e^3 \sqrt {e \cos (c+d x)}}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{5 d e (e \cos (c+d x))^{5/2}}+\frac {\left (3 a^2-2 b^2\right ) \int \frac {1}{(e \cos (c+d x))^{3/2}} \, dx}{5 e^2} \\ & = \frac {2 a b}{5 d e^3 \sqrt {e \cos (c+d x)}}+\frac {2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{5 d e^3 \sqrt {e \cos (c+d x)}}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{5 d e (e \cos (c+d x))^{5/2}}-\frac {\left (3 a^2-2 b^2\right ) \int \sqrt {e \cos (c+d x)} \, dx}{5 e^4} \\ & = \frac {2 a b}{5 d e^3 \sqrt {e \cos (c+d x)}}+\frac {2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{5 d e^3 \sqrt {e \cos (c+d x)}}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{5 d e (e \cos (c+d x))^{5/2}}-\frac {\left (\left (3 a^2-2 b^2\right ) \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 e^4 \sqrt {\cos (c+d x)}} \\ & = \frac {2 a b}{5 d e^3 \sqrt {e \cos (c+d x)}}-\frac {2 \left (3 a^2-2 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d e^4 \sqrt {\cos (c+d x)}}+\frac {2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{5 d e^3 \sqrt {e \cos (c+d x)}}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{5 d e (e \cos (c+d x))^{5/2}} \\ \end{align*}
Time = 0.97 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.66 \[ \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{7/2}} \, dx=\frac {8 a b-4 \left (3 a^2-2 b^2\right ) \cos ^{\frac {5}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\left (7 a^2+2 b^2\right ) \sin (c+d x)+3 a^2 \sin (3 (c+d x))-2 b^2 \sin (3 (c+d x))}{10 d e (e \cos (c+d x))^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(563\) vs. \(2(168)=336\).
Time = 7.34 (sec) , antiderivative size = 564, normalized size of antiderivative = 3.52
method | result | size |
default | \(\frac {\frac {48 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{5}-\frac {32 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{5}-\frac {24 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, E\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 ) a^{2}}{5}+\frac {16 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, E\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 ) b^{2}}{5}-\frac {48 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{5}+\frac {32 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{5}+\frac {24 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, E\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 ) a^{2}}{5}-\frac {16 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, E\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 ) b^{2}}{5}+\frac {16 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{5}-\frac {4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{5}-\frac {6 \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}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2}}{5}+\frac {4 \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}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{2}}{5}+\frac {4 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a b}{5}}{\left (4 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \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^{3} d}\) | \(564\) |
parts | \(\text {Expression too large to display}\) | \(757\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.01 \[ \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{7/2}} \, dx=\frac {\sqrt {2} {\left (-3 i \, a^{2} + 2 i \, b^{2}\right )} \sqrt {e} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + \sqrt {2} {\left (3 i \, a^{2} - 2 i \, b^{2}\right )} \sqrt {e} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (2 \, a b + {\left ({\left (3 \, a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + b^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{5 \, d e^{4} \cos \left (d x + c\right )^{3}} \]
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Timed out. \[ \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{7/2}} \, dx=\int { \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{2}}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{7/2}} \, dx=\int { \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{2}}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{7/2}} \, dx=\int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^2}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \]
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