Integrand size = 25, antiderivative size = 264 \[ \int \frac {(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{11/2}} \, dx=\frac {2 a b \left (21 a^2-22 b^2\right ) (e \cos (c+d x))^{3/2}}{45 d e^7}-\frac {2 \left (7 a^4-12 a^2 b^2+4 b^4\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d e^6 \sqrt {\cos (c+d x)}}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^3}{9 d e (e \cos (c+d x))^{9/2}}-\frac {2 (a+b \sin (c+d x)) \left (b \left (7 a^2-6 b^2\right )-a \left (21 a^2-22 b^2\right ) \sin (c+d x)\right )}{45 d e^5 \sqrt {e \cos (c+d x)}}+\frac {2 (a+b \sin (c+d x))^2 \left (a b+\left (7 a^2-6 b^2\right ) \sin (c+d x)\right )}{45 d e^3 (e \cos (c+d x))^{5/2}} \]
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Time = 0.34 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2770, 2940, 2748, 2721, 2719} \[ \int \frac {(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{11/2}} \, dx=\frac {2 a b \left (21 a^2-22 b^2\right ) (e \cos (c+d x))^{3/2}}{45 d e^7}-\frac {2 \left (b \left (7 a^2-6 b^2\right )-a \left (21 a^2-22 b^2\right ) \sin (c+d x)\right ) (a+b \sin (c+d x))}{45 d e^5 \sqrt {e \cos (c+d x)}}+\frac {2 \left (\left (7 a^2-6 b^2\right ) \sin (c+d x)+a b\right ) (a+b \sin (c+d x))^2}{45 d e^3 (e \cos (c+d x))^{5/2}}-\frac {2 \left (7 a^4-12 a^2 b^2+4 b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{15 d e^6 \sqrt {\cos (c+d x)}}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{9 d e (e \cos (c+d x))^{9/2}} \]
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Rule 2719
Rule 2721
Rule 2748
Rule 2770
Rule 2940
Rubi steps \begin{align*} \text {integral}& = \frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^3}{9 d e (e \cos (c+d x))^{9/2}}-\frac {2 \int \frac {(a+b \sin (c+d x))^2 \left (-\frac {7 a^2}{2}+3 b^2-\frac {1}{2} a b \sin (c+d x)\right )}{(e \cos (c+d x))^{7/2}} \, dx}{9 e^2} \\ & = \frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^3}{9 d e (e \cos (c+d x))^{9/2}}+\frac {2 (a+b \sin (c+d x))^2 \left (a b+\left (7 a^2-6 b^2\right ) \sin (c+d x)\right )}{45 d e^3 (e \cos (c+d x))^{5/2}}+\frac {4 \int \frac {(a+b \sin (c+d x)) \left (\frac {1}{4} a \left (21 a^2-22 b^2\right )-\frac {1}{4} b \left (7 a^2-6 b^2\right ) \sin (c+d x)\right )}{(e \cos (c+d x))^{3/2}} \, dx}{45 e^4} \\ & = \frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^3}{9 d e (e \cos (c+d x))^{9/2}}-\frac {2 (a+b \sin (c+d x)) \left (b \left (7 a^2-6 b^2\right )-a \left (21 a^2-22 b^2\right ) \sin (c+d x)\right )}{45 d e^5 \sqrt {e \cos (c+d x)}}+\frac {2 (a+b \sin (c+d x))^2 \left (a b+\left (7 a^2-6 b^2\right ) \sin (c+d x)\right )}{45 d e^3 (e \cos (c+d x))^{5/2}}-\frac {8 \int \sqrt {e \cos (c+d x)} \left (\frac {3}{8} \left (7 a^4-12 a^2 b^2+4 b^4\right )+\frac {3}{8} a b \left (21 a^2-22 b^2\right ) \sin (c+d x)\right ) \, dx}{45 e^6} \\ & = \frac {2 a b \left (21 a^2-22 b^2\right ) (e \cos (c+d x))^{3/2}}{45 d e^7}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^3}{9 d e (e \cos (c+d x))^{9/2}}-\frac {2 (a+b \sin (c+d x)) \left (b \left (7 a^2-6 b^2\right )-a \left (21 a^2-22 b^2\right ) \sin (c+d x)\right )}{45 d e^5 \sqrt {e \cos (c+d x)}}+\frac {2 (a+b \sin (c+d x))^2 \left (a b+\left (7 a^2-6 b^2\right ) \sin (c+d x)\right )}{45 d e^3 (e \cos (c+d x))^{5/2}}-\frac {\left (7 a^4-12 a^2 b^2+4 b^4\right ) \int \sqrt {e \cos (c+d x)} \, dx}{15 e^6} \\ & = \frac {2 a b \left (21 a^2-22 b^2\right ) (e \cos (c+d x))^{3/2}}{45 d e^7}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^3}{9 d e (e \cos (c+d x))^{9/2}}-\frac {2 (a+b \sin (c+d x)) \left (b \left (7 a^2-6 b^2\right )-a \left (21 a^2-22 b^2\right ) \sin (c+d x)\right )}{45 d e^5 \sqrt {e \cos (c+d x)}}+\frac {2 (a+b \sin (c+d x))^2 \left (a b+\left (7 a^2-6 b^2\right ) \sin (c+d x)\right )}{45 d e^3 (e \cos (c+d x))^{5/2}}-\frac {\left (\left (7 a^4-12 a^2 b^2+4 b^4\right ) \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{15 e^6 \sqrt {\cos (c+d x)}} \\ & = \frac {2 a b \left (21 a^2-22 b^2\right ) (e \cos (c+d x))^{3/2}}{45 d e^7}-\frac {2 \left (7 a^4-12 a^2 b^2+4 b^4\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d e^6 \sqrt {\cos (c+d x)}}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^3}{9 d e (e \cos (c+d x))^{9/2}}-\frac {2 (a+b \sin (c+d x)) \left (b \left (7 a^2-6 b^2\right )-a \left (21 a^2-22 b^2\right ) \sin (c+d x)\right )}{45 d e^5 \sqrt {e \cos (c+d x)}}+\frac {2 (a+b \sin (c+d x))^2 \left (a b+\left (7 a^2-6 b^2\right ) \sin (c+d x)\right )}{45 d e^3 (e \cos (c+d x))^{5/2}} \\ \end{align*}
Time = 2.91 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{11/2}} \, dx=\frac {\sqrt {e \cos (c+d x)} \sec ^5(c+d x) \left (320 a^3 b+32 a b^3-288 a b^3 \cos (2 (c+d x))-48 \left (7 a^4-12 a^2 b^2+4 b^4\right ) \cos ^{\frac {9}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+150 a^4 \sin (c+d x)+360 a^2 b^2 \sin (c+d x)+60 b^4 \sin (c+d x)+91 a^4 \sin (3 (c+d x))-156 a^2 b^2 \sin (3 (c+d x))-8 b^4 \sin (3 (c+d x))+21 a^4 \sin (5 (c+d x))-36 a^2 b^2 \sin (5 (c+d x))+12 b^4 \sin (5 (c+d x))\right )}{360 d e^6} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1314\) vs. \(2(268)=536\).
Time = 18.69 (sec) , antiderivative size = 1315, normalized size of antiderivative = 4.98
method | result | size |
parts | \(\text {Expression too large to display}\) | \(1315\) |
default | \(\text {Expression too large to display}\) | \(1416\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{11/2}} \, dx=-\frac {3 \, \sqrt {2} {\left (7 i \, a^{4} - 12 i \, a^{2} b^{2} + 4 i \, b^{4}\right )} \sqrt {e} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, \sqrt {2} {\left (-7 i \, a^{4} + 12 i \, a^{2} b^{2} - 4 i \, b^{4}\right )} \sqrt {e} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (36 \, a b^{3} \cos \left (d x + c\right )^{2} - 20 \, a^{3} b - 20 \, a b^{3} - {\left (3 \, {\left (7 \, a^{4} - 12 \, a^{2} b^{2} + 4 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 5 \, a^{4} + 30 \, a^{2} b^{2} + 5 \, b^{4} + {\left (7 \, a^{4} - 12 \, a^{2} b^{2} - 11 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{45 \, d e^{6} \cos \left (d x + c\right )^{5}} \]
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Timed out. \[ \int \frac {(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{11/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{11/2}} \, dx=\int { \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{4}}{\left (e \cos \left (d x + c\right )\right )^{\frac {11}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{11/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{11/2}} \, dx=\int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^4}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{11/2}} \,d x \]
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