Optimal. Leaf size=258 \[ -\frac {10 a b \left (115 a^2+94 b^2\right ) (e \cos (c+d x))^{7/2}}{3003 d e}-\frac {2 b \left (73 a^2+22 b^2\right ) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{429 d e}+\frac {2 e^2 \left (39 a^4+52 a^2 b^2+4 b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{65 d \sqrt {\cos (c+d x)}}+\frac {2 e \left (39 a^4+52 a^2 b^2+4 b^4\right ) \sin (c+d x) (e \cos (c+d x))^{3/2}}{195 d}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}-\frac {38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{143 d e} \]
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Rubi [A] time = 0.51, antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2692, 2862, 2669, 2635, 2640, 2639} \[ \frac {2 e^2 \left (52 a^2 b^2+39 a^4+4 b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{65 d \sqrt {\cos (c+d x)}}-\frac {10 a b \left (115 a^2+94 b^2\right ) (e \cos (c+d x))^{7/2}}{3003 d e}-\frac {2 b \left (73 a^2+22 b^2\right ) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{429 d e}+\frac {2 e \left (52 a^2 b^2+39 a^4+4 b^4\right ) \sin (c+d x) (e \cos (c+d x))^{3/2}}{195 d}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}-\frac {38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{143 d e} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2640
Rule 2669
Rule 2692
Rule 2862
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^4 \, dx &=-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}+\frac {2}{13} \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2 \left (\frac {13 a^2}{2}+3 b^2+\frac {19}{2} a b \sin (c+d x)\right ) \, dx\\ &=-\frac {38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{143 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}+\frac {4}{143} \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x)) \left (\frac {1}{4} a \left (143 a^2+142 b^2\right )+\frac {3}{4} b \left (73 a^2+22 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac {2 b \left (73 a^2+22 b^2\right ) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{429 d e}-\frac {38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{143 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}+\frac {8 \int (e \cos (c+d x))^{5/2} \left (\frac {33}{8} \left (39 a^4+52 a^2 b^2+4 b^4\right )+\frac {15}{8} a b \left (115 a^2+94 b^2\right ) \sin (c+d x)\right ) \, dx}{1287}\\ &=-\frac {10 a b \left (115 a^2+94 b^2\right ) (e \cos (c+d x))^{7/2}}{3003 d e}-\frac {2 b \left (73 a^2+22 b^2\right ) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{429 d e}-\frac {38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{143 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}+\frac {1}{39} \left (39 a^4+52 a^2 b^2+4 b^4\right ) \int (e \cos (c+d x))^{5/2} \, dx\\ &=-\frac {10 a b \left (115 a^2+94 b^2\right ) (e \cos (c+d x))^{7/2}}{3003 d e}+\frac {2 \left (39 a^4+52 a^2 b^2+4 b^4\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{195 d}-\frac {2 b \left (73 a^2+22 b^2\right ) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{429 d e}-\frac {38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{143 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}+\frac {1}{65} \left (\left (39 a^4+52 a^2 b^2+4 b^4\right ) e^2\right ) \int \sqrt {e \cos (c+d x)} \, dx\\ &=-\frac {10 a b \left (115 a^2+94 b^2\right ) (e \cos (c+d x))^{7/2}}{3003 d e}+\frac {2 \left (39 a^4+52 a^2 b^2+4 b^4\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{195 d}-\frac {2 b \left (73 a^2+22 b^2\right ) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{429 d e}-\frac {38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{143 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}+\frac {\left (\left (39 a^4+52 a^2 b^2+4 b^4\right ) e^2 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{65 \sqrt {\cos (c+d x)}}\\ &=-\frac {10 a b \left (115 a^2+94 b^2\right ) (e \cos (c+d x))^{7/2}}{3003 d e}+\frac {2 \left (39 a^4+52 a^2 b^2+4 b^4\right ) e^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{65 d \sqrt {\cos (c+d x)}}+\frac {2 \left (39 a^4+52 a^2 b^2+4 b^4\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{195 d}-\frac {2 b \left (73 a^2+22 b^2\right ) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{429 d e}-\frac {38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{143 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}\\ \end {align*}
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Mathematica [A] time = 2.15, size = 209, normalized size = 0.81 \[ \frac {(e \cos (c+d x))^{5/2} \left (2 \left (39 a^4+52 a^2 b^2+4 b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+65 \sqrt {\cos (c+d x)} \left (-\frac {1}{78} b^2 \left (13 a^2+b^2\right ) \sin (4 (c+d x))-\frac {1}{77} a b \left (66 a^2+31 b^2\right ) \cos (c+d x)-\frac {1}{154} a b \left (44 a^2+9 b^2\right ) \cos (3 (c+d x))+\frac {\left (624 a^4-208 a^2 b^2-61 b^4\right ) \sin (2 (c+d x))}{3120}+\frac {1}{22} a b^3 \cos (5 (c+d x))+\frac {1}{208} b^4 \sin (6 (c+d x))\right )\right )}{65 d \cos ^{\frac {5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{4} e^{2} \cos \left (d x + c\right )^{6} - 2 \, {\left (3 \, a^{2} b^{2} + b^{4}\right )} e^{2} \cos \left (d x + c\right )^{4} + {\left (a^{4} + 6 \, a^{2} b^{2} + b^{4}\right )} e^{2} \cos \left (d x + c\right )^{2} - 4 \, {\left (a b^{3} e^{2} \cos \left (d x + c\right )^{4} - {\left (a^{3} b + a b^{3}\right )} e^{2} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.00, size = 776, normalized size = 3.01 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^4 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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