Optimal. Leaf size=197 \[ \frac {2 a e^2 \left (7 a^2+6 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {e \cos (c+d x)}}-\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{315 d e}+\frac {2 a e \left (7 a^2+6 b^2\right ) \sin (c+d x) \sqrt {e \cos (c+d x)}}{21 d}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{63 d e} \]
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Rubi [A] time = 0.29, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2692, 2862, 2669, 2635, 2642, 2641} \[ \frac {2 a e^2 \left (7 a^2+6 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {e \cos (c+d x)}}-\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{315 d e}+\frac {2 a e \left (7 a^2+6 b^2\right ) \sin (c+d x) \sqrt {e \cos (c+d x)}}{21 d}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{63 d e} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rule 2642
Rule 2669
Rule 2692
Rule 2862
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3 \, dx &=-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}+\frac {2}{9} \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x)) \left (\frac {9 a^2}{2}+2 b^2+\frac {13}{2} a b \sin (c+d x)\right ) \, dx\\ &=-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{63 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}+\frac {4}{63} \int (e \cos (c+d x))^{3/2} \left (\frac {9}{4} a \left (7 a^2+6 b^2\right )+\frac {1}{4} b \left (89 a^2+28 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{315 d e}-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{63 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}+\frac {1}{7} \left (a \left (7 a^2+6 b^2\right )\right ) \int (e \cos (c+d x))^{3/2} \, dx\\ &=-\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{315 d e}+\frac {2 a \left (7 a^2+6 b^2\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{63 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}+\frac {1}{21} \left (a \left (7 a^2+6 b^2\right ) e^2\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{315 d e}+\frac {2 a \left (7 a^2+6 b^2\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{63 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}+\frac {\left (a \left (7 a^2+6 b^2\right ) e^2 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 \sqrt {e \cos (c+d x)}}\\ &=-\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{315 d e}+\frac {2 a \left (7 a^2+6 b^2\right ) e^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {e \cos (c+d x)}}+\frac {2 a \left (7 a^2+6 b^2\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{63 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\\ \end {align*}
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Mathematica [A] time = 1.45, size = 153, normalized size = 0.78 \[ \frac {(e \cos (c+d x))^{3/2} \left (80 \left (7 a^3+6 a b^2\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\frac {2}{3} \sqrt {\cos (c+d x)} \left (840 a^3 \sin (c+d x)-28 \left (27 a^2 b+4 b^3\right ) \cos (2 (c+d x))-756 a^2 b+450 a b^2 \sin (c+d x)-270 a b^2 \sin (3 (c+d x))+35 b^3 \cos (4 (c+d x))-147 b^3\right )\right )}{840 d \cos ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (3 \, a b^{2} e \cos \left (d x + c\right )^{3} - {\left (a^{3} + 3 \, a b^{2}\right )} e \cos \left (d x + c\right ) + {\left (b^{3} e \cos \left (d x + c\right )^{3} - {\left (3 \, a^{2} b + b^{3}\right )} e \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.62, size = 450, normalized size = 2.28 \[ -\frac {2 e^{2} \left (1120 b^{3} \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2160 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2800 b^{3} \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1512 a^{2} b \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3240 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2296 b^{3} \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+420 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2268 a^{2} b \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1260 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-644 b^{3} \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+105 \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}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{3}+90 \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}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a \,b^{2}-210 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1134 a^{2} b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+90 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-28 b^{3} \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+189 a^{2} b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+28 b^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315 \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}\, d} \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3 \,d x \]
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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