Optimal. Leaf size=218 \[ \frac {2 a b \left (15 a^2+62 b^2\right ) (e \cos (c+d x))^{3/2}}{15 d e^3}+\frac {2 b \left (5 a^2+6 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e^3}-\frac {2 \left (5 a^4+60 a^2 b^2+12 b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d e^2 \sqrt {\cos (c+d x)}}+\frac {2 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{d e^3}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{d e \sqrt {e \cos (c+d x)}} \]
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Rubi [A] time = 0.44, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2691, 2862, 2669, 2640, 2639} \[ \frac {2 a b \left (15 a^2+62 b^2\right ) (e \cos (c+d x))^{3/2}}{15 d e^3}+\frac {2 b \left (5 a^2+6 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e^3}-\frac {2 \left (60 a^2 b^2+5 a^4+12 b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d e^2 \sqrt {\cos (c+d x)}}+\frac {2 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{d e^3}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{d e \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2640
Rule 2669
Rule 2691
Rule 2862
Rubi steps
\begin {align*} \int \frac {(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{3/2}} \, dx &=\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^3}{d e \sqrt {e \cos (c+d x)}}-\frac {2 \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2 \left (\frac {a^2}{2}+3 b^2+\frac {7}{2} a b \sin (c+d x)\right ) \, dx}{e^2}\\ &=\frac {2 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{d e^3}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^3}{d e \sqrt {e \cos (c+d x)}}-\frac {4 \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x)) \left (\frac {7}{4} a \left (a^2+10 b^2\right )+\frac {7}{4} b \left (5 a^2+6 b^2\right ) \sin (c+d x)\right ) \, dx}{7 e^2}\\ &=\frac {2 b \left (5 a^2+6 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e^3}+\frac {2 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{d e^3}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^3}{d e \sqrt {e \cos (c+d x)}}-\frac {8 \int \sqrt {e \cos (c+d x)} \left (\frac {7}{8} \left (5 a^4+60 a^2 b^2+12 b^4\right )+\frac {7}{8} a b \left (15 a^2+62 b^2\right ) \sin (c+d x)\right ) \, dx}{35 e^2}\\ &=\frac {2 a b \left (15 a^2+62 b^2\right ) (e \cos (c+d x))^{3/2}}{15 d e^3}+\frac {2 b \left (5 a^2+6 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e^3}+\frac {2 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{d e^3}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^3}{d e \sqrt {e \cos (c+d x)}}-\frac {\left (5 a^4+60 a^2 b^2+12 b^4\right ) \int \sqrt {e \cos (c+d x)} \, dx}{5 e^2}\\ &=\frac {2 a b \left (15 a^2+62 b^2\right ) (e \cos (c+d x))^{3/2}}{15 d e^3}+\frac {2 b \left (5 a^2+6 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e^3}+\frac {2 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{d e^3}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^3}{d e \sqrt {e \cos (c+d x)}}-\frac {\left (\left (5 a^4+60 a^2 b^2+12 b^4\right ) \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 e^2 \sqrt {\cos (c+d x)}}\\ &=\frac {2 a b \left (15 a^2+62 b^2\right ) (e \cos (c+d x))^{3/2}}{15 d e^3}-\frac {2 \left (5 a^4+60 a^2 b^2+12 b^4\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d e^2 \sqrt {\cos (c+d x)}}+\frac {2 b \left (5 a^2+6 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e^3}+\frac {2 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{d e^3}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^3}{d e \sqrt {e \cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.60, size = 135, normalized size = 0.62 \[ \frac {\frac {1}{2} \left (240 a^3 b+\left (60 a^4+360 a^2 b^2+63 b^4\right ) \sin (c+d x)+40 a b^3 \cos (2 (c+d x))+280 a b^3+3 b^4 \sin (3 (c+d x))\right )-6 \left (5 a^4+60 a^2 b^2+12 b^4\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d e \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{4} \cos \left (d x + c\right )^{4} + a^{4} + 6 \, a^{2} b^{2} + b^{4} - 2 \, {\left (3 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2} - 4 \, {\left (a b^{3} \cos \left (d x + c\right )^{2} - a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{e^{2} \cos \left (d x + c\right )^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{4}}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 3.02, size = 378, normalized size = 1.73 \[ -\frac {2 \left (-24 b^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-80 a \,b^{3} \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 b^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{4}+180 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2} b^{2}+36 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{4}-30 a^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-180 a^{2} b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+80 a \,b^{3} \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-36 b^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-60 a^{3} b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-80 a \,b^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 e \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) d} \]
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 \[ \int \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{4}}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^4}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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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