Optimal. Leaf size=187 \[ \frac {2 b \left (3 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{5 d e^5}-\frac {6 a \left (a^2-2 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d e^4 \sqrt {\cos (c+d x)}}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}-\frac {2 (a+b \sin (c+d x)) \left (a b-\left (3 a^2-4 b^2\right ) \sin (c+d x)\right )}{5 d e^3 \sqrt {e \cos (c+d x)}} \]
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Rubi [A]
time = 0.17, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2770, 2940,
2748, 2721, 2719} \begin {gather*} \frac {2 b \left (3 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{5 d e^5}-\frac {6 a \left (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 (a b-\left (3 a^2-4 b^2\right ) \sin (c+d x)\right ) (a+b \sin (c+d x))}{5 d e^3 \sqrt {e \cos (c+d x)}}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2721
Rule 2748
Rule 2770
Rule 2940
Rubi steps
\begin {align*} \int \frac {(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx &=\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}-\frac {2 \int \frac {(a+b \sin (c+d x)) \left (-\frac {3 a^2}{2}+2 b^2+\frac {1}{2} a b \sin (c+d x)\right )}{(e \cos (c+d x))^{3/2}} \, dx}{5 e^2}\\ &=\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}-\frac {2 (a+b \sin (c+d x)) \left (a b-\left (3 a^2-4 b^2\right ) \sin (c+d x)\right )}{5 d e^3 \sqrt {e \cos (c+d x)}}+\frac {4 \int \sqrt {e \cos (c+d x)} \left (-\frac {3}{4} a \left (a^2-2 b^2\right )-\frac {3}{4} b \left (3 a^2-4 b^2\right ) \sin (c+d x)\right ) \, dx}{5 e^4}\\ &=\frac {2 b \left (3 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{5 d e^5}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}-\frac {2 (a+b \sin (c+d x)) \left (a b-\left (3 a^2-4 b^2\right ) \sin (c+d x)\right )}{5 d e^3 \sqrt {e \cos (c+d x)}}-\frac {\left (3 a \left (a^2-2 b^2\right )\right ) \int \sqrt {e \cos (c+d x)} \, dx}{5 e^4}\\ &=\frac {2 b \left (3 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{5 d e^5}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}-\frac {2 (a+b \sin (c+d x)) \left (a b-\left (3 a^2-4 b^2\right ) \sin (c+d x)\right )}{5 d e^3 \sqrt {e \cos (c+d x)}}-\frac {\left (3 a \left (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 b \left (3 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{5 d e^5}-\frac {6 a \left (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 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}-\frac {2 (a+b \sin (c+d x)) \left (a b-\left (3 a^2-4 b^2\right ) \sin (c+d x)\right )}{5 d e^3 \sqrt {e \cos (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.76, size = 126, normalized size = 0.67 \begin {gather*} \frac {2 \left (-5 b^3-3 a \left (a^2-2 b^2\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+b \left (3 a^2+b^2\right ) \sec ^2(c+d x)+3 a^3 \sin (c+d x)-6 a b^2 \sin (c+d x)+a \left (a^2+3 b^2\right ) \sec (c+d x) \tan (c+d x)\right )}{5 d e^3 \sqrt {e \cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(617\) vs.
\(2(195)=390\).
time = 17.23, size = 618, normalized size = 3.30
method | result | size |
default | \(-\frac {2 \left (12 \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^{3} \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \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 \,b^{2} \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+48 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 )-12 \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^{3} \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 \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 \,b^{2} \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 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 )-48 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 )+20 b^{3} \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \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^{3}-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}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a \,b^{2}-8 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 )+6 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 )-20 b^{3} \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 a^{2} b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+4 b^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{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}\) | \(618\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.11, size = 181, normalized size = 0.97 \begin {gather*} -\frac {{\left (3 \, \sqrt {2} {\left (i \, a^{3} - 2 i \, a b^{2}\right )} \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 ) + 3 \, \sqrt {2} {\left (-i \, a^{3} + 2 i \, a b^{2}\right )} \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 (5 \, b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{2} b - b^{3} - {\left (a^{3} + 3 \, a b^{2} + 3 \, {\left (a^{3} - 2 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}\right )} e^{\left (-\frac {7}{2}\right )}}{5 \, d \cos \left (d x + c\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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