Optimal. Leaf size=102 \[ \frac {8 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{7 b d^4 \sqrt {d \cos (a+b x)}}-\frac {4 \sin (a+b x)}{7 b d^3 (d \cos (a+b x))^{3/2}}+\frac {2 \sin ^3(a+b x)}{7 b d (d \cos (a+b x))^{7/2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2646, 2721,
2720} \begin {gather*} \frac {8 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{7 b d^4 \sqrt {d \cos (a+b x)}}-\frac {4 \sin (a+b x)}{7 b d^3 (d \cos (a+b x))^{3/2}}+\frac {2 \sin ^3(a+b x)}{7 b d (d \cos (a+b x))^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2646
Rule 2720
Rule 2721
Rubi steps
\begin {align*} \int \frac {\sin ^4(a+b x)}{(d \cos (a+b x))^{9/2}} \, dx &=\frac {2 \sin ^3(a+b x)}{7 b d (d \cos (a+b x))^{7/2}}-\frac {6 \int \frac {\sin ^2(a+b x)}{(d \cos (a+b x))^{5/2}} \, dx}{7 d^2}\\ &=-\frac {4 \sin (a+b x)}{7 b d^3 (d \cos (a+b x))^{3/2}}+\frac {2 \sin ^3(a+b x)}{7 b d (d \cos (a+b x))^{7/2}}+\frac {4 \int \frac {1}{\sqrt {d \cos (a+b x)}} \, dx}{7 d^4}\\ &=-\frac {4 \sin (a+b x)}{7 b d^3 (d \cos (a+b x))^{3/2}}+\frac {2 \sin ^3(a+b x)}{7 b d (d \cos (a+b x))^{7/2}}+\frac {\left (4 \sqrt {\cos (a+b x)}\right ) \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx}{7 d^4 \sqrt {d \cos (a+b x)}}\\ &=\frac {8 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{7 b d^4 \sqrt {d \cos (a+b x)}}-\frac {4 \sin (a+b x)}{7 b d^3 (d \cos (a+b x))^{3/2}}+\frac {2 \sin ^3(a+b x)}{7 b d (d \cos (a+b x))^{7/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.05, size = 65, normalized size = 0.64 \begin {gather*} \frac {\cos ^3(a+b x) \cos ^2(a+b x)^{3/4} \, _2F_1\left (\frac {5}{2},\frac {11}{4};\frac {7}{2};\sin ^2(a+b x)\right ) \sin ^5(a+b x)}{5 b (d \cos (a+b x))^{9/2}} \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. \(397\) vs.
\(2(114)=228\).
time = 0.29, size = 398, normalized size = 3.90
method | result | size |
default | \(\frac {8 \left (8 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-12 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-6 \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+6 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+6 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )-\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}}{7 d^{4} \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )^{3} \sqrt {-d \left (2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right )}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b}\) | \(398\) |
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 = 0.10, size = 114, normalized size = 1.12 \begin {gather*} -\frac {2 \, {\left (2 i \, \sqrt {2} \sqrt {d} \cos \left (b x + a\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - 2 i \, \sqrt {2} \sqrt {d} \cos \left (b x + a\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) + \sqrt {d \cos \left (b x + a\right )} {\left (3 \, \cos \left (b x + a\right )^{2} - 1\right )} \sin \left (b x + a\right )\right )}}{7 \, b d^{5} \cos \left (b x + a\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 {{\sin \left (a+b\,x\right )}^4}{{\left (d\,\cos \left (a+b\,x\right )\right )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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