Optimal. Leaf size=100 \[ -\frac {24 \sqrt {d \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b d^2 \sqrt {\cos (a+b x)}}+\frac {12 (d \cos (a+b x))^{3/2} \sin (a+b x)}{5 b d^3}+\frac {2 \sin ^3(a+b x)}{b d \sqrt {d \cos (a+b x)}} \]
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Rubi [A]
time = 0.07, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2646, 2648,
2721, 2719} \begin {gather*} \frac {12 \sin (a+b x) (d \cos (a+b x))^{3/2}}{5 b d^3}-\frac {24 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {d \cos (a+b x)}}{5 b d^2 \sqrt {\cos (a+b x)}}+\frac {2 \sin ^3(a+b x)}{b d \sqrt {d \cos (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2646
Rule 2648
Rule 2719
Rule 2721
Rubi steps
\begin {align*} \int \frac {\sin ^4(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx &=\frac {2 \sin ^3(a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {6 \int \sqrt {d \cos (a+b x)} \sin ^2(a+b x) \, dx}{d^2}\\ &=\frac {12 (d \cos (a+b x))^{3/2} \sin (a+b x)}{5 b d^3}+\frac {2 \sin ^3(a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {12 \int \sqrt {d \cos (a+b x)} \, dx}{5 d^2}\\ &=\frac {12 (d \cos (a+b x))^{3/2} \sin (a+b x)}{5 b d^3}+\frac {2 \sin ^3(a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {\left (12 \sqrt {d \cos (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \, dx}{5 d^2 \sqrt {\cos (a+b x)}}\\ &=-\frac {24 \sqrt {d \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b d^2 \sqrt {\cos (a+b x)}}+\frac {12 (d \cos (a+b x))^{3/2} \sin (a+b x)}{5 b d^3}+\frac {2 \sin ^3(a+b x)}{b d \sqrt {d \cos (a+b x)}}\\ \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 = 60, normalized size = 0.60 \begin {gather*} \frac {\sqrt [4]{\cos ^2(a+b x)} \, _2F_1\left (\frac {5}{4},\frac {5}{2};\frac {7}{2};\sin ^2(a+b x)\right ) \sin ^5(a+b x)}{5 b d \sqrt {d \cos (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 213, normalized size = 2.13
method | result | size |
default | \(-\frac {8 \sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d +\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d}\, \left (-2 \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+2 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+3 \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )-3 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{5 d \sqrt {-d \left (2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right )}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b}\) | \(213\) |
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 = 115, normalized size = 1.15 \begin {gather*} -\frac {2 \, {\left (6 i \, \sqrt {2} \sqrt {d} \cos \left (b x + a\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) - 6 i \, \sqrt {2} \sqrt {d} \cos \left (b x + a\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) - \sqrt {d \cos \left (b x + a\right )} {\left (\cos \left (b x + a\right )^{2} + 5\right )} \sin \left (b x + a\right )\right )}}{5 \, b d^{2} \cos \left (b x + a\right )} \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 )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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