Optimal. Leaf size=102 \[ \frac {24 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {d \cos (a+b x)}}{5 b d^4 \sqrt {\cos (a+b x)}}-\frac {12 \sin (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}}+\frac {2 \sin ^3(a+b x)}{5 b d (d \cos (a+b x))^{5/2}} \]
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Rubi [A] time = 0.11, 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 = {2566, 2640, 2639} \[ -\frac {12 \sin (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}}+\frac {24 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {d \cos (a+b x)}}{5 b d^4 \sqrt {\cos (a+b x)}}+\frac {2 \sin ^3(a+b x)}{5 b d (d \cos (a+b x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2566
Rule 2639
Rule 2640
Rubi steps
\begin {align*} \int \frac {\sin ^4(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx &=\frac {2 \sin ^3(a+b x)}{5 b d (d \cos (a+b x))^{5/2}}-\frac {6 \int \frac {\sin ^2(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx}{5 d^2}\\ &=-\frac {12 \sin (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}}+\frac {2 \sin ^3(a+b x)}{5 b d (d \cos (a+b x))^{5/2}}+\frac {12 \int \sqrt {d \cos (a+b x)} \, dx}{5 d^4}\\ &=-\frac {12 \sin (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}}+\frac {2 \sin ^3(a+b x)}{5 b d (d \cos (a+b x))^{5/2}}+\frac {\left (12 \sqrt {d \cos (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \, dx}{5 d^4 \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^4 \sqrt {\cos (a+b x)}}-\frac {12 \sin (a+b x)}{5 b d^3 \sqrt {d \cos (a+b x)}}+\frac {2 \sin ^3(a+b x)}{5 b d (d \cos (a+b x))^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 65, normalized size = 0.64 \[ \frac {\sin ^5(a+b x) \cos ^3(a+b x) \sqrt [4]{\cos ^2(a+b x)} \, _2F_1\left (\frac {9}{4},\frac {5}{2};\frac {7}{2};\sin ^2(a+b x)\right )}{5 b (d \cos (a+b x))^{7/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \sqrt {d \cos \left (b x + a\right )}}{d^{4} \cos \left (b x + a\right )^{4}}, x\right ) \]
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 \[ \int \frac {\sin \left (b x + a\right )^{4}}{\left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.16, size = 366, normalized size = 3.59 \[ -\frac {8 \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (12 \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-14 \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-12 \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+14 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+3 \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )-3 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d +\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d}}{5 d^{4} \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{3} \left (8 \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-12 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+6 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b} \]
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 \[ \int \frac {\sin \left (b x + a\right )^{4}}{\left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\sin \left (a+b\,x\right )}^4}{{\left (d\,\cos \left (a+b\,x\right )\right )}^{7/2}} \,d x \]
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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