Optimal. Leaf size=89 \[ \frac {4 \sqrt {2} \sqrt {1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{m+3} F_1\left (m+\frac {5}{2};-\frac {3}{2},4;m+\frac {7}{2};\frac {1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right )}{a^3 f (2 m+5)} \]
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Rubi [A] time = 0.10, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2719, 137, 136} \[ \frac {4 \sqrt {2} \sqrt {1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{m+3} F_1\left (m+\frac {5}{2};-\frac {3}{2},4;m+\frac {7}{2};\frac {1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right )}{a^3 f (2 m+5)} \]
Antiderivative was successfully verified.
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Rule 136
Rule 137
Rule 2719
Rubi steps
\begin {align*} \int \cot ^4(e+f x) (a+a \sin (e+f x))^m \, dx &=\frac {\left (\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {(a-x)^{3/2} (a+x)^{\frac {3}{2}+m}}{x^4} \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac {\left (2 \sqrt {2} \sec (e+f x) (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {(a+x)^{\frac {3}{2}+m} \left (\frac {1}{2}-\frac {x}{2 a}\right )^{3/2}}{x^4} \, dx,x,a \sin (e+f x)\right )}{f \sqrt {\frac {a-a \sin (e+f x)}{a}}}\\ &=\frac {4 \sqrt {2} F_1\left (\frac {5}{2}+m;-\frac {3}{2},4;\frac {7}{2}+m;\frac {1}{2} (1+\sin (e+f x)),1+\sin (e+f x)\right ) \sec (e+f x) \sqrt {1-\sin (e+f x)} (a+a \sin (e+f x))^{3+m}}{a^3 f (5+2 m)}\\ \end {align*}
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Mathematica [F] time = 0.78, size = 0, normalized size = 0.00 \[ \int \cot ^4(e+f x) (a+a \sin (e+f x))^m \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cot \left (f x + e\right )^{4}, x\right ) \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cot \left (f x + e\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.44, size = 0, normalized size = 0.00 \[ \int \left (\cot ^{4}\left (f x +e \right )\right ) \left (a +a \sin \left (f x +e \right )\right )^{m}\, dx \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cot \left (f x + e\right )^{4}\,{d x} \]
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 \[ \int {\mathrm {cot}\left (e+f\,x\right )}^4\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \cot ^{4}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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