Optimal. Leaf size=59 \[ \frac {\tan ^5(e+f x)}{5 a^3 c^3 f}+\frac {2 \tan ^3(e+f x)}{3 a^3 c^3 f}+\frac {\tan (e+f x)}{a^3 c^3 f} \]
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Rubi [A] time = 0.07, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2736, 3767} \[ \frac {\tan ^5(e+f x)}{5 a^3 c^3 f}+\frac {2 \tan ^3(e+f x)}{3 a^3 c^3 f}+\frac {\tan (e+f x)}{a^3 c^3 f} \]
Antiderivative was successfully verified.
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Rule 2736
Rule 3767
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^3} \, dx &=\frac {\int \sec ^6(e+f x) \, dx}{a^3 c^3}\\ &=-\frac {\operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (e+f x)\right )}{a^3 c^3 f}\\ &=\frac {\tan (e+f x)}{a^3 c^3 f}+\frac {2 \tan ^3(e+f x)}{3 a^3 c^3 f}+\frac {\tan ^5(e+f x)}{5 a^3 c^3 f}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 41, normalized size = 0.69 \[ \frac {\frac {1}{5} \tan ^5(e+f x)+\frac {2}{3} \tan ^3(e+f x)+\tan (e+f x)}{a^3 c^3 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 47, normalized size = 0.80 \[ \frac {{\left (8 \, \cos \left (f x + e\right )^{4} + 4 \, \cos \left (f x + e\right )^{2} + 3\right )} \sin \left (f x + e\right )}{15 \, a^{3} c^{3} f \cos \left (f x + e\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 43, normalized size = 0.73 \[ \frac {3 \, \tan \left (f x + e\right )^{5} + 10 \, \tan \left (f x + e\right )^{3} + 15 \, \tan \left (f x + e\right )}{15 \, a^{3} c^{3} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a +a \sin \left (f x +e \right )\right )^{3} \left (c -c \sin \left (f x +e \right )\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 40, normalized size = 0.68 \[ \frac {3 \, \tan \left (f x + e\right )^{5} + 10 \, \tan \left (f x + e\right )^{3} + 15 \, \tan \left (f x + e\right )}{15 \, a^{3} c^{3} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.34, size = 89, normalized size = 1.51 \[ -\frac {2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+58\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-20\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+15\right )}{15\,a^3\,c^3\,f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 14.79, size = 687, normalized size = 11.64 \[ \begin {cases} - \frac {30 \tan ^{9}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 a^{3} c^{3} f \tan ^{10}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 75 a^{3} c^{3} f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} c^{3} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 150 a^{3} c^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} c^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 15 a^{3} c^{3} f} + \frac {40 \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 a^{3} c^{3} f \tan ^{10}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 75 a^{3} c^{3} f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} c^{3} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 150 a^{3} c^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} c^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 15 a^{3} c^{3} f} - \frac {116 \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 a^{3} c^{3} f \tan ^{10}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 75 a^{3} c^{3} f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} c^{3} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 150 a^{3} c^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} c^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 15 a^{3} c^{3} f} + \frac {40 \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 a^{3} c^{3} f \tan ^{10}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 75 a^{3} c^{3} f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} c^{3} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 150 a^{3} c^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} c^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 15 a^{3} c^{3} f} - \frac {30 \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 a^{3} c^{3} f \tan ^{10}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 75 a^{3} c^{3} f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} c^{3} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 150 a^{3} c^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} c^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 15 a^{3} c^{3} f} & \text {for}\: f \neq 0 \\\frac {x}{\left (a \sin {\relax (e )} + a\right )^{3} \left (- c \sin {\relax (e )} + c\right )^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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