Optimal. Leaf size=551 \[ \frac {973 \sec (e+f x) (a \sin (e+f x)+a)^{2/3} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a \sin (e+f x)+a}\right ) \sqrt {\frac {2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a \sin (e+f x)+a}+(a \sin (e+f x)+a)^{2/3}}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a \sin (e+f x)+a}\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{2} \sqrt [3]{a}-\left (1-\sqrt {3}\right ) \sqrt [3]{\sin (e+f x) a+a}}{\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sin (e+f x) a+a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{495 \sqrt [3]{2} \sqrt [4]{3} a^{4/3} f \sqrt {-\frac {\sqrt [3]{a \sin (e+f x)+a} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a \sin (e+f x)+a}\right )}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a \sin (e+f x)+a}\right )^2}}}+\frac {3 a^2 \sin ^2(e+f x) \tan (e+f x)}{f (a-a \sin (e+f x)) (a \sin (e+f x)+a)^{4/3}}+\frac {3 a^2 \sin (e+f x) \tan (e+f x)}{4 f (a-a \sin (e+f x)) (a \sin (e+f x)+a)^{4/3}}-\frac {\sec (e+f x) (356 a \sin (e+f x)+95 a)}{132 f (1-\sin (e+f x)) (a \sin (e+f x)+a)^{4/3}}+\frac {973 \sec (e+f x)}{396 f \sqrt [3]{a \sin (e+f x)+a}}-\frac {973 (1-\sin (e+f x)) \sec (e+f x)}{495 f \sqrt [3]{a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.49, antiderivative size = 551, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2719, 100, 153, 144, 51, 63, 225} \[ \frac {3 a^2 \sin ^2(e+f x) \tan (e+f x)}{f (a-a \sin (e+f x)) (a \sin (e+f x)+a)^{4/3}}+\frac {3 a^2 \sin (e+f x) \tan (e+f x)}{4 f (a-a \sin (e+f x)) (a \sin (e+f x)+a)^{4/3}}+\frac {973 \sec (e+f x) (a \sin (e+f x)+a)^{2/3} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a \sin (e+f x)+a}\right ) \sqrt {\frac {2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a \sin (e+f x)+a}+(a \sin (e+f x)+a)^{2/3}}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a \sin (e+f x)+a}\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{2} \sqrt [3]{a}-\left (1-\sqrt {3}\right ) \sqrt [3]{\sin (e+f x) a+a}}{\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sin (e+f x) a+a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{495 \sqrt [3]{2} \sqrt [4]{3} a^{4/3} f \sqrt {-\frac {\sqrt [3]{a \sin (e+f x)+a} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a \sin (e+f x)+a}\right )}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a \sin (e+f x)+a}\right )^2}}}-\frac {\sec (e+f x) (356 a \sin (e+f x)+95 a)}{132 f (1-\sin (e+f x)) (a \sin (e+f x)+a)^{4/3}}+\frac {973 \sec (e+f x)}{396 f \sqrt [3]{a \sin (e+f x)+a}}-\frac {973 (1-\sin (e+f x)) \sec (e+f x)}{495 f \sqrt [3]{a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 100
Rule 144
Rule 153
Rule 225
Rule 2719
Rubi steps
\begin {align*} \int \frac {\tan ^4(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, 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 {x^4}{(a-x)^{5/2} (a+x)^{17/6}} \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac {3 a^2 \sin ^2(e+f x) \tan (e+f x)}{f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac {\left (3 \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-3 a^2+\frac {a x}{3}\right )}{(a-x)^{5/2} (a+x)^{17/6}} \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac {3 a^2 \sin (e+f x) \tan (e+f x)}{4 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac {3 a^2 \sin ^2(e+f x) \tan (e+f x)}{f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac {\left (9 \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {x \left (-\frac {2 a^3}{3}-\frac {35 a^2 x}{9}\right )}{(a-x)^{5/2} (a+x)^{17/6}} \, dx,x,a \sin (e+f x)\right )}{4 a f}\\ &=-\frac {\sec (e+f x) (95 a+356 a \sin (e+f x))}{132 f (1-\sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac {3 a^2 \sin (e+f x) \tan (e+f x)}{4 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac {3 a^2 \sin ^2(e+f x) \tan (e+f x)}{f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac {\left (973 a \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x)^{3/2} (a+x)^{11/6}} \, dx,x,a \sin (e+f x)\right )}{396 f}\\ &=\frac {973 \sec (e+f x)}{396 f \sqrt [3]{a+a \sin (e+f x)}}-\frac {\sec (e+f x) (95 a+356 a \sin (e+f x))}{132 f (1-\sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac {3 a^2 \sin (e+f x) \tan (e+f x)}{4 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac {3 a^2 \sin ^2(e+f x) \tan (e+f x)}{f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac {\left (973 \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-x} (a+x)^{11/6}} \, dx,x,a \sin (e+f x)\right )}{297 f}\\ &=\frac {973 \sec (e+f x)}{396 f \sqrt [3]{a+a \sin (e+f x)}}-\frac {973 \sec (e+f x) (1-\sin (e+f x))}{495 f \sqrt [3]{a+a \sin (e+f x)}}-\frac {\sec (e+f x) (95 a+356 a \sin (e+f x))}{132 f (1-\sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac {3 a^2 \sin (e+f x) \tan (e+f x)}{4 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac {3 a^2 \sin ^2(e+f x) \tan (e+f x)}{f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac {\left (973 \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-x} (a+x)^{5/6}} \, dx,x,a \sin (e+f x)\right )}{1485 a f}\\ &=\frac {973 \sec (e+f x)}{396 f \sqrt [3]{a+a \sin (e+f x)}}-\frac {973 \sec (e+f x) (1-\sin (e+f x))}{495 f \sqrt [3]{a+a \sin (e+f x)}}-\frac {\sec (e+f x) (95 a+356 a \sin (e+f x))}{132 f (1-\sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac {3 a^2 \sin (e+f x) \tan (e+f x)}{4 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac {3 a^2 \sin ^2(e+f x) \tan (e+f x)}{f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac {\left (1946 \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 a-x^6}} \, dx,x,\sqrt [6]{a+a \sin (e+f x)}\right )}{495 a f}\\ &=\frac {973 \sec (e+f x)}{396 f \sqrt [3]{a+a \sin (e+f x)}}-\frac {973 \sec (e+f x) (1-\sin (e+f x))}{495 f \sqrt [3]{a+a \sin (e+f x)}}-\frac {\sec (e+f x) (95 a+356 a \sin (e+f x))}{132 f (1-\sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac {973 F\left (\cos ^{-1}\left (\frac {\sqrt [3]{2} \sqrt [3]{a}-\left (1-\sqrt {3}\right ) \sqrt [3]{a+a \sin (e+f x)}}{\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+a \sin (e+f x)}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) \sec (e+f x) (a+a \sin (e+f x))^{2/3} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+a \sin (e+f x)}\right ) \sqrt {\frac {2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a+a \sin (e+f x)}+(a+a \sin (e+f x))^{2/3}}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+a \sin (e+f x)}\right )^2}}}{495 \sqrt [3]{2} \sqrt [4]{3} a^{4/3} f \sqrt {-\frac {\sqrt [3]{a+a \sin (e+f x)} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+a \sin (e+f x)}\right )}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+a \sin (e+f x)}\right )^2}}}+\frac {3 a^2 \sin (e+f x) \tan (e+f x)}{4 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac {3 a^2 \sin ^2(e+f x) \tan (e+f x)}{f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}\\ \end {align*}
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Mathematica [C] time = 0.79, size = 128, normalized size = 0.23 \[ \frac {973 \sqrt {2} \cos (e+f x) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\sin ^2\left (\frac {1}{4} (2 e+2 f x+\pi )\right )\right )+\sqrt {1-\sin (e+f x)} \sec ^3(e+f x) (22 \sin (e+f x)-128 \sin (3 (e+f x))-64 \cos (2 (e+f x))-49)}{495 f \sqrt {1-\sin (e+f x)} \sqrt [3]{a (\sin (e+f x)+1)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.31, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\tan \left (f x + e\right )^{4}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {1}{3}}}, x\right ) \]
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 \[ \int \frac {\tan \left (f x + e\right )^{4}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{4}\left (f x +e \right )}{\left (a +a \sin \left (f x +e \right )\right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {\tan \left (f x + e\right )^{4}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {tan}\left (e+f\,x\right )}^4}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{1/3}} \,d x \]
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 \[ \int \frac {\tan ^{4}{\left (e + f x \right )}}{\sqrt [3]{a \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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