Optimal. Leaf size=982 \[ -\frac {3 \sin ^2(e+f x) \tan (e+f x) a^2}{f (a-a \sin (e+f x)) (\sin (e+f x) a+a)^{2/3}}+\frac {3 \sin (e+f x) \tan (e+f x) a^2}{2 f (a-a \sin (e+f x)) (\sin (e+f x) a+a)^{2/3}}-\frac {\sec (e+f x) \left (65 a^2-142 a^2 \sin (e+f x)\right )}{42 f (a-a \sin (e+f x)) (\sin (e+f x) a+a)^{2/3}}-\frac {361 \sec (e+f x) \sqrt [3]{\sin (e+f x) a+a}}{126 f}+\frac {361 \sec (e+f x) (1-\sin (e+f x)) \sqrt [3]{\sin (e+f x) a+a}}{63 f}+\frac {361 \left (1+\sqrt {3}\right ) \sec (e+f x) (1-\sin (e+f x)) (\sin (e+f x) a+a)^{2/3}}{63 f \left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sin (e+f x) a+a}\right )}-\frac {361 \sqrt [3]{2} E\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 ) \sec (e+f x) (\sin (e+f x) a+a)^{2/3} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{\sin (e+f x) a+a}\right ) \sqrt {\frac {2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{\sin (e+f x) a+a} \sqrt [3]{a}+(\sin (e+f x) a+a)^{2/3}}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sin (e+f x) a+a}\right )^2}}}{21\ 3^{3/4} f \sqrt {-\frac {\sqrt [3]{\sin (e+f x) a+a} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{\sin (e+f x) a+a}\right )}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sin (e+f x) a+a}\right )^2}} a^{2/3}}-\frac {361 \left (1-\sqrt {3}\right ) 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 ) \sec (e+f x) (\sin (e+f x) a+a)^{2/3} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{\sin (e+f x) a+a}\right ) \sqrt {\frac {2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{\sin (e+f x) a+a} \sqrt [3]{a}+(\sin (e+f x) a+a)^{2/3}}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sin (e+f x) a+a}\right )^2}}}{63\ 2^{2/3} \sqrt [4]{3} f \sqrt {-\frac {\sqrt [3]{\sin (e+f x) a+a} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{\sin (e+f x) a+a}\right )}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sin (e+f x) a+a}\right )^2}} a^{2/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.27, antiderivative size = 982, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {2719, 100, 153, 144, 51, 63, 308, 225, 1881} \[ -\frac {3 \sin ^2(e+f x) \tan (e+f x) a^2}{f (a-a \sin (e+f x)) (\sin (e+f x) a+a)^{2/3}}+\frac {3 \sin (e+f x) \tan (e+f x) a^2}{2 f (a-a \sin (e+f x)) (\sin (e+f x) a+a)^{2/3}}-\frac {\sec (e+f x) \left (65 a^2-142 a^2 \sin (e+f x)\right )}{42 f (a-a \sin (e+f x)) (\sin (e+f x) a+a)^{2/3}}-\frac {361 \sec (e+f x) \sqrt [3]{\sin (e+f x) a+a}}{126 f}+\frac {361 \sec (e+f x) (1-\sin (e+f x)) \sqrt [3]{\sin (e+f x) a+a}}{63 f}+\frac {361 \left (1+\sqrt {3}\right ) \sec (e+f x) (1-\sin (e+f x)) (\sin (e+f x) a+a)^{2/3}}{63 f \left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sin (e+f x) a+a}\right )}-\frac {361 \sqrt [3]{2} E\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 ) \sec (e+f x) (\sin (e+f x) a+a)^{2/3} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{\sin (e+f x) a+a}\right ) \sqrt {\frac {2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{\sin (e+f x) a+a} \sqrt [3]{a}+(\sin (e+f x) a+a)^{2/3}}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sin (e+f x) a+a}\right )^2}}}{21\ 3^{3/4} f \sqrt {-\frac {\sqrt [3]{\sin (e+f x) a+a} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{\sin (e+f x) a+a}\right )}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sin (e+f x) a+a}\right )^2}} a^{2/3}}-\frac {361 \left (1-\sqrt {3}\right ) 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 ) \sec (e+f x) (\sin (e+f x) a+a)^{2/3} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{\sin (e+f x) a+a}\right ) \sqrt {\frac {2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{\sin (e+f x) a+a} \sqrt [3]{a}+(\sin (e+f x) a+a)^{2/3}}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sin (e+f x) a+a}\right )^2}}}{63\ 2^{2/3} \sqrt [4]{3} f \sqrt {-\frac {\sqrt [3]{\sin (e+f x) a+a} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{\sin (e+f x) a+a}\right )}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sin (e+f x) a+a}\right )^2}} a^{2/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 100
Rule 144
Rule 153
Rule 225
Rule 308
Rule 1881
Rule 2719
Rubi steps
\begin {align*} \int \sqrt [3]{a+a \sin (e+f x)} \tan ^4(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)^{13/6}} \, 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))^{2/3}}-\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)^{13/6}} \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac {3 a^2 \sin (e+f x) \tan (e+f x)}{2 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}-\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))^{2/3}}-\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 {17 a^2 x}{9}\right )}{(a-x)^{5/2} (a+x)^{13/6}} \, dx,x,a \sin (e+f x)\right )}{2 a f}\\ &=-\frac {\sec (e+f x) \left (65 a^2-142 a^2 \sin (e+f x)\right )}{42 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}+\frac {3 a^2 \sin (e+f x) \tan (e+f x)}{2 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}-\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))^{2/3}}-\frac {\left (361 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)^{7/6}} \, dx,x,a \sin (e+f x)\right )}{126 f}\\ &=-\frac {361 \sec (e+f x) \sqrt [3]{a+a \sin (e+f x)}}{126 f}-\frac {\sec (e+f x) \left (65 a^2-142 a^2 \sin (e+f x)\right )}{42 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}+\frac {3 a^2 \sin (e+f x) \tan (e+f x)}{2 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}-\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))^{2/3}}-\frac {\left (361 \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)^{7/6}} \, dx,x,a \sin (e+f x)\right )}{189 f}\\ &=-\frac {361 \sec (e+f x) \sqrt [3]{a+a \sin (e+f x)}}{126 f}+\frac {361 \sec (e+f x) (1-\sin (e+f x)) \sqrt [3]{a+a \sin (e+f x)}}{63 f}-\frac {\sec (e+f x) \left (65 a^2-142 a^2 \sin (e+f x)\right )}{42 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}+\frac {3 a^2 \sin (e+f x) \tan (e+f x)}{2 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}-\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))^{2/3}}+\frac {\left (361 \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} \sqrt [6]{a+x}} \, dx,x,a \sin (e+f x)\right )}{189 a f}\\ &=-\frac {361 \sec (e+f x) \sqrt [3]{a+a \sin (e+f x)}}{126 f}+\frac {361 \sec (e+f x) (1-\sin (e+f x)) \sqrt [3]{a+a \sin (e+f x)}}{63 f}-\frac {\sec (e+f x) \left (65 a^2-142 a^2 \sin (e+f x)\right )}{42 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}+\frac {3 a^2 \sin (e+f x) \tan (e+f x)}{2 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}-\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))^{2/3}}+\frac {\left (722 \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}{\sqrt {2 a-x^6}} \, dx,x,\sqrt [6]{a+a \sin (e+f x)}\right )}{63 a f}\\ &=-\frac {361 \sec (e+f x) \sqrt [3]{a+a \sin (e+f x)}}{126 f}+\frac {361 \sec (e+f x) (1-\sin (e+f x)) \sqrt [3]{a+a \sin (e+f x)}}{63 f}-\frac {\sec (e+f x) \left (65 a^2-142 a^2 \sin (e+f x)\right )}{42 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}+\frac {3 a^2 \sin (e+f x) \tan (e+f x)}{2 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}-\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))^{2/3}}-\frac {\left (361 \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {2^{2/3} \left (-1+\sqrt {3}\right ) a^{2/3}-2 x^4}{\sqrt {2 a-x^6}} \, dx,x,\sqrt [6]{a+a \sin (e+f x)}\right )}{63 a f}-\frac {\left (361\ 2^{2/3} \left (1-\sqrt {3}\right ) \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 )}{63 \sqrt [3]{a} f}\\ &=-\frac {361 \sec (e+f x) \sqrt [3]{a+a \sin (e+f x)}}{126 f}+\frac {361 \sec (e+f x) (1-\sin (e+f x)) \sqrt [3]{a+a \sin (e+f x)}}{63 f}-\frac {\sec (e+f x) \left (65 a^2-142 a^2 \sin (e+f x)\right )}{42 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}+\frac {361 \left (1+\sqrt {3}\right ) \sec (e+f x) (1-\sin (e+f x)) (a+a \sin (e+f x))^{2/3}}{63 f \left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+a \sin (e+f x)}\right )}-\frac {361 \sqrt [3]{2} E\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}}}{21\ 3^{3/4} a^{2/3} 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 {361 \left (1-\sqrt {3}\right ) 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}}}{63\ 2^{2/3} \sqrt [4]{3} a^{2/3} 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)}{2 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}-\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))^{2/3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 3.24, size = 318, normalized size = 0.32 \[ \frac {\sqrt [3]{a (\sin (e+f x)+1)} \left (3 \left (-172 \tan (e+f x)-3 \sec ^3(e+f x)+86 \sec (e+f x)+24 \tan (e+f x) \sec ^2(e+f x)+361\right )+\frac {\left (\frac {1083}{10}+\frac {1083 i}{10}\right ) (-1)^{3/4} e^{-i (e+f x)} \left (-2 \left (1+i e^{-i (e+f x)}\right )^{2/3} \left (1+e^{2 i (e+f x)}\right ) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};\sin ^2\left (\frac {1}{4} (2 e+2 f x+\pi )\right )\right )+5 i \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};-i e^{-i (e+f x)}\right ) \sqrt {2-2 \sin (e+f x)}+20 e^{i (e+f x)} \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {2}{3};-i e^{-i (e+f x)}\right ) \sqrt {\cos ^2\left (\frac {1}{4} (2 e+2 f x+\pi )\right )}\right )}{\sqrt {2} \left (1+i e^{-i (e+f x)}\right )^{2/3} \sqrt {i e^{-i (e+f x)} \left (e^{i (e+f x)}-i\right )^2}}\right )}{189 f} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {1}{3}} \tan \left (f x + e\right )^{4}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {1}{3}} \tan \left (f x + e\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.27, size = 0, normalized size = 0.00 \[ \int \left (a +a \sin \left (f x +e \right )\right )^{\frac {1}{3}} \left (\tan ^{4}\left (f x +e \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {1}{3}} \tan \left (f x + e\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {tan}\left (e+f\,x\right )}^4\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{1/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt [3]{a \left (\sin {\left (e + f x \right )} + 1\right )} \tan ^{4}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________