Optimal. Leaf size=78 \[ \frac {3 \tan (c+d x) F_1\left (-\frac {2}{3};\frac {1}{2},1;\frac {1}{3};\sec (c+d x),-\sec (c+d x)\right )}{2 d \sqrt {1-\sec (c+d x)} \sqrt {a \sec (c+d x)+a} (e \sec (c+d x))^{2/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3828, 3827, 130, 510} \[ \frac {3 \tan (c+d x) F_1\left (-\frac {2}{3};\frac {1}{2},1;\frac {1}{3};\sec (c+d x),-\sec (c+d x)\right )}{2 d \sqrt {1-\sec (c+d x)} \sqrt {a \sec (c+d x)+a} (e \sec (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 130
Rule 510
Rule 3827
Rule 3828
Rubi steps
\begin {align*} \int \frac {1}{(e \sec (c+d x))^{2/3} \sqrt {a+a \sec (c+d x)}} \, dx &=\frac {\sqrt {1+\sec (c+d x)} \int \frac {1}{(e \sec (c+d x))^{2/3} \sqrt {1+\sec (c+d x)}} \, dx}{\sqrt {a+a \sec (c+d x)}}\\ &=-\frac {(e \tan (c+d x)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} (e x)^{5/3} (1+x)} \, dx,x,\sec (c+d x)\right )}{d \sqrt {1-\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}\\ &=-\frac {(3 \tan (c+d x)) \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {1-\frac {x^3}{e}} \left (1+\frac {x^3}{e}\right )} \, dx,x,\sqrt [3]{e \sec (c+d x)}\right )}{d \sqrt {1-\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}\\ &=\frac {3 F_1\left (-\frac {2}{3};\frac {1}{2},1;\frac {1}{3};\sec (c+d x),-\sec (c+d x)\right ) \tan (c+d x)}{2 d \sqrt {1-\sec (c+d x)} (e \sec (c+d x))^{2/3} \sqrt {a+a \sec (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 7.47, size = 585, normalized size = 7.50 \[ \frac {\sec ^{\frac {7}{6}}(c+d x) \left (\frac {5 \sin \left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {1}{\cos (c+d x)+1}} (3 \cos (c+d x)-1) \left (\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)\right )^{5/6} \left (2 \tan ^2\left (\frac {1}{2} (c+d x)\right ) F_1\left (\frac {3}{2};\frac {5}{6},\frac {2}{3};\frac {5}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )^{5/6}-3 \cos ^{\frac {5}{6}}(c+d x) \sqrt [3]{\sec ^2\left (\frac {1}{2} (c+d x)\right )} \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {3}{2};2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )\right )}{5 \sqrt {2} \cos \left (\frac {1}{2} (c+d x)\right ) \left (3-4 \sqrt {2} \left (\frac {1}{\cos (c+d x)+1}\right )^{2/3} \left (\frac {\cos (c+d x)}{\cos (c+d x)+1}\right )^{5/6} \tan ^4\left (\frac {1}{2} (c+d x)\right ) F_1\left (\frac {5}{2};\frac {11}{6},\frac {2}{3};\frac {7}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right )-120 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\frac {1}{\cos (c+d x)+1}\right )^{2/3} \left (\frac {\cos (c+d x)}{\cos (c+d x)+1}\right )^{5/6} \tan \left (\frac {1}{2} (c+d x)\right ) F_1\left (\frac {3}{2};\frac {5}{6},\frac {2}{3};\frac {5}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )+32 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\frac {1}{\cos (c+d x)+1}\right )^{2/3} \left (\frac {\cos (c+d x)}{\cos (c+d x)+1}\right )^{5/6} \tan ^3\left (\frac {1}{2} (c+d x)\right ) F_1\left (\frac {5}{2};\frac {5}{6},\frac {5}{3};\frac {7}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}-\frac {3}{2} \left (\sin \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {3}{2} (c+d x)\right )\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \sec ^{\frac {5}{6}}(c+d x)\right )}{d \sqrt {a (\sec (c+d x)+1)} (e \sec (c+d x))^{2/3}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.19, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \sec \left (d x +c \right )\right )^{\frac {2}{3}} \sqrt {a +a \sec \left (d x +c \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \sec \left (d x + c\right ) + a} \left (e \sec \left (d x + c\right )\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}\,{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{2/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )} \left (e \sec {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________