Optimal. Leaf size=177 \[ \frac {2 a^3 \left (16 n^2+24 n+3\right ) \tan (e+f x) \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};1-\sec (e+f x)\right )}{f (2 n+1) (2 n+3) \sqrt {a \sec (e+f x)+a}}+\frac {2 a^3 (4 n+7) \sin (e+f x) \sec ^{n+1}(e+f x)}{f (2 n+1) (2 n+3) \sqrt {a \sec (e+f x)+a}}+\frac {2 a^2 \sin (e+f x) \sqrt {a \sec (e+f x)+a} \sec ^{n+1}(e+f x)}{f (2 n+3)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.30, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3814, 4016, 3806, 65} \[ \frac {2 a^3 \left (16 n^2+24 n+3\right ) \tan (e+f x) \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};1-\sec (e+f x)\right )}{f (2 n+1) (2 n+3) \sqrt {a \sec (e+f x)+a}}+\frac {2 a^2 \sin (e+f x) \sqrt {a \sec (e+f x)+a} \sec ^{n+1}(e+f x)}{f (2 n+3)}+\frac {2 a^3 (4 n+7) \sin (e+f x) \sec ^{n+1}(e+f x)}{f (2 n+1) (2 n+3) \sqrt {a \sec (e+f x)+a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 3806
Rule 3814
Rule 4016
Rubi steps
\begin {align*} \int \sec ^n(e+f x) (a+a \sec (e+f x))^{5/2} \, dx &=\frac {2 a^2 \sec ^{1+n}(e+f x) \sqrt {a+a \sec (e+f x)} \sin (e+f x)}{f (3+2 n)}+\frac {(2 a) \int \sec ^n(e+f x) \sqrt {a+a \sec (e+f x)} \left (a \left (\frac {3}{2}+2 n\right )+a \left (\frac {7}{2}+2 n\right ) \sec (e+f x)\right ) \, dx}{3+2 n}\\ &=\frac {2 a^3 (7+4 n) \sec ^{1+n}(e+f x) \sin (e+f x)}{f (1+2 n) (3+2 n) \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 \sec ^{1+n}(e+f x) \sqrt {a+a \sec (e+f x)} \sin (e+f x)}{f (3+2 n)}+\frac {\left (a^2 \left (3+24 n+16 n^2\right )\right ) \int \sec ^n(e+f x) \sqrt {a+a \sec (e+f x)} \, dx}{3+8 n+4 n^2}\\ &=\frac {2 a^3 (7+4 n) \sec ^{1+n}(e+f x) \sin (e+f x)}{f (1+2 n) (3+2 n) \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 \sec ^{1+n}(e+f x) \sqrt {a+a \sec (e+f x)} \sin (e+f x)}{f (3+2 n)}-\frac {\left (a^4 \left (3+24 n+16 n^2\right ) \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {x^{-1+n}}{\sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{f \left (3+8 n+4 n^2\right ) \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {2 a^3 (7+4 n) \sec ^{1+n}(e+f x) \sin (e+f x)}{f (1+2 n) (3+2 n) \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 \sec ^{1+n}(e+f x) \sqrt {a+a \sec (e+f x)} \sin (e+f x)}{f (3+2 n)}+\frac {2 a^3 \left (3+24 n+16 n^2\right ) \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};1-\sec (e+f x)\right ) \tan (e+f x)}{f \left (3+8 n+4 n^2\right ) \sqrt {a+a \sec (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 8.12, size = 400, normalized size = 2.26 \[ -\frac {i 2^{n-\frac {5}{2}} e^{-\frac {1}{2} i (2 n+3) (e+f x)} \left (\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^{n+\frac {3}{2}} \sec ^5\left (\frac {1}{2} (e+f x)\right ) (a (\sec (e+f x)+1))^{5/2} \left (\frac {10 e^{i (n+2) (e+f x)} \, _2F_1\left (1,\frac {1}{2} (-n-1);\frac {n+4}{2};-e^{2 i (e+f x)}\right )}{n+2}+\frac {5 e^{i (n+4) (e+f x)} \, _2F_1\left (1,\frac {1-n}{2};\frac {n+6}{2};-e^{2 i (e+f x)}\right )}{n+4}+\frac {e^{i n (e+f x)} \, _2F_1\left (1,-\frac {n}{2}-\frac {3}{2};\frac {n}{2}+1;-e^{2 i (e+f x)}\right )}{n}+\frac {5 e^{i (n+1) (e+f x)} \, _2F_1\left (1,-\frac {n}{2}-1;\frac {n+3}{2};-e^{2 i (e+f x)}\right )}{n+1}+\frac {e^{i (n+5) (e+f x)} \, _2F_1\left (1,1-\frac {n}{2};\frac {n+7}{2};-e^{2 i (e+f x)}\right )}{n+5}+\frac {10 e^{i (n+3) (e+f x)} \, _2F_1\left (1,-\frac {n}{2};\frac {n+5}{2};-e^{2 i (e+f x)}\right )}{n+3}\right )}{f \sec ^{\frac {5}{2}}(e+f x)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} \sec \left (f x + e\right )^{2} + 2 \, a^{2} \sec \left (f x + e\right ) + a^{2}\right )} \sqrt {a \sec \left (f x + e\right ) + a} \sec \left (f x + e\right )^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \sec \left (f x + e\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.19, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{n}\left (f x +e \right )\right ) \left (a +a \sec \left (f x +e \right )\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \sec \left (f x + e\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}\,{\left (\frac {1}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [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]
________________________________________________________________________________________