Optimal. Leaf size=178 \[ \frac {2 a^3 \left (3+24 n+16 n^2\right ) \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};1+\sec (e+f x)\right ) \tan (e+f x)}{f (1+2 n) (3+2 n) \sqrt {a-a \sec (e+f x)}}+\frac {2 a^3 (7+4 n) (-\sec (e+f x))^n \tan (e+f x)}{f (1+2 n) (3+2 n) \sqrt {a-a \sec (e+f x)}}+\frac {2 a^2 (-\sec (e+f x))^n \sqrt {a-a \sec (e+f x)} \tan (e+f x)}{f (3+2 n)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.25, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3899, 4101,
3891, 67} \begin {gather*} \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};\sec (e+f x)+1\right )}{f (2 n+1) (2 n+3) \sqrt {a-a \sec (e+f x)}}+\frac {2 a^3 (4 n+7) \tan (e+f x) (-\sec (e+f x))^n}{f (2 n+1) (2 n+3) \sqrt {a-a \sec (e+f x)}}+\frac {2 a^2 \tan (e+f x) \sqrt {a-a \sec (e+f x)} (-\sec (e+f x))^n}{f (2 n+3)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 67
Rule 3891
Rule 3899
Rule 4101
Rubi steps
\begin {align*} \int (-\sec (e+f x))^n (a-a \sec (e+f x))^{5/2} \, dx &=\frac {2 a^2 (-\sec (e+f x))^n \sqrt {a-a \sec (e+f x)} \tan (e+f x)}{f (3+2 n)}-\frac {(2 a) \int (-\sec (e+f x))^n \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 (e+f x))^n \tan (e+f x)}{f (1+2 n) (3+2 n) \sqrt {a-a \sec (e+f x)}}+\frac {2 a^2 (-\sec (e+f x))^n \sqrt {a-a \sec (e+f x)} \tan (e+f x)}{f (3+2 n)}+\frac {\left (a^2 \left (3+24 n+16 n^2\right )\right ) \int (-\sec (e+f x))^n \sqrt {a-a \sec (e+f x)} \, dx}{3+8 n+4 n^2}\\ &=\frac {2 a^3 (7+4 n) (-\sec (e+f x))^n \tan (e+f x)}{f (1+2 n) (3+2 n) \sqrt {a-a \sec (e+f x)}}+\frac {2 a^2 (-\sec (e+f x))^n \sqrt {a-a \sec (e+f x)} \tan (e+f x)}{f (3+2 n)}+\frac {\left (a^4 \left (3+24 n+16 n^2\right ) \tan (e+f x)\right ) \text {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 \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)}}+\frac {2 a^3 (7+4 n) (-\sec (e+f x))^n \tan (e+f x)}{f (1+2 n) (3+2 n) \sqrt {a-a \sec (e+f x)}}+\frac {2 a^2 (-\sec (e+f x))^n \sqrt {a-a \sec (e+f x)} \tan (e+f x)}{f (3+2 n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 26.26, size = 458, normalized size = 2.57 \begin {gather*} \frac {2^{-\frac {5}{2}+n} e^{\frac {1}{2} i (e+f (1-2 n) x)} \left (\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^{-\frac {1}{2}+n} \left (1+e^{2 i (e+f x)}\right )^{-\frac {1}{2}+n} \csc ^5\left (\frac {e}{2}+\frac {f x}{2}\right ) \left (\frac {e^{i f n x} \, _2F_1\left (\frac {n}{2},\frac {5}{2}+n;\frac {2+n}{2};-e^{2 i (e+f x)}\right )}{n}-\frac {5 e^{i (e+f (1+n) x)} \, _2F_1\left (\frac {1+n}{2},\frac {5}{2}+n;\frac {3+n}{2};-e^{2 i (e+f x)}\right )}{1+n}+\frac {10 e^{i (2 e+f (2+n) x)} \, _2F_1\left (\frac {2+n}{2},\frac {5}{2}+n;\frac {4+n}{2};-e^{2 i (e+f x)}\right )}{2+n}-\frac {10 e^{i (3 e+f (3+n) x)} \, _2F_1\left (\frac {5}{2}+n,\frac {3+n}{2};\frac {5+n}{2};-e^{2 i (e+f x)}\right )}{3+n}+\frac {5 e^{i (4 e+f (4+n) x)} \, _2F_1\left (\frac {5}{2}+n,\frac {4+n}{2};\frac {6+n}{2};-e^{2 i (e+f x)}\right )}{4+n}-\frac {e^{i (5 e+f (5+n) x)} \, _2F_1\left (\frac {5}{2}+n,\frac {5+n}{2};\frac {7+n}{2};-e^{2 i (e+f x)}\right )}{5+n}\right ) (-\sec (e+f x))^n \sec ^{-\frac {5}{2}-n}(e+f x) (a-a \sec (e+f x))^{5/2}}{f} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \left (-\sec \left (f x +e \right )\right )^{n} \left (a -a \sec \left (f x +e \right )\right )^{\frac {5}{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________