Optimal. Leaf size=205 \[ -\frac {(5-2 n) \, _2F_1\left (1,\frac {1}{2}+n;\frac {3}{2}+n;\frac {1}{2} (1-\sec (e+f x))\right ) (c-c \sec (e+f x))^n \tan (e+f x)}{4 a f (1+2 n) \sqrt {a+a \sec (e+f x)}}+\frac {2 \, _2F_1\left (1,\frac {1}{2}+n;\frac {3}{2}+n;1-\sec (e+f x)\right ) (c-c \sec (e+f x))^n \tan (e+f x)}{a f (1+2 n) \sqrt {a+a \sec (e+f x)}}-\frac {(c-c \sec (e+f x))^n \tan (e+f x)}{2 a f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3997, 105, 162,
67, 70} \begin {gather*} -\frac {(5-2 n) \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left (1,n+\frac {1}{2};n+\frac {3}{2};\frac {1}{2} (1-\sec (e+f x))\right )}{4 a f (2 n+1) \sqrt {a \sec (e+f x)+a}}+\frac {2 \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left (1,n+\frac {1}{2};n+\frac {3}{2};1-\sec (e+f x)\right )}{a f (2 n+1) \sqrt {a \sec (e+f x)+a}}-\frac {\tan (e+f x) (c-c \sec (e+f x))^n}{2 a f (\sec (e+f x)+1) \sqrt {a \sec (e+f x)+a}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 67
Rule 70
Rule 105
Rule 162
Rule 3997
Rubi steps
\begin {align*} \int \frac {(c-c \sec (e+f x))^n}{(a+a \sec (e+f x))^{3/2}} \, dx &=-\frac {(a c \tan (e+f x)) \text {Subst}\left (\int \frac {(c-c x)^{-\frac {1}{2}+n}}{x (a+a x)^2} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=-\frac {(c-c \sec (e+f x))^n \tan (e+f x)}{2 a f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}-\frac {\tan (e+f x) \text {Subst}\left (\int \frac {(c-c x)^{-\frac {1}{2}+n} \left (2 a c-\frac {1}{2} a c (1-2 n) x\right )}{x (a+a x)} \, dx,x,\sec (e+f x)\right )}{2 a f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=-\frac {(c-c \sec (e+f x))^n \tan (e+f x)}{2 a f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}-\frac {(c \tan (e+f x)) \text {Subst}\left (\int \frac {(c-c x)^{-\frac {1}{2}+n}}{x} \, dx,x,\sec (e+f x)\right )}{a f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}+\frac {(c (5-2 n) \tan (e+f x)) \text {Subst}\left (\int \frac {(c-c x)^{-\frac {1}{2}+n}}{a+a x} \, dx,x,\sec (e+f x)\right )}{4 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=-\frac {(5-2 n) \, _2F_1\left (1,\frac {1}{2}+n;\frac {3}{2}+n;\frac {1}{2} (1-\sec (e+f x))\right ) (c-c \sec (e+f x))^n \tan (e+f x)}{4 a f (1+2 n) \sqrt {a+a \sec (e+f x)}}+\frac {2 \, _2F_1\left (1,\frac {1}{2}+n;\frac {3}{2}+n;1-\sec (e+f x)\right ) (c-c \sec (e+f x))^n \tan (e+f x)}{a f (1+2 n) \sqrt {a+a \sec (e+f x)}}-\frac {(c-c \sec (e+f x))^n \tan (e+f x)}{2 a f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F]
time = 1.49, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c-c \sec (e+f x))^n}{(a+a \sec (e+f x))^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.17, size = 0, normalized size = 0.00 \[\int \frac {\left (c -c \sec \left (f x +e \right )\right )^{n}}{\left (a +a \sec \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- c \left (\sec {\left (e + f x \right )} - 1\right )\right )^{n}}{\left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \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 = -0.00 \begin {gather*} \int \frac {{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^n}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________