Optimal. Leaf size=111 \[ \frac {2^{\frac {3}{2}+n} F_1\left (\frac {1}{4};-\frac {1}{2}+n,1;\frac {5}{4};-\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)},\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)}\right ) \left (\frac {1}{1+\sec (c+d x)}\right )^{\frac {1}{2}+n} (a+a \sec (c+d x))^n \sqrt {\tan (c+d x)}}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {3974}
\begin {gather*} \frac {2^{n+\frac {3}{2}} \sqrt {\tan (c+d x)} \left (\frac {1}{\sec (c+d x)+1}\right )^{n+\frac {1}{2}} (a \sec (c+d x)+a)^n F_1\left (\frac {1}{4};n-\frac {1}{2},1;\frac {5}{4};-\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3974
Rubi steps
\begin {align*} \int \frac {(a+a \sec (c+d x))^n}{\sqrt {\tan (c+d x)}} \, dx &=\frac {2^{\frac {3}{2}+n} F_1\left (\frac {1}{4};-\frac {1}{2}+n,1;\frac {5}{4};-\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)},\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)}\right ) \left (\frac {1}{1+\sec (c+d x)}\right )^{\frac {1}{2}+n} (a+a \sec (c+d x))^n \sqrt {\tan (c+d x)}}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(229\) vs. \(2(111)=222\).
time = 2.42, size = 229, normalized size = 2.06 \begin {gather*} \frac {10 F_1\left (\frac {1}{4};-\frac {1}{2}+n,1;\frac {5}{4};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \cos (c+d x) (1+\cos (c+d x)) (a (1+\sec (c+d x)))^n \sqrt {\tan (c+d x)}}{d \left (2 \left (2 F_1\left (\frac {5}{4};-\frac {1}{2}+n,2;\frac {9}{4};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )+(1-2 n) F_1\left (\frac {5}{4};\frac {1}{2}+n,1;\frac {9}{4};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right ) (-1+\cos (c+d x))+5 F_1\left (\frac {1}{4};-\frac {1}{2}+n,1;\frac {5}{4};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) (1+\cos (c+d x))\right )} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int \frac {\left (a +a \sec \left (d x +c \right )\right )^{n}}{\sqrt {\tan \left (d x +c \right )}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \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 (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{n}}{\sqrt {\tan {\left (c + d x \right )}}}\, 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 = -0.01 \begin {gather*} \int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n}{\sqrt {\mathrm {tan}\left (c+d\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________