Optimal. Leaf size=111 \[ \frac {2^{n+\frac {3}{2}} \sqrt {\tan (c+d x)} \left (\frac {1}{\sec (c+d x)+1}\right )^{n+\frac {1}{2}} (a \sec (c+d x)+a)^n F_1\left (\frac {1}{4};n-\frac {1}{2},1;\frac {5}{4};-\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{d} \]
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Rubi [A] time = 0.06, 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 = {3889} \[ \frac {2^{n+\frac {3}{2}} \sqrt {\tan (c+d x)} \left (\frac {1}{\sec (c+d x)+1}\right )^{n+\frac {1}{2}} (a \sec (c+d x)+a)^n F_1\left (\frac {1}{4};n-\frac {1}{2},1;\frac {5}{4};-\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 3889
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*}
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Mathematica [B] time = 1.61, size = 229, normalized size = 2.06 \[ \frac {10 \cos (c+d x) (\cos (c+d x)+1) \sqrt {\tan (c+d x)} (a (\sec (c+d x)+1))^n F_1\left (\frac {1}{4};n-\frac {1}{2},1;\frac {5}{4};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}{d \left (2 (\cos (c+d x)-1) \left (2 F_1\left (\frac {5}{4};n-\frac {1}{2},2;\frac {9}{4};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )+(1-2 n) F_1\left (\frac {5}{4};n+\frac {1}{2},1;\frac {9}{4};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right )+5 (\cos (c+d x)+1) F_1\left (\frac {1}{4};n-\frac {1}{2},1;\frac {5}{4};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a \sec \left (d x + c\right ) + a\right )}^{n}}{\sqrt {\tan \left (d x + c\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \sec \left (d x + c\right ) + a\right )}^{n}}{\sqrt {\tan \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.54, 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.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \sec \left (d x + c\right ) + a\right )}^{n}}{\sqrt {\tan \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n}{\sqrt {\mathrm {tan}\left (c+d\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{n}}{\sqrt {\tan {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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