Optimal. Leaf size=91 \[ -\frac {i 2^{n+\frac {3}{4}} (1+i \tan (c+d x))^{\frac {1}{4}-n} (a+i a \tan (c+d x))^n \, _2F_1\left (-\frac {1}{4},\frac {5}{4}-n;\frac {3}{4};\frac {1}{2} (1-i \tan (c+d x))\right )}{d \sqrt {e \sec (c+d x)}} \]
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Rubi [A] time = 0.18, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3505, 3523, 70, 69} \[ -\frac {i 2^{n+\frac {3}{4}} (1+i \tan (c+d x))^{\frac {1}{4}-n} (a+i a \tan (c+d x))^n \text {Hypergeometric2F1}\left (-\frac {1}{4},\frac {5}{4}-n,\frac {3}{4},\frac {1}{2} (1-i \tan (c+d x))\right )}{d \sqrt {e \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 3505
Rule 3523
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (c+d x))^n}{\sqrt {e \sec (c+d x)}} \, dx &=\frac {\left (\sqrt [4]{a-i a \tan (c+d x)} \sqrt [4]{a+i a \tan (c+d x)}\right ) \int \frac {(a+i a \tan (c+d x))^{-\frac {1}{4}+n}}{\sqrt [4]{a-i a \tan (c+d x)}} \, dx}{\sqrt {e \sec (c+d x)}}\\ &=\frac {\left (a^2 \sqrt [4]{a-i a \tan (c+d x)} \sqrt [4]{a+i a \tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {(a+i a x)^{-\frac {5}{4}+n}}{(a-i a x)^{5/4}} \, dx,x,\tan (c+d x)\right )}{d \sqrt {e \sec (c+d x)}}\\ &=\frac {\left (2^{-\frac {5}{4}+n} a \sqrt [4]{a-i a \tan (c+d x)} (a+i a \tan (c+d x))^n \left (\frac {a+i a \tan (c+d x)}{a}\right )^{\frac {1}{4}-n}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1}{2}+\frac {i x}{2}\right )^{-\frac {5}{4}+n}}{(a-i a x)^{5/4}} \, dx,x,\tan (c+d x)\right )}{d \sqrt {e \sec (c+d x)}}\\ &=-\frac {i 2^{\frac {3}{4}+n} \, _2F_1\left (-\frac {1}{4},\frac {5}{4}-n;\frac {3}{4};\frac {1}{2} (1-i \tan (c+d x))\right ) (1+i \tan (c+d x))^{\frac {1}{4}-n} (a+i a \tan (c+d x))^n}{d \sqrt {e \sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 9.75, size = 129, normalized size = 1.42 \[ -\frac {i 2^{n+\frac {1}{2}} e^{i (c+d x)} \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{n-\frac {3}{2}} \, _2F_1\left (1,\frac {5}{4};n+\frac {3}{4};-e^{2 i (c+d x)}\right ) \sec ^{\frac {1}{2}-n}(c+d x) (a+i a \tan (c+d x))^n}{d (4 n-1) \sqrt {e \sec (c+d x)}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {2} \left (\frac {2 \, a e^{\left (2 i \, d x + 2 i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{n} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{2 \, e}, x\right ) \]
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 \[ \int \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n}}{\sqrt {e \sec \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.88, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +i a \tan \left (d x +c \right )\right )^{n}}{\sqrt {e \sec \left (d x +c \right )}}\, dx \]
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 \[ \int \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n}}{\sqrt {e \sec \left (d x + c\right )}}\,{d x} \]
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 \[ \int \frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^n}{\sqrt {\frac {e}{\cos \left (c+d\,x\right )}}} \,d x \]
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 \[ \int \frac {\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{n}}{\sqrt {e \sec {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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