Optimal. Leaf size=52 \[ \frac {i \, _2F_1\left (3,n;1+n;\frac {1}{2} (1-i \tan (e+f x))\right ) (c-i c \tan (e+f x))^n}{8 a^2 f n} \]
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Rubi [A]
time = 0.09, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3603, 3568, 70}
\begin {gather*} \frac {i (c-i c \tan (e+f x))^n \, _2F_1\left (3,n;n+1;\frac {1}{2} (1-i \tan (e+f x))\right )}{8 a^2 f n} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 3568
Rule 3603
Rubi steps
\begin {align*} \int \frac {(c-i c \tan (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx &=\frac {\int \cos ^4(e+f x) (c-i c \tan (e+f x))^{2+n} \, dx}{a^2 c^2}\\ &=\frac {\left (i c^3\right ) \text {Subst}\left (\int \frac {(c+x)^{-1+n}}{(c-x)^3} \, dx,x,-i c \tan (e+f x)\right )}{a^2 f}\\ &=\frac {i \, _2F_1\left (3,n;1+n;\frac {1}{2} (1-i \tan (e+f x))\right ) (c-i c \tan (e+f x))^n}{8 a^2 f n}\\ \end {align*}
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Mathematica [F]
time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 2.69, size = 0, normalized size = 0.00 \[\int \frac {\left (c -i c \tan \left (f x +e \right )\right )^{n}}{\left (a +i a \tan \left (f x +e \right )\right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {\left (- i c \tan {\left (e + f x \right )} + c\right )^{n}}{\tan ^{2}{\left (e + f x \right )} - 2 i \tan {\left (e + f x \right )} - 1}\, dx}{a^{2}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^n}{{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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