Optimal. Leaf size=115 \[ \frac {(i A (3-n)+B (3+n)) \, _2F_1\left (3,n;1+n;\frac {1}{2} (1-i \tan (e+f x))\right ) (c-i c \tan (e+f x))^n}{48 a^3 f n}+\frac {(i A-B) (c-i c \tan (e+f x))^n}{6 a^3 f (1+i \tan (e+f x))^3} \]
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Rubi [A]
time = 0.12, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {3669, 79, 70}
\begin {gather*} \frac {(B (n+3)+i A (3-n)) (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 )}{48 a^3 f n}+\frac {(-B+i A) (c-i c \tan (e+f x))^n}{6 a^3 f (1+i \tan (e+f x))^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 79
Rule 3669
Rubi steps
\begin {align*} \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^n}{(a+i a \tan (e+f x))^3} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {(A+B x) (c-i c x)^{-1+n}}{(a+i a x)^4} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(i A-B) (c-i c \tan (e+f x))^n}{6 a^3 f (1+i \tan (e+f x))^3}+\frac {(c (A (3-n)-i B (3+n))) \text {Subst}\left (\int \frac {(c-i c x)^{-1+n}}{(a+i a x)^3} \, dx,x,\tan (e+f x)\right )}{6 f}\\ &=\frac {(i A (3-n)+B (3+n)) \, _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}{48 a^3 f n}+\frac {(i A-B) (c-i c \tan (e+f x))^n}{6 a^3 f (1+i \tan (e+f x))^3}\\ \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 = 1.39, size = 0, normalized size = 0.00 \[\int \frac {\left (A +B \tan \left (f x +e \right )\right ) \left (c -i c \tan \left (f x +e \right )\right )^{n}}{\left (a +i a \tan \left (f x +e \right )\right )^{3}}\, dx\]
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 {i \left (\int \frac {A \left (- i c \tan {\left (e + f x \right )} + c\right )^{n}}{\tan ^{3}{\left (e + f x \right )} - 3 i \tan ^{2}{\left (e + f x \right )} - 3 \tan {\left (e + f x \right )} + i}\, dx + \int \frac {B \left (- i c \tan {\left (e + f x \right )} + c\right )^{n} \tan {\left (e + f x \right )}}{\tan ^{3}{\left (e + f x \right )} - 3 i \tan ^{2}{\left (e + f x \right )} - 3 \tan {\left (e + f x \right )} + i}\, dx\right )}{a^{3}} \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 = -0.01 \begin {gather*} \int \frac {\left (A+B\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\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 )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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