Optimal. Leaf size=131 \[ \frac {(B+i A) (a+i a \tan (c+d x))^n \, _2F_1\left (1,n;n+1;\frac {1}{2} (i \tan (c+d x)+1)\right )}{2 d n}-\frac {(B+i A n) (a+i a \tan (c+d x))^n \, _2F_1(1,n;n+1;i \tan (c+d x)+1)}{d n}-\frac {A \cot (c+d x) (a+i a \tan (c+d x))^n}{d} \]
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Rubi [A] time = 0.33, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3598, 3600, 3481, 68, 3599, 65} \[ \frac {(B+i A) (a+i a \tan (c+d x))^n \, _2F_1\left (1,n;n+1;\frac {1}{2} (i \tan (c+d x)+1)\right )}{2 d n}-\frac {(B+i A n) (a+i a \tan (c+d x))^n \, _2F_1(1,n;n+1;i \tan (c+d x)+1)}{d n}-\frac {A \cot (c+d x) (a+i a \tan (c+d x))^n}{d} \]
Antiderivative was successfully verified.
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Rule 65
Rule 68
Rule 3481
Rule 3598
Rule 3599
Rule 3600
Rubi steps
\begin {align*} \int \cot ^2(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx &=-\frac {A \cot (c+d x) (a+i a \tan (c+d x))^n}{d}+\frac {\int \cot (c+d x) (a+i a \tan (c+d x))^n (a (B+i A n)-a A (1-n) \tan (c+d x)) \, dx}{a}\\ &=-\frac {A \cot (c+d x) (a+i a \tan (c+d x))^n}{d}+(-A+i B) \int (a+i a \tan (c+d x))^n \, dx+\frac {(B+i A n) \int \cot (c+d x) (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^n \, dx}{a}\\ &=-\frac {A \cot (c+d x) (a+i a \tan (c+d x))^n}{d}+\frac {(a (i A+B)) \operatorname {Subst}\left (\int \frac {(a+x)^{-1+n}}{a-x} \, dx,x,i a \tan (c+d x)\right )}{d}+\frac {(a (B+i A n)) \operatorname {Subst}\left (\int \frac {(a+i a x)^{-1+n}}{x} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {A \cot (c+d x) (a+i a \tan (c+d x))^n}{d}+\frac {(i A+B) \, _2F_1\left (1,n;1+n;\frac {1}{2} (1+i \tan (c+d x))\right ) (a+i a \tan (c+d x))^n}{2 d n}-\frac {(B+i A n) \, _2F_1(1,n;1+n;1+i \tan (c+d x)) (a+i a \tan (c+d x))^n}{d n}\\ \end {align*}
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Mathematica [F] time = 47.03, size = 0, normalized size = 0.00 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left ({\left (A - i \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, A e^{\left (2 i \, d x + 2 i \, c\right )} + A + i \, B\right )} \left (\frac {2 \, a e^{\left (2 i \, d x + 2 i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{n}}{e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1}, x\right ) \]
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 \[ \int {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.44, size = 0, normalized size = 0.00 \[ \int \left (\cot ^{2}\left (d x +c \right )\right ) \left (a +i a \tan \left (d x +c \right )\right )^{n} \left (A +B \tan \left (d x +c \right )\right )\, dx \]
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 \[ \int {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{2}\,{d x} \]
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 \[ \int {\mathrm {cot}\left (c+d\,x\right )}^2\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^n \,d x \]
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 \[ \int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{n} \left (A + B \tan {\left (c + d x \right )}\right ) \cot ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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