Optimal. Leaf size=291 \[ -\frac {2 (A-i B) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n F_1\left (\frac {1}{2};1-n,1;\frac {3}{2};-i \tan (c+d x),i \tan (c+d x)\right )}{d \sqrt {\cot (c+d x)}}+\frac {2 \left (2 A n (2 n+3)-i B \left (4 n^2+6 n+3\right )\right ) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};-i \tan (c+d x)\right )}{d (2 n+1) (2 n+3) \sqrt {\cot (c+d x)}}-\frac {2 (-A (2 n+3)+2 i B n) (a+i a \tan (c+d x))^n}{d (2 n+1) (2 n+3) \sqrt {\cot (c+d x)}}+\frac {2 B (a+i a \tan (c+d x))^n}{d (2 n+3) \cot ^{\frac {3}{2}}(c+d x)} \]
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Rubi [A] time = 0.90, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {4241, 3597, 3601, 3564, 130, 430, 429, 3599, 66, 64} \[ -\frac {2 (A-i B) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n F_1\left (\frac {1}{2};1-n,1;\frac {3}{2};-i \tan (c+d x),i \tan (c+d x)\right )}{d \sqrt {\cot (c+d x)}}+\frac {2 \left (2 A n (2 n+3)-i B \left (4 n^2+6 n+3\right )\right ) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};-i \tan (c+d x)\right )}{d (2 n+1) (2 n+3) \sqrt {\cot (c+d x)}}-\frac {2 (-A (2 n+3)+2 i B n) (a+i a \tan (c+d x))^n}{d (2 n+1) (2 n+3) \sqrt {\cot (c+d x)}}+\frac {2 B (a+i a \tan (c+d x))^n}{d (2 n+3) \cot ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 64
Rule 66
Rule 130
Rule 429
Rule 430
Rule 3564
Rule 3597
Rule 3599
Rule 3601
Rule 4241
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (c+d x))^n (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx &=\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx\\ &=\frac {2 B (a+i a \tan (c+d x))^n}{d (3+2 n) \cot ^{\frac {3}{2}}(c+d x)}+\frac {\left (2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^n \left (-\frac {3 a B}{2}-\frac {1}{2} a (2 i B n-A (3+2 n)) \tan (c+d x)\right ) \, dx}{a (3+2 n)}\\ &=\frac {2 B (a+i a \tan (c+d x))^n}{d (3+2 n) \cot ^{\frac {3}{2}}(c+d x)}-\frac {2 (2 i B n-A (3+2 n)) (a+i a \tan (c+d x))^n}{d (1+2 n) (3+2 n) \sqrt {\cot (c+d x)}}+\frac {\left (4 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {(a+i a \tan (c+d x))^n \left (\frac {1}{4} a^2 (2 i B n-A (3+2 n))-\frac {1}{4} a^2 \left (2 i A n (3+2 n)+B \left (3+6 n+4 n^2\right )\right ) \tan (c+d x)\right )}{\sqrt {\tan (c+d x)}} \, dx}{a^2 (1+2 n) (3+2 n)}\\ &=\frac {2 B (a+i a \tan (c+d x))^n}{d (3+2 n) \cot ^{\frac {3}{2}}(c+d x)}-\frac {2 (2 i B n-A (3+2 n)) (a+i a \tan (c+d x))^n}{d (1+2 n) (3+2 n) \sqrt {\cot (c+d x)}}-\frac {\left ((A-i B) \left (3+8 n+4 n^2\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {(a+i a \tan (c+d x))^n}{\sqrt {\tan (c+d x)}} \, dx}{(1+2 n) (3+2 n)}+\frac {\left (\left (2 A n (3+2 n)-i B \left (3+6 n+4 n^2\right )\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {(a-i a \tan (c+d x)) (a+i a \tan (c+d x))^n}{\sqrt {\tan (c+d x)}} \, dx}{a (1+2 n) (3+2 n)}\\ &=\frac {2 B (a+i a \tan (c+d x))^n}{d (3+2 n) \cot ^{\frac {3}{2}}(c+d x)}-\frac {2 (2 i B n-A (3+2 n)) (a+i a \tan (c+d x))^n}{d (1+2 n) (3+2 n) \sqrt {\cot (c+d x)}}-\frac {\left (i a^2 (A-i B) \left (3+8 n+4 n^2\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {(a+x)^{-1+n}}{\sqrt {-\frac {i x}{a}} \left (-a^2+a x\right )} \, dx,x,i a \tan (c+d x)\right )}{d (1+2 n) (3+2 n)}+\frac {\left (a \left (2 A n (3+2 n)-i B \left (3+6 n+4 n^2\right )\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {(a+i a x)^{-1+n}}{\sqrt {x}} \, dx,x,\tan (c+d x)\right )}{d (1+2 n) (3+2 n)}\\ &=\frac {2 B (a+i a \tan (c+d x))^n}{d (3+2 n) \cot ^{\frac {3}{2}}(c+d x)}-\frac {2 (2 i B n-A (3+2 n)) (a+i a \tan (c+d x))^n}{d (1+2 n) (3+2 n) \sqrt {\cot (c+d x)}}+\frac {\left (2 a^3 (A-i B) \left (3+8 n+4 n^2\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {\left (a+i a x^2\right )^{-1+n}}{-a^2+i a^2 x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d (1+2 n) (3+2 n)}+\frac {\left (\left (2 A n (3+2 n)-i B \left (3+6 n+4 n^2\right )\right ) \sqrt {\cot (c+d x)} (1+i \tan (c+d x))^{-n} \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^n\right ) \operatorname {Subst}\left (\int \frac {(1+i x)^{-1+n}}{\sqrt {x}} \, dx,x,\tan (c+d x)\right )}{d (1+2 n) (3+2 n)}\\ &=\frac {2 B (a+i a \tan (c+d x))^n}{d (3+2 n) \cot ^{\frac {3}{2}}(c+d x)}-\frac {2 (2 i B n-A (3+2 n)) (a+i a \tan (c+d x))^n}{d (1+2 n) (3+2 n) \sqrt {\cot (c+d x)}}+\frac {2 \left (2 A n (3+2 n)-i B \left (3+6 n+4 n^2\right )\right ) \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};-i \tan (c+d x)\right ) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n}{d (1+2 n) (3+2 n) \sqrt {\cot (c+d x)}}+\frac {\left (2 a^2 (A-i B) \left (3+8 n+4 n^2\right ) \sqrt {\cot (c+d x)} (1+i \tan (c+d x))^{-n} \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^n\right ) \operatorname {Subst}\left (\int \frac {\left (1+i x^2\right )^{-1+n}}{-a^2+i a^2 x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d (1+2 n) (3+2 n)}\\ &=\frac {2 B (a+i a \tan (c+d x))^n}{d (3+2 n) \cot ^{\frac {3}{2}}(c+d x)}-\frac {2 (2 i B n-A (3+2 n)) (a+i a \tan (c+d x))^n}{d (1+2 n) (3+2 n) \sqrt {\cot (c+d x)}}-\frac {2 (A-i B) F_1\left (\frac {1}{2};1-n,1;\frac {3}{2};-i \tan (c+d x),i \tan (c+d x)\right ) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n}{d \sqrt {\cot (c+d x)}}+\frac {2 \left (2 A n (3+2 n)-i B \left (3+6 n+4 n^2\right )\right ) \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};-i \tan (c+d x)\right ) (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n}{d (1+2 n) (3+2 n) \sqrt {\cot (c+d x)}}\\ \end {align*}
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Mathematica [F] time = 18.21, size = 0, normalized size = 0.00 \[ \int \frac {(a+i a \tan (c+d x))^n (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left ({\left (A - i \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} - {\left (A - 3 i \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - {\left (A + 3 i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + A + i \, B\right )} \left (\frac {2 \, a e^{\left (2 i \, d x + 2 i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{n} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}}{e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1}, x\right ) \]
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 \[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n}}{\cot \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.79, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +i a \tan \left (d x +c \right )\right )^{n} \left (A +B \tan \left (d x +c \right )\right )}{\cot \left (d x +c \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n}}{\cot \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^n}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{n} \left (A + B \tan {\left (c + d x \right )}\right )}{\cot ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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