Optimal. Leaf size=226 \[ \frac {(1-m) (A (1-m)-i B (m+1)) \tan ^{m+1}(c+d x) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\tan ^2(c+d x)\right )}{4 a^2 d (m+1)}+\frac {m (B m+i A (2-m)) \tan ^{m+2}(c+d x) \, _2F_1\left (1,\frac {m+2}{2};\frac {m+4}{2};-\tan ^2(c+d x)\right )}{4 a^2 d (m+2)}+\frac {(A (2-m)-i B m) \tan ^{m+1}(c+d x)}{4 a^2 d (1+i \tan (c+d x))}+\frac {(A+i B) \tan ^{m+1}(c+d x)}{4 d (a+i a \tan (c+d x))^2} \]
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Rubi [A] time = 0.48, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3596, 3538, 3476, 364} \[ \frac {(1-m) (A (1-m)-i B (m+1)) \tan ^{m+1}(c+d x) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\tan ^2(c+d x)\right )}{4 a^2 d (m+1)}+\frac {m (B m+i A (2-m)) \tan ^{m+2}(c+d x) \, _2F_1\left (1,\frac {m+2}{2};\frac {m+4}{2};-\tan ^2(c+d x)\right )}{4 a^2 d (m+2)}+\frac {(A (2-m)-i B m) \tan ^{m+1}(c+d x)}{4 a^2 d (1+i \tan (c+d x))}+\frac {(A+i B) \tan ^{m+1}(c+d x)}{4 d (a+i a \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 364
Rule 3476
Rule 3538
Rule 3596
Rubi steps
\begin {align*} \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx &=\frac {(A+i B) \tan ^{1+m}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {\int \frac {\tan ^m(c+d x) (a (A (3-m)-i B (1+m))-a (i A-B) (1-m) \tan (c+d x))}{a+i a \tan (c+d x)} \, dx}{4 a^2}\\ &=\frac {(A (2-m)-i B m) \tan ^{1+m}(c+d x)}{4 a^2 d (1+i \tan (c+d x))}+\frac {(A+i B) \tan ^{1+m}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {\int \tan ^m(c+d x) \left (2 a^2 (1-m) (A (1-m)-i B (1+m))+2 a^2 m (i A (2-m)+B m) \tan (c+d x)\right ) \, dx}{8 a^4}\\ &=\frac {(A (2-m)-i B m) \tan ^{1+m}(c+d x)}{4 a^2 d (1+i \tan (c+d x))}+\frac {(A+i B) \tan ^{1+m}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {(m (i A (2-m)+B m)) \int \tan ^{1+m}(c+d x) \, dx}{4 a^2}+\frac {((1-m) (A (1-m)-i B (1+m))) \int \tan ^m(c+d x) \, dx}{4 a^2}\\ &=\frac {(A (2-m)-i B m) \tan ^{1+m}(c+d x)}{4 a^2 d (1+i \tan (c+d x))}+\frac {(A+i B) \tan ^{1+m}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {(m (i A (2-m)+B m)) \operatorname {Subst}\left (\int \frac {x^{1+m}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{4 a^2 d}+\frac {((1-m) (A (1-m)-i B (1+m))) \operatorname {Subst}\left (\int \frac {x^m}{1+x^2} \, dx,x,\tan (c+d x)\right )}{4 a^2 d}\\ &=\frac {(1-m) (A (1-m)-i B (1+m)) \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\tan ^2(c+d x)\right ) \tan ^{1+m}(c+d x)}{4 a^2 d (1+m)}+\frac {(A (2-m)-i B m) \tan ^{1+m}(c+d x)}{4 a^2 d (1+i \tan (c+d x))}+\frac {m (i A (2-m)+B m) \, _2F_1\left (1,\frac {2+m}{2};\frac {4+m}{2};-\tan ^2(c+d x)\right ) \tan ^{2+m}(c+d x)}{4 a^2 d (2+m)}+\frac {(A+i B) \tan ^{1+m}(c+d x)}{4 d (a+i a \tan (c+d x))^2}\\ \end {align*}
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Mathematica [B] time = 9.19, size = 565, normalized size = 2.50 \[ -\frac {i e^{-2 i c} \left (-\frac {i \left (-1+e^{2 i (c+d x)}\right )}{1+e^{2 i (c+d x)}}\right )^m \left (\frac {-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{-m} \sec (c+d x) (\cos (d x)+i \sin (d x))^2 (A+B \tan (c+d x)) \left (\frac {e^{4 i c} 2^{1-m} \left (A \left (2 m^2-4 m+1\right )+i B \left (2 m^2-1\right )\right ) \left (\frac {-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^m \left (\left (1+e^{2 i (c+d x)}\right )^m \left ((m+1) \, _2F_1\left (m,m;m+1;\frac {1}{2} \left (1-e^{2 i (c+d x)}\right )\right )+m \left (-1+e^{2 i (c+d x)}\right ) \, _2F_1\left (m,m+1;m+2;\frac {1}{2} \left (1-e^{2 i (c+d x)}\right )\right )\right )-2^m (m+1) \, _2F_1\left (1,m;m+1;\frac {1-e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right )\right )}{m (m+1)}+\frac {e^{4 i c} 2^{2-m} (A (2 m-3)+i (2 B m+B)) \left (-1+e^{2 i (c+d x)}\right )^{m+1} \, _2F_1\left (m-1,m+1;m+2;\frac {1}{2} \left (1-e^{2 i (c+d x)}\right )\right )}{m+1}+(A+i B) e^{-4 i d x} \left (-1+e^{2 i (c+d x)}\right )^{m+1} \left (1+e^{2 i (c+d x)}\right )^{2-m}+(A (3-2 m)-i (2 B m+B)) e^{2 i (c-d x)} \left (-1+e^{2 i (c+d x)}\right )^{m+1} \left (1+e^{2 i (c+d x)}\right )^{2-m}\right )}{16 d (a+i a \tan (c+d x))^2 (A \cos (c+d x)+B \sin (c+d x))} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left ({\left (A - i \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, A e^{\left (2 i \, d x + 2 i \, c\right )} + A + i \, B\right )} \left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{m} e^{\left (-4 i \, d x - 4 i \, c\right )}}{4 \, a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \tan \left (d x + c\right )^{m}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.65, size = 0, normalized size = 0.00 \[ \int \frac {\left (\tan ^{m}\left (d x +c \right )\right ) \left (A +B \tan \left (d x +c \right )\right )}{\left (a +i a \tan \left (d x +c \right )\right )^{2}}\, dx \]
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 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {{\mathrm {tan}\left (c+d\,x\right )}^m\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]
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 \[ - \frac {\int \frac {A \tan ^{m}{\left (c + d x \right )}}{\tan ^{2}{\left (c + d x \right )} - 2 i \tan {\left (c + d x \right )} - 1}\, dx + \int \frac {B \tan {\left (c + d x \right )} \tan ^{m}{\left (c + d x \right )}}{\tan ^{2}{\left (c + d x \right )} - 2 i \tan {\left (c + d x \right )} - 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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