Integrand size = 36, antiderivative size = 285 \[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {(A+i B) \tan ^{1+m}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(A (5-4 m)-i (B+4 B m)) \tan ^{1+m}(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {(A-i B) \operatorname {AppellF1}\left (1+m,\frac {1}{2},1,2+m,-i \tan (c+d x),i \tan (c+d x)\right ) \sqrt {1+i \tan (c+d x)} \tan ^{1+m}(c+d x)}{4 a d (1+m) \sqrt {a+i a \tan (c+d x)}}+\frac {(1+2 m) (B+i A (5-4 m)+4 B m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},1+i \tan (c+d x)\right ) (-i \tan (c+d x))^{-m} \tan ^m(c+d x) \sqrt {a+i a \tan (c+d x)}}{6 a^2 d} \]
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Time = 1.17 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3677, 3682, 3645, 140, 138, 3680, 69, 67} \[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {(2 m+1) (i A (5-4 m)+4 B m+B) \sqrt {a+i a \tan (c+d x)} \tan ^m(c+d x) (-i \tan (c+d x))^{-m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},i \tan (c+d x)+1\right )}{6 a^2 d}+\frac {(A-i B) \sqrt {1+i \tan (c+d x)} \tan ^{m+1}(c+d x) \operatorname {AppellF1}\left (m+1,\frac {1}{2},1,m+2,-i \tan (c+d x),i \tan (c+d x)\right )}{4 a d (m+1) \sqrt {a+i a \tan (c+d x)}}+\frac {(A (5-4 m)-i (4 B m+B)) \tan ^{m+1}(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {(A+i B) \tan ^{m+1}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}} \]
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Rule 67
Rule 69
Rule 138
Rule 140
Rule 3645
Rule 3677
Rule 3680
Rule 3682
Rubi steps \begin{align*} \text {integral}& = \frac {(A+i B) \tan ^{1+m}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {\int \frac {\tan ^m(c+d x) \left (a (A (2-m)-i B (1+m))-\frac {1}{2} a (i A-B) (1-2 m) \tan (c+d x)\right )}{\sqrt {a+i a \tan (c+d x)}} \, dx}{3 a^2} \\ & = \frac {(A+i B) \tan ^{1+m}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(A (5-4 m)-i (B+4 B m)) \tan ^{1+m}(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {\int \tan ^m(c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {1}{2} a^2 \left (i B \left (1-3 m-4 m^2\right )+A \left (1+3 m-4 m^2\right )\right )+\frac {1}{4} a^2 (1+2 m) (B+i A (5-4 m)+4 B m) \tan (c+d x)\right ) \, dx}{3 a^4} \\ & = \frac {(A+i B) \tan ^{1+m}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(A (5-4 m)-i (B+4 B m)) \tan ^{1+m}(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {(A-i B) \int \tan ^m(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx}{4 a^2}-\frac {((1+2 m) (A (5-4 m)-i (B+4 B m))) \int \tan ^m(c+d x) (a-i a \tan (c+d x)) \sqrt {a+i a \tan (c+d x)} \, dx}{12 a^3} \\ & = \frac {(A+i B) \tan ^{1+m}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(A (5-4 m)-i (B+4 B m)) \tan ^{1+m}(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {(i A+B) \text {Subst}\left (\int \frac {\left (-\frac {i x}{a}\right )^m}{\sqrt {a+x} \left (-a^2+a x\right )} \, dx,x,i a \tan (c+d x)\right )}{4 d}-\frac {((1+2 m) (A (5-4 m)-i (B+4 B m))) \text {Subst}\left (\int \frac {x^m}{\sqrt {a+i a x}} \, dx,x,\tan (c+d x)\right )}{12 a d} \\ & = \frac {(A+i B) \tan ^{1+m}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(A (5-4 m)-i (B+4 B m)) \tan ^{1+m}(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {\left ((1+2 m) (A (5-4 m)-i (B+4 B m)) (-i \tan (c+d x))^{-m} \tan ^m(c+d x)\right ) \text {Subst}\left (\int \frac {(-i x)^m}{\sqrt {a+i a x}} \, dx,x,\tan (c+d x)\right )}{12 a d}+\frac {\left ((i A+B) \sqrt {1+i \tan (c+d x)}\right ) \text {Subst}\left (\int \frac {\left (-\frac {i x}{a}\right )^m}{\sqrt {1+\frac {x}{a}} \left (-a^2+a x\right )} \, dx,x,i a \tan (c+d x)\right )}{4 d \sqrt {a+i a \tan (c+d x)}} \\ & = \frac {(A+i B) \tan ^{1+m}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(A (5-4 m)-i (B+4 B m)) \tan ^{1+m}(c+d x)}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {(A-i B) \operatorname {AppellF1}\left (1+m,\frac {1}{2},1,2+m,-i \tan (c+d x),i \tan (c+d x)\right ) \sqrt {1+i \tan (c+d x)} \tan ^{1+m}(c+d x)}{4 a d (1+m) \sqrt {a+i a \tan (c+d x)}}+\frac {(1+2 m) (B+i A (5-4 m)+4 B m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},1+i \tan (c+d x)\right ) (-i \tan (c+d x))^{-m} \tan ^m(c+d x) \sqrt {a+i a \tan (c+d x)}}{6 a^2 d} \\ \end{align*}
\[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx \]
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\[\int \frac {\left (\tan ^{m}\left (d x +c \right )\right ) \left (A +B \tan \left (d x +c \right )\right )}{\left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \tan \left (d x + c\right )^{m}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \tan ^{m}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \tan \left (d x + c\right )^{m}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\int \frac {{\mathrm {tan}\left (c+d\,x\right )}^m\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]
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