Optimal. Leaf size=285 \[ \frac{(2 m+1) (i A (5-4 m)+4 B m+B) \sqrt{a+i a \tan (c+d x)} \tan ^m(c+d x) (-i \tan (c+d x))^{-m} \text{Hypergeometric2F1}\left (\frac{1}{2},-m,\frac{3}{2},1+i \tan (c+d x)\right )}{6 a^2 d}+\frac{(A-i B) \sqrt{1+i \tan (c+d x)} \tan ^{m+1}(c+d x) F_1\left (m+1;\frac{1}{2},1;m+2;-i \tan (c+d x),i \tan (c+d x)\right )}{4 a d (m+1) \sqrt{a+i a \tan (c+d x)}}+\frac{(A (5-4 m)-i (4 B m+B)) \tan ^{m+1}(c+d x)}{6 a d \sqrt{a+i a \tan (c+d x)}}+\frac{(A+i B) \tan ^{m+1}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}} \]
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Rubi [A] time = 0.97602, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3596, 3601, 3564, 135, 133, 3599, 67, 65} \[ \frac{(2 m+1) (i A (5-4 m)+4 B m+B) \sqrt{a+i a \tan (c+d x)} \tan ^m(c+d x) (-i \tan (c+d x))^{-m} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};i \tan (c+d x)+1\right )}{6 a^2 d}+\frac{(A-i B) \sqrt{1+i \tan (c+d x)} \tan ^{m+1}(c+d x) F_1\left (m+1;\frac{1}{2},1;m+2;-i \tan (c+d x),i \tan (c+d x)\right )}{4 a d (m+1) \sqrt{a+i a \tan (c+d x)}}+\frac{(A (5-4 m)-i (4 B m+B)) \tan ^{m+1}(c+d x)}{6 a d \sqrt{a+i a \tan (c+d x)}}+\frac{(A+i B) \tan ^{m+1}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3596
Rule 3601
Rule 3564
Rule 135
Rule 133
Rule 3599
Rule 67
Rule 65
Rubi steps
\begin{align*} \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{\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) \operatorname{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))) \operatorname{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 ) \operatorname{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 ) \operatorname{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) F_1\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) \, _2F_1\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*}
Mathematica [F] time = 15.6552, size = 0, normalized size = 0. \[ \int \frac{\tan ^m(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.482, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \tan \left ( dx+c \right ) \right ) ^{m} \left ( A+B\tan \left ( dx+c \right ) \right ) \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{2}{\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} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-3 i \, d x - 3 i \, c\right )}}{4 \, a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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