Optimal. Leaf size=227 \[ \frac{2 a (B+(2 m+3) (B+i A)) \sqrt{a+i a \tan (c+d x)} \tan ^m(c+d x) (-i \tan (c+d x))^{-m} \text{Hypergeometric2F1}\left (\frac{1}{2},-m,\frac{3}{2},1+i \tan (c+d x)\right )}{d (2 m+3)}+\frac{2 a^2 (A-i B) \sqrt{1+i \tan (c+d x)} \tan ^{m+1}(c+d x) F_1\left (m+1;\frac{1}{2},1;m+2;-i \tan (c+d x),i \tan (c+d x)\right )}{d (m+1) \sqrt{a+i a \tan (c+d x)}}+\frac{2 i a B \sqrt{a+i a \tan (c+d x)} \tan ^{m+1}(c+d x)}{d (2 m+3)} \]
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Rubi [A] time = 0.703497, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3594, 3601, 3564, 135, 133, 3599, 67, 65} \[ \frac{2 a^2 (A-i B) \sqrt{1+i \tan (c+d x)} \tan ^{m+1}(c+d x) F_1\left (m+1;\frac{1}{2},1;m+2;-i \tan (c+d x),i \tan (c+d x)\right )}{d (m+1) \sqrt{a+i a \tan (c+d x)}}+\frac{2 a (B+(2 m+3) (B+i A)) \sqrt{a+i a \tan (c+d x)} \tan ^m(c+d x) (-i \tan (c+d x))^{-m} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};i \tan (c+d x)+1\right )}{d (2 m+3)}+\frac{2 i a B \sqrt{a+i a \tan (c+d x)} \tan ^{m+1}(c+d x)}{d (2 m+3)} \]
Antiderivative was successfully verified.
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Rule 3594
Rule 3601
Rule 3564
Rule 135
Rule 133
Rule 3599
Rule 67
Rule 65
Rubi steps
\begin{align*} \int \tan ^m(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx &=\frac{2 i a B \tan ^{1+m}(c+d x) \sqrt{a+i a \tan (c+d x)}}{d (3+2 m)}+\frac{2 \int \tan ^m(c+d x) \sqrt{a+i a \tan (c+d x)} \left (-\frac{1}{2} a \left (2 i B (1+m)-2 A \left (\frac{3}{2}+m\right )\right )+\frac{1}{2} a (B+(i A+B) (3+2 m)) \tan (c+d x)\right ) \, dx}{3+2 m}\\ &=\frac{2 i a B \tan ^{1+m}(c+d x) \sqrt{a+i a \tan (c+d x)}}{d (3+2 m)}+(2 a (A-i B)) \int \tan ^m(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx+\frac{(i (B+(i A+B) (3+2 m))) \int \tan ^m(c+d x) (a-i a \tan (c+d x)) \sqrt{a+i a \tan (c+d x)} \, dx}{3+2 m}\\ &=\frac{2 i a B \tan ^{1+m}(c+d x) \sqrt{a+i a \tan (c+d x)}}{d (3+2 m)}+\frac{\left (2 a^3 (i A+B)\right ) \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 )}{d}+\frac{\left (i a^2 (B+(i A+B) (3+2 m))\right ) \operatorname{Subst}\left (\int \frac{x^m}{\sqrt{a+i a x}} \, dx,x,\tan (c+d x)\right )}{d (3+2 m)}\\ &=\frac{2 i a B \tan ^{1+m}(c+d x) \sqrt{a+i a \tan (c+d x)}}{d (3+2 m)}+\frac{\left (i a^2 (B+(i A+B) (3+2 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 )}{d (3+2 m)}+\frac{\left (2 a^3 (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 )}{d \sqrt{a+i a \tan (c+d x)}}\\ &=\frac{2 a^2 (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)}{d (1+m) \sqrt{a+i a \tan (c+d x)}}+\frac{2 a (B+(i A+B) (3+2 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)}}{d (3+2 m)}+\frac{2 i a B \tan ^{1+m}(c+d x) \sqrt{a+i a \tan (c+d x)}}{d (3+2 m)}\\ \end{align*}
Mathematica [F] time = 4.70488, size = 0, normalized size = 0. \[ \int \tan ^m(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.488, size = 0, normalized size = 0. \begin{align*} \int \left ( \tan \left ( dx+c \right ) \right ) ^{m} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}} \left ( A+B\tan \left ( dx+c \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \tan \left (d x + c\right ) + A\right )}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \tan \left (d x + c\right )^{m}\,{d x} \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{2 \, \sqrt{2}{\left ({\left (A - i \, B\right )} a e^{\left (4 i \, d x + 4 i \, c\right )} +{\left (A + i \, B\right )} a e^{\left (2 i \, d x + 2 i \, c\right )}\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 (i \, d x + i \, c\right )}}{e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\left (B \tan \left (d x + c\right ) + A\right )}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \tan \left (d x + c\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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