Optimal. Leaf size=187 \[ \frac{(A+i B) \tan ^{m+1}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left (m+1;\frac{1}{2},1;m+2;-\frac{b \tan (c+d x)}{a},-i \tan (c+d x)\right )}{2 d (m+1) \sqrt{a+b \tan (c+d x)}}+\frac{(A-i B) \tan ^{m+1}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left (m+1;\frac{1}{2},1;m+2;-\frac{b \tan (c+d x)}{a},i \tan (c+d x)\right )}{2 d (m+1) \sqrt{a+b \tan (c+d x)}} \]
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Rubi [A] time = 0.409318, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {3603, 3602, 135, 133} \[ \frac{(A+i B) \tan ^{m+1}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left (m+1;\frac{1}{2},1;m+2;-\frac{b \tan (c+d x)}{a},-i \tan (c+d x)\right )}{2 d (m+1) \sqrt{a+b \tan (c+d x)}}+\frac{(A-i B) \tan ^{m+1}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left (m+1;\frac{1}{2},1;m+2;-\frac{b \tan (c+d x)}{a},i \tan (c+d x)\right )}{2 d (m+1) \sqrt{a+b \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3603
Rule 3602
Rule 135
Rule 133
Rubi steps
\begin{align*} \int \frac{\tan ^m(c+d x) (A+B \tan (c+d x))}{\sqrt{a+b \tan (c+d x)}} \, dx &=\frac{1}{2} (A-i B) \int \frac{(1+i \tan (c+d x)) \tan ^m(c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx+\frac{1}{2} (A+i B) \int \frac{(1-i \tan (c+d x)) \tan ^m(c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx\\ &=\frac{(A-i B) \operatorname{Subst}\left (\int \frac{x^m}{(1-i x) \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac{(A+i B) \operatorname{Subst}\left (\int \frac{x^m}{(1+i x) \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac{\left ((A-i B) \sqrt{1+\frac{b \tan (c+d x)}{a}}\right ) \operatorname{Subst}\left (\int \frac{x^m}{(1-i x) \sqrt{1+\frac{b x}{a}}} \, dx,x,\tan (c+d x)\right )}{2 d \sqrt{a+b \tan (c+d x)}}+\frac{\left ((A+i B) \sqrt{1+\frac{b \tan (c+d x)}{a}}\right ) \operatorname{Subst}\left (\int \frac{x^m}{(1+i x) \sqrt{1+\frac{b x}{a}}} \, dx,x,\tan (c+d x)\right )}{2 d \sqrt{a+b \tan (c+d x)}}\\ &=\frac{(A+i B) F_1\left (1+m;\frac{1}{2},1;2+m;-\frac{b \tan (c+d x)}{a},-i \tan (c+d x)\right ) \tan ^{1+m}(c+d x) \sqrt{1+\frac{b \tan (c+d x)}{a}}}{2 d (1+m) \sqrt{a+b \tan (c+d x)}}+\frac{(A-i B) F_1\left (1+m;\frac{1}{2},1;2+m;-\frac{b \tan (c+d x)}{a},i \tan (c+d x)\right ) \tan ^{1+m}(c+d x) \sqrt{1+\frac{b \tan (c+d x)}{a}}}{2 d (1+m) \sqrt{a+b \tan (c+d x)}}\\ \end{align*}
Mathematica [F] time = 9.20934, size = 0, normalized size = 0. \[ \int \frac{\tan ^m(c+d x) (A+B \tan (c+d x))}{\sqrt{a+b \tan (c+d x)}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.568, 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 ){\frac{1}{\sqrt{a+b\tan \left ( dx+c \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 \frac{{\left (B \tan \left (d x + c\right ) + A\right )} \tan \left (d x + c\right )^{m}}{\sqrt{b \tan \left (d x + c\right ) + a}}\,{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{{\left (B \tan \left (d x + c\right ) + A\right )} \tan \left (d x + c\right )^{m}}{\sqrt{b \tan \left (d x + c\right ) + a}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B \tan{\left (c + d x \right )}\right ) \tan ^{m}{\left (c + d x \right )}}{\sqrt{a + b \tan{\left (c + d x \right )}}}\, dx \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}}{\sqrt{b \tan \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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