Optimal. Leaf size=261 \[ \frac{(a+b \tan (e+f x))^{m+1} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} F_1\left (m+1;\frac{1}{2},1;m+2;-\frac{d (a+b \tan (e+f x))}{b c-a d},\frac{a+b \tan (e+f x)}{a-i b}\right )}{2 f (m+1) (b+i a) \sqrt{c+d \tan (e+f x)}}-\frac{(a+b \tan (e+f x))^{m+1} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} F_1\left (m+1;\frac{1}{2},1;m+2;-\frac{d (a+b \tan (e+f x))}{b c-a d},\frac{a+b \tan (e+f x)}{a+i b}\right )}{2 f (m+1) (-b+i a) \sqrt{c+d \tan (e+f x)}} \]
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Rubi [A] time = 0.277998, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {3575, 912, 137, 136} \[ \frac{(a+b \tan (e+f x))^{m+1} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} F_1\left (m+1;\frac{1}{2},1;m+2;-\frac{d (a+b \tan (e+f x))}{b c-a d},\frac{a+b \tan (e+f x)}{a-i b}\right )}{2 f (m+1) (b+i a) \sqrt{c+d \tan (e+f x)}}-\frac{(a+b \tan (e+f x))^{m+1} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} F_1\left (m+1;\frac{1}{2},1;m+2;-\frac{d (a+b \tan (e+f x))}{b c-a d},\frac{a+b \tan (e+f x)}{a+i b}\right )}{2 f (m+1) (-b+i a) \sqrt{c+d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3575
Rule 912
Rule 137
Rule 136
Rubi steps
\begin{align*} \int \frac{(a+b \tan (e+f x))^m}{\sqrt{c+d \tan (e+f x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^m}{\sqrt{c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{i (a+b x)^m}{2 (i-x) \sqrt{c+d x}}+\frac{i (a+b x)^m}{2 (i+x) \sqrt{c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{i \operatorname{Subst}\left (\int \frac{(a+b x)^m}{(i-x) \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 f}+\frac{i \operatorname{Subst}\left (\int \frac{(a+b x)^m}{(i+x) \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 f}\\ &=\frac{\left (i \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^m}{(i-x) \sqrt{\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}}} \, dx,x,\tan (e+f x)\right )}{2 f \sqrt{c+d \tan (e+f x)}}+\frac{\left (i \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^m}{(i+x) \sqrt{\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}}} \, dx,x,\tan (e+f x)\right )}{2 f \sqrt{c+d \tan (e+f x)}}\\ &=\frac{F_1\left (1+m;\frac{1}{2},1;2+m;-\frac{d (a+b \tan (e+f x))}{b c-a d},\frac{a+b \tan (e+f x)}{a-i b}\right ) (a+b \tan (e+f x))^{1+m} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}}}{2 (i a+b) f (1+m) \sqrt{c+d \tan (e+f x)}}-\frac{F_1\left (1+m;\frac{1}{2},1;2+m;-\frac{d (a+b \tan (e+f x))}{b c-a d},\frac{a+b \tan (e+f x)}{a+i b}\right ) (a+b \tan (e+f x))^{1+m} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}}}{2 (i a-b) f (1+m) \sqrt{c+d \tan (e+f x)}}\\ \end{align*}
Mathematica [F] time = 4.3566, size = 0, normalized size = 0. \[ \int \frac{(a+b \tan (e+f x))^m}{\sqrt{c+d \tan (e+f x)}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.37, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\tan \left ( fx+e \right ) \right ) ^{m}{\frac{1}{\sqrt{c+d\tan \left ( fx+e \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 (f x + e\right ) + a\right )}^{m}}{\sqrt{d \tan \left (f x + e\right ) + c}}\,{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 (f x + e\right ) + a\right )}^{m}}{\sqrt{d \tan \left (f x + e\right ) + c}}, 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 (e + f x \right )}\right )^{m}}{\sqrt{c + d \tan{\left (e + f 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 (f x + e\right ) + a\right )}^{m}}{\sqrt{d \tan \left (f x + e\right ) + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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