Optimal. Leaf size=181 \[ \frac{b (d \sec (e+f x))^m \left (\frac{a+b \tan (e+f x)}{\sqrt{-b^2}-a}+1\right )^{-m/2} \left (1-\frac{a+b \tan (e+f x)}{a+\sqrt{-b^2}}\right )^{-m/2} (a+b \tan (e+f x))^{n+1} F_1\left (n+1;1-\frac{m}{2},1-\frac{m}{2};n+2;\frac{a+b \tan (e+f x)}{a+\sqrt{-b^2}},\frac{a+b \tan (e+f x)}{a-\sqrt{-b^2}}\right )}{f (n+1) \left (a^2+b^2\right )} \]
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Rubi [A] time = 0.167858, antiderivative size = 187, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3512, 760, 133} \[ \frac{\cos ^2(e+f x) (d \sec (e+f x))^m \left (1-\frac{a+b \tan (e+f x)}{a-\sqrt{-b^2}}\right )^{1-\frac{m}{2}} \left (1-\frac{a+b \tan (e+f x)}{a+\sqrt{-b^2}}\right )^{1-\frac{m}{2}} (a+b \tan (e+f x))^{n+1} F_1\left (n+1;1-\frac{m}{2},1-\frac{m}{2};n+2;\frac{a+b \tan (e+f x)}{a-\sqrt{-b^2}},\frac{a+b \tan (e+f x)}{a+\sqrt{-b^2}}\right )}{b f (n+1)} \]
Warning: Unable to verify antiderivative.
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Rule 3512
Rule 760
Rule 133
Rubi steps
\begin{align*} \int (d \sec (e+f x))^m (a+b \tan (e+f x))^n \, dx &=\frac{\left ((d \sec (e+f x))^m \sec ^2(e+f x)^{-m/2}\right ) \operatorname{Subst}\left (\int (a+x)^n \left (1+\frac{x^2}{b^2}\right )^{-1+\frac{m}{2}} \, dx,x,b \tan (e+f x)\right )}{b f}\\ &=\frac{\left (\cos ^2(e+f x) (d \sec (e+f x))^m \left (1-\frac{a+b \tan (e+f x)}{a-\frac{b^2}{\sqrt{-b^2}}}\right )^{1-\frac{m}{2}} \left (1-\frac{a+b \tan (e+f x)}{a+\frac{b^2}{\sqrt{-b^2}}}\right )^{1-\frac{m}{2}}\right ) \operatorname{Subst}\left (\int x^n \left (1-\frac{x}{a-\sqrt{-b^2}}\right )^{-1+\frac{m}{2}} \left (1-\frac{x}{a+\sqrt{-b^2}}\right )^{-1+\frac{m}{2}} \, dx,x,a+b \tan (e+f x)\right )}{b f}\\ &=\frac{F_1\left (1+n;1-\frac{m}{2},1-\frac{m}{2};2+n;\frac{a+b \tan (e+f x)}{a-\sqrt{-b^2}},\frac{a+b \tan (e+f x)}{a+\sqrt{-b^2}}\right ) \cos ^2(e+f x) (d \sec (e+f x))^m (a+b \tan (e+f x))^{1+n} \left (1-\frac{a+b \tan (e+f x)}{a-\sqrt{-b^2}}\right )^{1-\frac{m}{2}} \left (1-\frac{a+b \tan (e+f x)}{a+\sqrt{-b^2}}\right )^{1-\frac{m}{2}}}{b f (1+n)}\\ \end{align*}
Mathematica [C] time = 6.2025, size = 699, normalized size = 3.86 \[ \frac{2 (d \sec (e+f x))^m (a+b \tan (e+f x))^{n+1} F_1\left (n+1;1-\frac{m}{2},1-\frac{m}{2};n+2;\frac{a+b \tan (e+f x)}{a-i b},\frac{a+b \tan (e+f x)}{a+i b}\right )}{f \left (2 n (b-a \tan (e+f x)) F_1\left (n+1;1-\frac{m}{2},1-\frac{m}{2};n+2;\frac{a+b \tan (e+f x)}{a-i b},\frac{a+b \tan (e+f x)}{a+i b}\right )+2 (m+n) \tan (e+f x) (a+b \tan (e+f x)) F_1\left (n+1;1-\frac{m}{2},1-\frac{m}{2};n+2;\frac{a+b \tan (e+f x)}{a-i b},\frac{a+b \tan (e+f x)}{a+i b}\right )+2 b \sec ^2(e+f x) F_1\left (n+1;1-\frac{m}{2},1-\frac{m}{2};n+2;\frac{a+b \tan (e+f x)}{a-i b},\frac{a+b \tan (e+f x)}{a+i b}\right )-\frac{b (m-2) (n+1) \sec ^2(e+f x) (a+b \tan (e+f x)) \left ((a-i b) F_1\left (n+2;1-\frac{m}{2},2-\frac{m}{2};n+3;\frac{a+b \tan (e+f x)}{a-i b},\frac{a+b \tan (e+f x)}{a+i b}\right )+(a+i b) F_1\left (n+2;2-\frac{m}{2},1-\frac{m}{2};n+3;\frac{a+b \tan (e+f x)}{a-i b},\frac{a+b \tan (e+f x)}{a+i b}\right )\right )}{(n+2) (a-i b) (a+i b)}-\frac{m \sec ^2(e+f x) (a+b \tan (e+f x)) F_1\left (n+1;1-\frac{m}{2},1-\frac{m}{2};n+2;\frac{a+b \tan (e+f x)}{a-i b},\frac{a+b \tan (e+f x)}{a+i b}\right )}{\tan (e+f x)-i}-\frac{m \sec ^2(e+f x) (a+b \tan (e+f x)) F_1\left (n+1;1-\frac{m}{2},1-\frac{m}{2};n+2;\frac{a+b \tan (e+f x)}{a-i b},\frac{a+b \tan (e+f x)}{a+i b}\right )}{\tan (e+f x)+i}\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.284, size = 0, normalized size = 0. \begin{align*} \int \left ( d\sec \left ( fx+e \right ) \right ) ^{m} \left ( a+b\tan \left ( fx+e \right ) \right ) ^{n}\, 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 (d \sec \left (f x + e\right )\right )^{m}{\left (b \tan \left (f x + e\right ) + a\right )}^{n}\,{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 (\left (d \sec \left (f x + e\right )\right )^{m}{\left (b \tan \left (f x + e\right ) + a\right )}^{n}, 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 (d \sec \left (f x + e\right )\right )^{m}{\left (b \tan \left (f x + e\right ) + a\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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