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 = 0.17, antiderivative size = 187, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, 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 133
Rule 760
Rule 3512
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*}
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Mathematica [C] time = 6.55, 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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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d \sec \left (f x + e\right )\right )^{m} {\left (b \tan \left (f x + e\right ) + a\right )}^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \sec \left (f x + e\right )\right )^{m} {\left (b \tan \left (f x + e\right ) + a\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.93, size = 0, normalized size = 0.00 \[ \int \left (d \sec \left (f x +e \right )\right )^{m} \left (a +b \tan \left (f x +e \right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \sec \left (f x + e\right )\right )^{m} {\left (b \tan \left (f x + e\right ) + a\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^m\,{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \sec {\left (e + f x \right )}\right )^{m} \left (a + b \tan {\left (e + f x \right )}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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