Optimal. Leaf size=187 \[ \frac {F_1\left (1+n;\frac {2+m}{2},\frac {2+m}{2};2+n;\frac {a+b \tan (e+f x)}{a-\sqrt {-b^2}},\frac {a+b \tan (e+f x)}{a+\sqrt {-b^2}}\right ) \cos ^2(e+f x) (d \cos (e+f x))^m (a+b \tan (e+f x))^{1+n} \left (1-\frac {a+b \tan (e+f x)}{a-\sqrt {-b^2}}\right )^{\frac {2+m}{2}} \left (1-\frac {a+b \tan (e+f x)}{a+\sqrt {-b^2}}\right )^{\frac {2+m}{2}}}{b f (1+n)} \]
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Rubi [A]
time = 0.17, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3596, 3593,
774, 138} \begin {gather*} \frac {\cos ^2(e+f x) (d \cos (e+f x))^m \left (1-\frac {a+b \tan (e+f x)}{a-\sqrt {-b^2}}\right )^{\frac {m+2}{2}} \left (1-\frac {a+b \tan (e+f x)}{a+\sqrt {-b^2}}\right )^{\frac {m+2}{2}} (a+b \tan (e+f x))^{n+1} F_1\left (n+1;\frac {m+2}{2},\frac {m+2}{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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 774
Rule 3593
Rule 3596
Rubi steps
\begin {align*} \int (d \cos (e+f x))^m (a+b \tan (e+f x))^n \, dx &=\left ((d \cos (e+f x))^m (d \sec (e+f x))^m\right ) \int (d \sec (e+f x))^{-m} (a+b \tan (e+f x))^n \, dx\\ &=\frac {\left ((d \cos (e+f x))^m \sec ^2(e+f x)^{m/2}\right ) \text {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 \cos (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 ) \text {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;\frac {2+m}{2},\frac {2+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 \cos (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 )^{\frac {2+m}{2}} \left (1-\frac {a+b \tan (e+f x)}{a+\sqrt {-b^2}}\right )^{\frac {2+m}{2}}}{b f (1+n)}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 22.25, size = 365, normalized size = 1.95 \begin {gather*} \frac {2 (a-i b) (a+i b) (2+n) F_1\left (1+n;1+\frac {m}{2},1+\frac {m}{2};2+n;\frac {a+b \tan (e+f x)}{a-i b},\frac {a+b \tan (e+f x)}{a+i b}\right ) \cos (e+f x) (d \cos (e+f x))^m (a \cos (e+f x)+b \sin (e+f x)) (a+b \tan (e+f x))^n}{b f (1+n) \left (2 \left (a^2+b^2\right ) (2+n) F_1\left (1+n;1+\frac {m}{2},1+\frac {m}{2};2+n;\frac {a+b \tan (e+f x)}{a-i b},\frac {a+b \tan (e+f x)}{a+i b}\right )+(2+m) \left ((a-i b) F_1\left (2+n;1+\frac {m}{2},2+\frac {m}{2};3+n;\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 (2+n;2+\frac {m}{2},1+\frac {m}{2};3+n;\frac {a+b \tan (e+f x)}{a-i b},\frac {a+b \tan (e+f x)}{a+i b}\right )\right ) (a+b \tan (e+f x))\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.36, size = 0, normalized size = 0.00 \[\int \left (d \cos \left (f x +e \right )\right )^{m} \left (a +b \tan \left (f x +e \right )\right )^{n}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (d \cos {\left (e + f x \right )}\right )^{m} \left (a + b \tan {\left (e + f x \right )}\right )^{n}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (d\,\cos \left (e+f\,x\right )\right )}^m\,{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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