Optimal. Leaf size=299 \[ -\frac {(A b-a B) F_1\left (\frac {1}{2};\frac {m}{2},1;\frac {3}{2};\sin ^2(e+f x),-\frac {b^2 \sin ^2(e+f x)}{a^2-b^2}\right ) \cos (e+f x) \cos ^2(e+f x)^{m/2} (c \sec (e+f x))^{1+m} \sin (e+f x)}{\left (a^2-b^2\right ) c f}+\frac {a (A b-a B) F_1\left (\frac {1}{2};\frac {1+m}{2},1;\frac {3}{2};\sin ^2(e+f x),-\frac {b^2 \sin ^2(e+f x)}{a^2-b^2}\right ) \cos ^2(e+f x)^{\frac {1+m}{2}} (c \sec (e+f x))^{1+m} \sin (e+f x)}{b \left (a^2-b^2\right ) c f}-\frac {B c \, _2F_1\left (\frac {1}{2},\frac {1-m}{2};\frac {3-m}{2};\cos ^2(e+f x)\right ) (c \sec (e+f x))^{-1+m} \sin (e+f x)}{b f (1-m) \sqrt {\sin ^2(e+f x)}} \]
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Rubi [A]
time = 0.37, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3039, 4123,
3857, 2722, 3954, 2902, 3268, 440} \begin {gather*} -\frac {(A b-a B) \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{m/2} (c \sec (e+f x))^{m+1} F_1\left (\frac {1}{2};\frac {m}{2},1;\frac {3}{2};\sin ^2(e+f x),-\frac {b^2 \sin ^2(e+f x)}{a^2-b^2}\right )}{c f \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (e+f x) \cos ^2(e+f x)^{\frac {m+1}{2}} (c \sec (e+f x))^{m+1} F_1\left (\frac {1}{2};\frac {m+1}{2},1;\frac {3}{2};\sin ^2(e+f x),-\frac {b^2 \sin ^2(e+f x)}{a^2-b^2}\right )}{b c f \left (a^2-b^2\right )}-\frac {B c \sin (e+f x) (c \sec (e+f x))^{m-1} \, _2F_1\left (\frac {1}{2},\frac {1-m}{2};\frac {3-m}{2};\cos ^2(e+f x)\right )}{b f (1-m) \sqrt {\sin ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 440
Rule 2722
Rule 2902
Rule 3039
Rule 3268
Rule 3857
Rule 3954
Rule 4123
Rubi steps
\begin {align*} \int \frac {(A+B \cos (e+f x)) (c \sec (e+f x))^m}{a+b \cos (e+f x)} \, dx &=\int \frac {(c \sec (e+f x))^m (B+A \sec (e+f x))}{b+a \sec (e+f x)} \, dx\\ &=\frac {B \int (c \sec (e+f x))^m \, dx}{b}+\frac {(A b-a B) \int \frac {(c \sec (e+f x))^{1+m}}{b+a \sec (e+f x)} \, dx}{b c}\\ &=\frac {\left (B \left (\frac {\cos (e+f x)}{c}\right )^m (c \sec (e+f x))^m\right ) \int \left (\frac {\cos (e+f x)}{c}\right )^{-m} \, dx}{b}+\frac {\left ((A b-a B) \cos ^{1+m}(e+f x) (c \sec (e+f x))^{1+m}\right ) \int \frac {\cos ^{-m}(e+f x)}{a+b \cos (e+f x)} \, dx}{b c}\\ &=-\frac {B \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1-m}{2};\frac {3-m}{2};\cos ^2(e+f x)\right ) (c \sec (e+f x))^m \sin (e+f x)}{b f (1-m) \sqrt {\sin ^2(e+f x)}}-\frac {\left ((A b-a B) \cos ^{1+m}(e+f x) (c \sec (e+f x))^{1+m}\right ) \int \frac {\cos ^{1-m}(e+f x)}{a^2-b^2 \cos ^2(e+f x)} \, dx}{c}+\frac {\left (a (A b-a B) \cos ^{1+m}(e+f x) (c \sec (e+f x))^{1+m}\right ) \int \frac {\cos ^{-m}(e+f x)}{a^2-b^2 \cos ^2(e+f x)} \, dx}{b c}\\ &=-\frac {B \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1-m}{2};\frac {3-m}{2};\cos ^2(e+f x)\right ) (c \sec (e+f x))^m \sin (e+f x)}{b f (1-m) \sqrt {\sin ^2(e+f x)}}+\frac {\left (a (A b-a B) \cos ^{1+2 \left (-\frac {1}{2}-\frac {m}{2}\right )+m}(e+f x) \cos ^2(e+f x)^{\frac {1}{2}+\frac {m}{2}} (c \sec (e+f x))^{1+m}\right ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^{\frac {1}{2} (-1-m)}}{a^2-b^2+b^2 x^2} \, dx,x,\sin (e+f x)\right )}{b c f}-\frac {\left ((A b-a B) \cos (e+f x) \cos ^2(e+f x)^{m/2} (c \sec (e+f x))^{1+m}\right ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^{-m/2}}{a^2-b^2+b^2 x^2} \, dx,x,\sin (e+f x)\right )}{c f}\\ &=-\frac {(A b-a B) F_1\left (\frac {1}{2};\frac {m}{2},1;\frac {3}{2};\sin ^2(e+f x),-\frac {b^2 \sin ^2(e+f x)}{a^2-b^2}\right ) \cos (e+f x) \cos ^2(e+f x)^{m/2} (c \sec (e+f x))^{1+m} \sin (e+f x)}{\left (a^2-b^2\right ) c f}+\frac {a (A b-a B) F_1\left (\frac {1}{2};\frac {1+m}{2},1;\frac {3}{2};\sin ^2(e+f x),-\frac {b^2 \sin ^2(e+f x)}{a^2-b^2}\right ) \cos ^2(e+f x)^{\frac {1+m}{2}} (c \sec (e+f x))^{1+m} \sin (e+f x)}{b \left (a^2-b^2\right ) c f}-\frac {B \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1-m}{2};\frac {3-m}{2};\cos ^2(e+f x)\right ) (c \sec (e+f x))^m \sin (e+f x)}{b f (1-m) \sqrt {\sin ^2(e+f x)}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(10630\) vs. \(2(299)=598\).
time = 26.34, size = 10630, normalized size = 35.55 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.40, size = 0, normalized size = 0.00 \[\int \frac {\left (A +B \cos \left (f x +e \right )\right ) \left (c \sec \left (f x +e \right )\right )^{m}}{a +b \cos \left (f x +e \right )}\, dx\]
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 \frac {\left (c \sec {\left (e + f x \right )}\right )^{m} \left (A + B \cos {\left (e + f x \right )}\right )}{a + b \cos {\left (e + f x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 = -0.00 \begin {gather*} \int \frac {{\left (\frac {c}{\cos \left (e+f\,x\right )}\right )}^m\,\left (A+B\,\cos \left (e+f\,x\right )\right )}{a+b\,\cos \left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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