Optimal. Leaf size=196 \[ \frac {a \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac {1}{2} (-n-1)} (d \cos (e+f x))^n F_1\left (\frac {1}{2};\frac {1}{2} (-n-1),1;\frac {3}{2};\sin ^2(e+f x),\frac {a^2 \sin ^2(e+f x)}{a^2-b^2}\right )}{f \left (a^2-b^2\right )}-\frac {b \sin (e+f x) \cos ^2(e+f x)^{-n/2} (d \cos (e+f x))^n F_1\left (\frac {1}{2};-\frac {n}{2},1;\frac {3}{2};\sin ^2(e+f x),\frac {a^2 \sin ^2(e+f x)}{a^2-b^2}\right )}{f \left (a^2-b^2\right )} \]
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Rubi [A] time = 0.37, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4264, 3869, 2823, 3189, 429} \[ \frac {a \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac {1}{2} (-n-1)} (d \cos (e+f x))^n F_1\left (\frac {1}{2};\frac {1}{2} (-n-1),1;\frac {3}{2};\sin ^2(e+f x),\frac {a^2 \sin ^2(e+f x)}{a^2-b^2}\right )}{f \left (a^2-b^2\right )}-\frac {b \sin (e+f x) \cos ^2(e+f x)^{-n/2} (d \cos (e+f x))^n F_1\left (\frac {1}{2};-\frac {n}{2},1;\frac {3}{2};\sin ^2(e+f x),\frac {a^2 \sin ^2(e+f x)}{a^2-b^2}\right )}{f \left (a^2-b^2\right )} \]
Antiderivative was successfully verified.
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Rule 429
Rule 2823
Rule 3189
Rule 3869
Rule 4264
Rubi steps
\begin {align*} \int \frac {(d \cos (e+f x))^n}{a+b \sec (e+f x)} \, dx &=\left ((d \cos (e+f x))^n (d \sec (e+f x))^n\right ) \int \frac {(d \sec (e+f x))^{-n}}{a+b \sec (e+f x)} \, dx\\ &=\left (\cos ^{-n}(e+f x) (d \cos (e+f x))^n\right ) \int \frac {\cos ^{1+n}(e+f x)}{b+a \cos (e+f x)} \, dx\\ &=-\left (\left (a \cos ^{-n}(e+f x) (d \cos (e+f x))^n\right ) \int \frac {\cos ^{2+n}(e+f x)}{b^2-a^2 \cos ^2(e+f x)} \, dx\right )+\left (b \cos ^{-n}(e+f x) (d \cos (e+f x))^n\right ) \int \frac {\cos ^{1+n}(e+f x)}{b^2-a^2 \cos ^2(e+f x)} \, dx\\ &=-\frac {\left (a \cos ^{2 \left (\frac {1}{2}+\frac {n}{2}\right )-n}(e+f x) (d \cos (e+f x))^n \cos ^2(e+f x)^{-\frac {1}{2}-\frac {n}{2}}\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^{\frac {1+n}{2}}}{-a^2+b^2+a^2 x^2} \, dx,x,\sin (e+f x)\right )}{f}+\frac {\left (b (d \cos (e+f x))^n \cos ^2(e+f x)^{-n/2}\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^{n/2}}{-a^2+b^2+a^2 x^2} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {a F_1\left (\frac {1}{2};\frac {1}{2} (-1-n),1;\frac {3}{2};\sin ^2(e+f x),\frac {a^2 \sin ^2(e+f x)}{a^2-b^2}\right ) \cos (e+f x) (d \cos (e+f x))^n \cos ^2(e+f x)^{\frac {1}{2} (-1-n)} \sin (e+f x)}{\left (a^2-b^2\right ) f}-\frac {b F_1\left (\frac {1}{2};-\frac {n}{2},1;\frac {3}{2};\sin ^2(e+f x),\frac {a^2 \sin ^2(e+f x)}{a^2-b^2}\right ) (d \cos (e+f x))^n \cos ^2(e+f x)^{-n/2} \sin (e+f x)}{\left (a^2-b^2\right ) f}\\ \end {align*}
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Mathematica [B] time = 25.73, size = 5216, normalized size = 26.61 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (d \cos \left (f x + e\right )\right )^{n}}{b \sec \left (f x + e\right ) + a}, x\right ) \]
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 \[ \int \frac {\left (d \cos \left (f x + e\right )\right )^{n}}{b \sec \left (f x + e\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.54, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \cos \left (f x +e \right )\right )^{n}}{a +b \sec \left (f x +e \right )}\, dx \]
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 \[ \int \frac {\left (d \cos \left (f x + e\right )\right )^{n}}{b \sec \left (f x + e\right ) + a}\,{d x} \]
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 \[ \int \frac {{\left (d\,\cos \left (e+f\,x\right )\right )}^n}{a+\frac {b}{\cos \left (e+f\,x\right )}} \,d x \]
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 \[ \int \frac {\left (d \cos {\left (e + f x \right )}\right )^{n}}{a + b \sec {\left (e + f x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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