Optimal. Leaf size=144 \[ -\frac {a^2 \cos ^{m+2}(c+d x) \, _2F_1\left (1,m+2;m+3;-\frac {a \cos (c+d x)}{b}\right )}{b d (m+2) \left (a^2-b^2\right )}+\frac {\cos ^{m+2}(c+d x) \, _2F_1(1,m+2;m+3;-\cos (c+d x))}{2 d (m+2) (a-b)}-\frac {\cos ^{m+2}(c+d x) \, _2F_1(1,m+2;m+3;\cos (c+d x))}{2 d (m+2) (a+b)} \]
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Rubi [A] time = 0.43, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4397, 2837, 961, 64} \[ -\frac {a^2 \cos ^{m+2}(c+d x) \, _2F_1\left (1,m+2;m+3;-\frac {a \cos (c+d x)}{b}\right )}{b d (m+2) \left (a^2-b^2\right )}+\frac {\cos ^{m+2}(c+d x) \, _2F_1(1,m+2;m+3;-\cos (c+d x))}{2 d (m+2) (a-b)}-\frac {\cos ^{m+2}(c+d x) \, _2F_1(1,m+2;m+3;\cos (c+d x))}{2 d (m+2) (a+b)} \]
Antiderivative was successfully verified.
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Rule 64
Rule 961
Rule 2837
Rule 4397
Rubi steps
\begin {align*} \int \frac {\cos ^m(c+d x)}{a \sin (c+d x)+b \tan (c+d x)} \, dx &=\int \frac {\cos ^{1+m}(c+d x) \csc (c+d x)}{b+a \cos (c+d x)} \, dx\\ &=-\frac {a \operatorname {Subst}\left (\int \frac {\left (\frac {x}{a}\right )^{1+m}}{(b+x) \left (a^2-x^2\right )} \, dx,x,a \cos (c+d x)\right )}{d}\\ &=-\frac {a \operatorname {Subst}\left (\int \left (\frac {\left (\frac {x}{a}\right )^{1+m}}{2 a (a+b) (a-x)}-\frac {\left (\frac {x}{a}\right )^{1+m}}{2 a (a-b) (a+x)}+\frac {\left (\frac {x}{a}\right )^{1+m}}{(a-b) (a+b) (b+x)}\right ) \, dx,x,a \cos (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (\frac {x}{a}\right )^{1+m}}{a+x} \, dx,x,a \cos (c+d x)\right )}{2 (a-b) d}-\frac {\operatorname {Subst}\left (\int \frac {\left (\frac {x}{a}\right )^{1+m}}{a-x} \, dx,x,a \cos (c+d x)\right )}{2 (a+b) d}-\frac {a \operatorname {Subst}\left (\int \frac {\left (\frac {x}{a}\right )^{1+m}}{b+x} \, dx,x,a \cos (c+d x)\right )}{\left (a^2-b^2\right ) d}\\ &=\frac {\cos ^{2+m}(c+d x) \, _2F_1(1,2+m;3+m;-\cos (c+d x))}{2 (a-b) d (2+m)}-\frac {\cos ^{2+m}(c+d x) \, _2F_1(1,2+m;3+m;\cos (c+d x))}{2 (a+b) d (2+m)}-\frac {a^2 \cos ^{2+m}(c+d x) \, _2F_1\left (1,2+m;3+m;-\frac {a \cos (c+d x)}{b}\right )}{b \left (a^2-b^2\right ) d (2+m)}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 106, normalized size = 0.74 \[ \frac {\cos ^{m+2}(c+d x) \left (-2 a^2 \, _2F_1\left (1,m+2;m+3;-\frac {a \cos (c+d x)}{b}\right )+b (a+b) \, _2F_1(1,m+2;m+3;-\cos (c+d x))-b (a-b) \, _2F_1(1,m+2;m+3;\cos (c+d x))\right )}{2 b d (m+2) (a-b) (a+b)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (d x + c\right )^{m}}{a \sin \left (d x + c\right ) + b \tan \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {\cos \left (d x + c\right )^{m}}{a \sin \left (d x + c\right ) + b \tan \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {\cos ^{m}\left (d x +c \right )}{a \sin \left (d x +c \right )+b \tan \left (d x +c \right )}\, dx \]
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 \[ \int \frac {\cos \left (d x + c\right )^{m}}{a \sin \left (d x + c\right ) + b \tan \left (d x + c\right )}\,{d x} \]
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 \[ \int \frac {\cos \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^m}{\sin \left (c+d\,x\right )\,\left (b+a\,\cos \left (c+d\,x\right )\right )} \,d x \]
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 \[ \int \frac {\cos ^{m}{\left (c + d x \right )}}{a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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