Optimal. Leaf size=372 \[ \frac {a \sin (c+d x) \left (A b^2-a (b B-a C)\right ) \cos ^{m-1}(c+d x) \cos ^2(c+d x)^{\frac {1-m}{2}} F_1\left (\frac {1}{2};\frac {1-m}{2},1;\frac {3}{2};\sin ^2(c+d x),-\frac {b^2 \sin ^2(c+d x)}{a^2-b^2}\right )}{b^2 d \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right ) \cos ^m(c+d x) \cos ^2(c+d x)^{-m/2} F_1\left (\frac {1}{2};-\frac {m}{2},1;\frac {3}{2};\sin ^2(c+d x),-\frac {b^2 \sin ^2(c+d x)}{a^2-b^2}\right )}{b d \left (a^2-b^2\right )}-\frac {(b B-a C) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(c+d x)\right )}{b^2 d (m+1) \sqrt {\sin ^2(c+d x)}}-\frac {C \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\cos ^2(c+d x)\right )}{b d (m+2) \sqrt {\sin ^2(c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.42, antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {3063, 2643, 2823, 3189, 429} \[ \frac {a \sin (c+d x) \left (A b^2-a (b B-a C)\right ) \cos ^{m-1}(c+d x) \cos ^2(c+d x)^{\frac {1-m}{2}} F_1\left (\frac {1}{2};\frac {1-m}{2},1;\frac {3}{2};\sin ^2(c+d x),-\frac {b^2 \sin ^2(c+d x)}{a^2-b^2}\right )}{b^2 d \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right ) \cos ^m(c+d x) \cos ^2(c+d x)^{-m/2} F_1\left (\frac {1}{2};-\frac {m}{2},1;\frac {3}{2};\sin ^2(c+d x),-\frac {b^2 \sin ^2(c+d x)}{a^2-b^2}\right )}{b d \left (a^2-b^2\right )}-\frac {(b B-a C) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(c+d x)\right )}{b^2 d (m+1) \sqrt {\sin ^2(c+d x)}}-\frac {C \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\cos ^2(c+d x)\right )}{b d (m+2) \sqrt {\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 429
Rule 2643
Rule 2823
Rule 3063
Rule 3189
Rubi steps
\begin {align*} \int \frac {\cos ^m(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx &=\frac {C \int \cos ^{1+m}(c+d x) \, dx}{b}+\frac {(b B-a C) \int \cos ^m(c+d x) \, dx}{b^2}+\left (A-\frac {a (b B-a C)}{b^2}\right ) \int \frac {\cos ^m(c+d x)}{a+b \cos (c+d x)} \, dx\\ &=-\frac {(b B-a C) \cos ^{1+m}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{b^2 d (1+m) \sqrt {\sin ^2(c+d x)}}-\frac {C \cos ^{2+m}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{b d (2+m) \sqrt {\sin ^2(c+d x)}}+\left (a \left (A-\frac {a (b B-a C)}{b^2}\right )\right ) \int \frac {\cos ^m(c+d x)}{a^2-b^2 \cos ^2(c+d x)} \, dx+\left (b \left (-A+\frac {a (b B-a C)}{b^2}\right )\right ) \int \frac {\cos ^{1+m}(c+d x)}{a^2-b^2 \cos ^2(c+d x)} \, dx\\ &=-\frac {(b B-a C) \cos ^{1+m}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{b^2 d (1+m) \sqrt {\sin ^2(c+d x)}}-\frac {C \cos ^{2+m}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{b d (2+m) \sqrt {\sin ^2(c+d x)}}+\frac {\left (a \left (A-\frac {a (b B-a C)}{b^2}\right ) \cos ^{2 \left (-\frac {1}{2}+\frac {m}{2}\right )}(c+d x) \cos ^2(c+d x)^{\frac {1}{2}-\frac {m}{2}}\right ) \operatorname {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 (c+d x)\right )}{d}+\frac {\left (b \left (-A+\frac {a (b B-a C)}{b^2}\right ) \cos ^m(c+d x) \cos ^2(c+d x)^{-m/2}\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^{m/2}}{a^2-b^2+b^2 x^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {a \left (A-\frac {a (b B-a C)}{b^2}\right ) F_1\left (\frac {1}{2};\frac {1-m}{2},1;\frac {3}{2};\sin ^2(c+d x),-\frac {b^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \cos ^{-1+m}(c+d x) \cos ^2(c+d x)^{\frac {1-m}{2}} \sin (c+d x)}{\left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) F_1\left (\frac {1}{2};-\frac {m}{2},1;\frac {3}{2};\sin ^2(c+d x),-\frac {b^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \cos ^m(c+d x) \cos ^2(c+d x)^{-m/2} \sin (c+d x)}{b \left (a^2-b^2\right ) d}-\frac {(b B-a C) \cos ^{1+m}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{b^2 d (1+m) \sqrt {\sin ^2(c+d x)}}-\frac {C \cos ^{2+m}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{b d (2+m) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 30.13, size = 15557, normalized size = 41.82 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{m}}{b \cos \left (d x + c\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{m}}{b \cos \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 5.04, size = 0, normalized size = 0.00 \[ \int \frac {\left (\cos ^{m}\left (d x +c \right )\right ) \left (A +B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )\right )}{a +b \cos \left (d x +c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{m}}{b \cos \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^m\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{a+b\,\cos \left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________