Optimal. Leaf size=117 \[ \frac{C \sin (c+d x) (b \cos (c+d x))^{n+1}}{b d (n+2)}-\frac{(A (n+2)+C (n+1)) \sin (c+d x) (b \cos (c+d x))^{n+1} \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(c+d x)\right )}{b d (n+1) (n+2) \sqrt{\sin ^2(c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0717034, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {3014, 2643} \[ \frac{C \sin (c+d x) (b \cos (c+d x))^{n+1}}{b d (n+2)}-\frac{(A (n+2)+C (n+1)) \sin (c+d x) (b \cos (c+d x))^{n+1} \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(c+d x)\right )}{b d (n+1) (n+2) \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3014
Rule 2643
Rubi steps
\begin{align*} \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{C (b \cos (c+d x))^{1+n} \sin (c+d x)}{b d (2+n)}+\left (A+\frac{C (1+n)}{2+n}\right ) \int (b \cos (c+d x))^n \, dx\\ &=\frac{C (b \cos (c+d x))^{1+n} \sin (c+d x)}{b d (2+n)}-\frac{\left (A+\frac{C (1+n)}{2+n}\right ) (b \cos (c+d x))^{1+n} \, _2F_1\left (\frac{1}{2},\frac{1+n}{2};\frac{3+n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{b d (1+n) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.162769, size = 114, normalized size = 0.97 \[ -\frac{\sqrt{\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^n \left (A (n+3) \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(c+d x)\right )+C (n+1) \cos ^2(c+d x) \, _2F_1\left (\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(c+d x)\right )\right )}{d (n+1) (n+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 1.42, size = 0, normalized size = 0. \begin{align*} \int \left ( b\cos \left ( dx+c \right ) \right ) ^{n} \left ( A+C \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]