Optimal. Leaf size=196 \[ \frac{b^3 (A (2-n)+C (3-n)) \sin (c+d x) (b \cos (c+d x))^{n-3} \, _2F_1\left (\frac{1}{2},\frac{n-3}{2};\frac{n-1}{2};\cos ^2(c+d x)\right )}{d (2-n) (3-n) \sqrt{\sin ^2(c+d x)}}+\frac{b^2 B \sin (c+d x) (b \cos (c+d x))^{n-2} \, _2F_1\left (\frac{1}{2},\frac{n-2}{2};\frac{n}{2};\cos ^2(c+d x)\right )}{d (2-n) \sqrt{\sin ^2(c+d x)}}-\frac{b^3 C \sin (c+d x) (b \cos (c+d x))^{n-3}}{d (2-n)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.261561, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {16, 3023, 2748, 2643} \[ \frac{b^3 (A (2-n)+C (3-n)) \sin (c+d x) (b \cos (c+d x))^{n-3} \, _2F_1\left (\frac{1}{2},\frac{n-3}{2};\frac{n-1}{2};\cos ^2(c+d x)\right )}{d (2-n) (3-n) \sqrt{\sin ^2(c+d x)}}+\frac{b^2 B \sin (c+d x) (b \cos (c+d x))^{n-2} \, _2F_1\left (\frac{1}{2},\frac{n-2}{2};\frac{n}{2};\cos ^2(c+d x)\right )}{d (2-n) \sqrt{\sin ^2(c+d x)}}-\frac{b^3 C \sin (c+d x) (b \cos (c+d x))^{n-3}}{d (2-n)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 16
Rule 3023
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int (b \cos (c+d x))^n \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx &=b^4 \int (b \cos (c+d x))^{-4+n} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\\ &=-\frac{b^3 C (b \cos (c+d x))^{-3+n} \sin (c+d x)}{d (2-n)}-\frac{b^3 \int (b \cos (c+d x))^{-4+n} (-b (A (2-n)+C (3-n))-b B (2-n) \cos (c+d x)) \, dx}{2-n}\\ &=-\frac{b^3 C (b \cos (c+d x))^{-3+n} \sin (c+d x)}{d (2-n)}+\left (b^3 B\right ) \int (b \cos (c+d x))^{-3+n} \, dx+\frac{\left (b^4 (A (2-n)+C (3-n))\right ) \int (b \cos (c+d x))^{-4+n} \, dx}{2-n}\\ &=-\frac{b^3 C (b \cos (c+d x))^{-3+n} \sin (c+d x)}{d (2-n)}+\frac{b^3 (A (2-n)+C (3-n)) (b \cos (c+d x))^{-3+n} \, _2F_1\left (\frac{1}{2},\frac{1}{2} (-3+n);\frac{1}{2} (-1+n);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (2-n) (3-n) \sqrt{\sin ^2(c+d x)}}+\frac{b^2 B (b \cos (c+d x))^{-2+n} \, _2F_1\left (\frac{1}{2},\frac{1}{2} (-2+n);\frac{n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{d (2-n) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.331017, size = 142, normalized size = 0.72 \[ -\frac{\tan (c+d x) \sec ^2(c+d x) (b \cos (c+d x))^n \left ((A (n-2)+C (n-3)) \, _2F_1\left (\frac{1}{2},\frac{n-3}{2};\frac{n-1}{2};\cos ^2(c+d x)\right )+(n-3) \left (B \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{n-2}{2};\frac{n}{2};\cos ^2(c+d x)\right )-C \sqrt{\sin ^2(c+d x)}\right )\right )}{d (n-3) (n-2) \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 1.433, size = 0, normalized size = 0. \begin{align*} \int \left ( b\cos \left ( dx+c \right ) \right ) ^{n} \left ( A+B\cos \left ( dx+c \right ) +C \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{4}\, 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} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{4}\,{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} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{4}, 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} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]