Optimal. Leaf size=74 \[ -\frac {a^2 (A-B) \sin (c+d x)}{d}+\frac {a^2 (2 A+B) \tanh ^{-1}(\sin (c+d x))}{d}+a^2 x (A+2 B)+\frac {A \tan (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.21, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2975, 2968, 3023, 2735, 3770} \[ -\frac {a^2 (A-B) \sin (c+d x)}{d}+\frac {a^2 (2 A+B) \tanh ^{-1}(\sin (c+d x))}{d}+a^2 x (A+2 B)+\frac {A \tan (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2735
Rule 2968
Rule 2975
Rule 3023
Rule 3770
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx &=\frac {A \left (a^2+a^2 \cos (c+d x)\right ) \tan (c+d x)}{d}+\int (a+a \cos (c+d x)) (a (2 A+B)-a (A-B) \cos (c+d x)) \sec (c+d x) \, dx\\ &=\frac {A \left (a^2+a^2 \cos (c+d x)\right ) \tan (c+d x)}{d}+\int \left (a^2 (2 A+B)+\left (-a^2 (A-B)+a^2 (2 A+B)\right ) \cos (c+d x)-a^2 (A-B) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac {a^2 (A-B) \sin (c+d x)}{d}+\frac {A \left (a^2+a^2 \cos (c+d x)\right ) \tan (c+d x)}{d}+\int \left (a^2 (2 A+B)+a^2 (A+2 B) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=a^2 (A+2 B) x-\frac {a^2 (A-B) \sin (c+d x)}{d}+\frac {A \left (a^2+a^2 \cos (c+d x)\right ) \tan (c+d x)}{d}+\left (a^2 (2 A+B)\right ) \int \sec (c+d x) \, dx\\ &=a^2 (A+2 B) x+\frac {a^2 (2 A+B) \tanh ^{-1}(\sin (c+d x))}{d}-\frac {a^2 (A-B) \sin (c+d x)}{d}+\frac {A \left (a^2+a^2 \cos (c+d x)\right ) \tan (c+d x)}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.37, size = 143, normalized size = 1.93 \[ \frac {a^2 \left (A \tan (c+d x)-2 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 A \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+A c+A d x+B \sin (c+d x)-B \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+B \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+2 B c+2 B d x\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.67, size = 108, normalized size = 1.46 \[ \frac {2 \, {\left (A + 2 \, B\right )} a^{2} d x \cos \left (d x + c\right ) + {\left (2 \, A + B\right )} a^{2} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (2 \, A + B\right )} a^{2} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (B a^{2} \cos \left (d x + c\right ) + A a^{2}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.06, size = 155, normalized size = 2.09 \[ \frac {{\left (A a^{2} + 2 \, B a^{2}\right )} {\left (d x + c\right )} + {\left (2 \, A a^{2} + B a^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (2 \, A a^{2} + B a^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.14, size = 107, normalized size = 1.45 \[ a^{2} A x +2 a^{2} B x +\frac {a^{2} A \tan \left (d x +c \right )}{d}+\frac {2 a^{2} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {A \,a^{2} c}{d}+\frac {B \,a^{2} \sin \left (d x +c \right )}{d}+\frac {B \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {2 B \,a^{2} c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 105, normalized size = 1.42 \[ \frac {2 \, {\left (d x + c\right )} A a^{2} + 4 \, {\left (d x + c\right )} B a^{2} + 2 \, A a^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + B a^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, B a^{2} \sin \left (d x + c\right ) + 2 \, A a^{2} \tan \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.32, size = 161, normalized size = 2.18 \[ \frac {B\,a^2\,\sin \left (c+d\,x\right )}{d}+\frac {2\,A\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {4\,A\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {4\,B\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,B\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {A\,a^2\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{2} \left (\int A \sec ^{2}{\left (c + d x \right )}\, dx + \int 2 A \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int A \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int B \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 2 B \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int B \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________