Optimal. Leaf size=107 \[ \frac{2 (5 B-2 C) \tan (c+d x)}{3 a^2 d}-\frac{(2 B-C) \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(2 B-C) \tan (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{(B-C) \tan (c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.379749, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {3029, 2978, 2748, 3767, 8, 3770} \[ \frac{2 (5 B-2 C) \tan (c+d x)}{3 a^2 d}-\frac{(2 B-C) \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(2 B-C) \tan (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{(B-C) \tan (c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3029
Rule 2978
Rule 2748
Rule 3767
Rule 8
Rule 3770
Rubi steps
\begin{align*} \int \frac{\left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx &=\int \frac{(B+C \cos (c+d x)) \sec ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx\\ &=-\frac{(B-C) \tan (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{\int \frac{(a (4 B-C)-2 a (B-C) \cos (c+d x)) \sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx}{3 a^2}\\ &=-\frac{(2 B-C) \tan (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac{(B-C) \tan (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{\int \left (2 a^2 (5 B-2 C)-3 a^2 (2 B-C) \cos (c+d x)\right ) \sec ^2(c+d x) \, dx}{3 a^4}\\ &=-\frac{(2 B-C) \tan (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac{(B-C) \tan (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{(2 (5 B-2 C)) \int \sec ^2(c+d x) \, dx}{3 a^2}-\frac{(2 B-C) \int \sec (c+d x) \, dx}{a^2}\\ &=-\frac{(2 B-C) \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(2 B-C) \tan (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac{(B-C) \tan (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac{(2 (5 B-2 C)) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 a^2 d}\\ &=-\frac{(2 B-C) \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{2 (5 B-2 C) \tan (c+d x)}{3 a^2 d}-\frac{(2 B-C) \tan (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac{(B-C) \tan (c+d x)}{3 d (a+a \cos (c+d x))^2}\\ \end{align*}
Mathematica [B] time = 1.53007, size = 264, normalized size = 2.47 \[ \frac{2 \cos \left (\frac{1}{2} (c+d x)\right ) \left ((B-C) \tan \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right )+(B-C) \sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right )+6 \cos ^3\left (\frac{1}{2} (c+d x)\right ) \left ((2 B-C) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+\frac{B \sin (d x)}{\left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}\right )+2 (7 B-4 C) \sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \cos ^2\left (\frac{1}{2} (c+d x)\right )\right )}{3 a^2 d (\cos (c+d x)+1)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.06, size = 205, normalized size = 1.9 \begin{align*}{\frac{B}{6\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{C}{6\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{5\,B}{2\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{3\,C}{2\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\frac{B\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) }{d{a}^{2}}}-{\frac{C}{d{a}^{2}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }-{\frac{B}{d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-2\,{\frac{B\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }{d{a}^{2}}}+{\frac{C}{d{a}^{2}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }-{\frac{B}{d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.36401, size = 329, normalized size = 3.07 \begin{align*} \frac{B{\left (\frac{\frac{15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac{12 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{2}} + \frac{12 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{2}} + \frac{12 \, \sin \left (d x + c\right )}{{\left (a^{2} - \frac{a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - C{\left (\frac{\frac{9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac{6 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{2}} + \frac{6 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{2}}\right )}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.72181, size = 502, normalized size = 4.69 \begin{align*} -\frac{3 \,{\left ({\left (2 \, B - C\right )} \cos \left (d x + c\right )^{3} + 2 \,{\left (2 \, B - C\right )} \cos \left (d x + c\right )^{2} +{\left (2 \, B - C\right )} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left ({\left (2 \, B - C\right )} \cos \left (d x + c\right )^{3} + 2 \,{\left (2 \, B - C\right )} \cos \left (d x + c\right )^{2} +{\left (2 \, B - C\right )} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (2 \,{\left (5 \, B - 2 \, C\right )} \cos \left (d x + c\right )^{2} +{\left (14 \, B - 5 \, C\right )} \cos \left (d x + c\right ) + 3 \, B\right )} \sin \left (d x + c\right )}{6 \,{\left (a^{2} d \cos \left (d x + c\right )^{3} + 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d \cos \left (d x + c\right )\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 [A] time = 1.74763, size = 209, normalized size = 1.95 \begin{align*} -\frac{\frac{6 \,{\left (2 \, B - C\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{2}} - \frac{6 \,{\left (2 \, B - C\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{2}} + \frac{12 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} a^{2}} - \frac{B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 15 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 9 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{6}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]