Optimal. Leaf size=123 \[ \frac {(c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}-\frac {d \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f^2}+\frac {2 d \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3 a^2 f^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3318, 4185, 4184, 3475} \[ \frac {(c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}-\frac {d \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f^2}+\frac {2 d \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3 a^2 f^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3318
Rule 3475
Rule 4184
Rule 4185
Rubi steps
\begin {align*} \int \frac {c+d x}{(a+a \cos (e+f x))^2} \, dx &=\frac {\int (c+d x) \csc ^4\left (\frac {e+\pi }{2}+\frac {f x}{2}\right ) \, dx}{4 a^2}\\ &=-\frac {d \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f^2}+\frac {(c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {\int (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{6 a^2}\\ &=-\frac {d \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f^2}+\frac {(c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}-\frac {d \int \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{3 a^2 f}\\ &=\frac {2 d \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3 a^2 f^2}-\frac {d \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f^2}+\frac {(c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.52, size = 113, normalized size = 0.92 \[ \frac {\cos \left (\frac {1}{2} (e+f x)\right ) \left (f (c+d x) \left (3 \sin \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {3}{2} (e+f x)\right )\right )+2 d \cos \left (\frac {3}{2} (e+f x)\right ) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )+2 d \cos \left (\frac {1}{2} (e+f x)\right ) \left (3 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )-1\right )\right )}{3 a^2 f^2 (\cos (e+f x)+1)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.55, size = 118, normalized size = 0.96 \[ -\frac {d \cos \left (f x + e\right ) - {\left (d \cos \left (f x + e\right )^{2} + 2 \, d \cos \left (f x + e\right ) + d\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) - {\left (2 \, d f x + 2 \, c f + {\left (d f x + c f\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right ) + d}{3 \, {\left (a^{2} f^{2} \cos \left (f x + e\right )^{2} + 2 \, a^{2} f^{2} \cos \left (f x + e\right ) + a^{2} f^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 3.03, size = 757, normalized size = 6.15 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.18, size = 123, normalized size = 1.00 \[ \frac {c \left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{6 a^{2} f}+\frac {c \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^{2} f}-\frac {d \left (\tan ^{2}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{6 a^{2} f^{2}}+\frac {x d \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^{2} f}+\frac {x d \left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{6 a^{2} f}-\frac {d \ln \left (1+\tan ^{2}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3 a^{2} f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.99, size = 763, normalized size = 6.20 \[ -\frac {\frac {2 \, {\left (2 \, {\left (3 \, {\left (f x + e\right )} \sin \left (f x + e\right ) + \cos \left (2 \, f x + 2 \, e\right ) + \cos \left (f x + e\right )\right )} \cos \left (3 \, f x + 3 \, e\right ) + 2 \, {\left (9 \, {\left (f x + e\right )} \sin \left (f x + e\right ) + 6 \, \cos \left (f x + e\right ) + 1\right )} \cos \left (2 \, f x + 2 \, e\right ) + 6 \, \cos \left (2 \, f x + 2 \, e\right )^{2} + 6 \, \cos \left (f x + e\right )^{2} - {\left (2 \, {\left (3 \, \cos \left (2 \, f x + 2 \, e\right ) + 3 \, \cos \left (f x + e\right ) + 1\right )} \cos \left (3 \, f x + 3 \, e\right ) + \cos \left (3 \, f x + 3 \, e\right )^{2} + 6 \, {\left (3 \, \cos \left (f x + e\right ) + 1\right )} \cos \left (2 \, f x + 2 \, e\right ) + 9 \, \cos \left (2 \, f x + 2 \, e\right )^{2} + 9 \, \cos \left (f x + e\right )^{2} + 6 \, {\left (\sin \left (2 \, f x + 2 \, e\right ) + \sin \left (f x + e\right )\right )} \sin \left (3 \, f x + 3 \, e\right ) + \sin \left (3 \, f x + 3 \, e\right )^{2} + 9 \, \sin \left (2 \, f x + 2 \, e\right )^{2} + 18 \, \sin \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + 9 \, \sin \left (f x + e\right )^{2} + 6 \, \cos \left (f x + e\right ) + 1\right )} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right ) - 2 \, {\left (f x + 3 \, {\left (f x + e\right )} \cos \left (f x + e\right ) + e - \sin \left (2 \, f x + 2 \, e\right ) - \sin \left (f x + e\right )\right )} \sin \left (3 \, f x + 3 \, e\right ) - 6 \, {\left (f x + 3 \, {\left (f x + e\right )} \cos \left (f x + e\right ) + e - 2 \, \sin \left (f x + e\right )\right )} \sin \left (2 \, f x + 2 \, e\right ) + 6 \, \sin \left (2 \, f x + 2 \, e\right )^{2} + 6 \, \sin \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right )\right )} d}{a^{2} f \cos \left (3 \, f x + 3 \, e\right )^{2} + 9 \, a^{2} f \cos \left (2 \, f x + 2 \, e\right )^{2} + 9 \, a^{2} f \cos \left (f x + e\right )^{2} + a^{2} f \sin \left (3 \, f x + 3 \, e\right )^{2} + 9 \, a^{2} f \sin \left (2 \, f x + 2 \, e\right )^{2} + 18 \, a^{2} f \sin \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + 9 \, a^{2} f \sin \left (f x + e\right )^{2} + 6 \, a^{2} f \cos \left (f x + e\right ) + a^{2} f + 2 \, {\left (3 \, a^{2} f \cos \left (2 \, f x + 2 \, e\right ) + 3 \, a^{2} f \cos \left (f x + e\right ) + a^{2} f\right )} \cos \left (3 \, f x + 3 \, e\right ) + 6 \, {\left (3 \, a^{2} f \cos \left (f x + e\right ) + a^{2} f\right )} \cos \left (2 \, f x + 2 \, e\right ) + 6 \, {\left (a^{2} f \sin \left (2 \, f x + 2 \, e\right ) + a^{2} f \sin \left (f x + e\right )\right )} \sin \left (3 \, f x + 3 \, e\right )} - \frac {c {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2}} + \frac {d e {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2} f}}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.29, size = 175, normalized size = 1.42 \[ \frac {2\,d\,\ln \left ({\mathrm {e}}^{e\,1{}\mathrm {i}}\,{\mathrm {e}}^{f\,x\,1{}\mathrm {i}}+1\right )}{3\,a^2\,f^2}+\frac {\left (c\,f+d\,f\,x-d\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{3\,a^2\,f^2\,\left (2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}-\frac {d\,x\,2{}\mathrm {i}}{3\,a^2\,f}-\frac {2\,d}{3\,a^2\,f^2\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+1\right )}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\left (c+d\,x\right )\,4{}\mathrm {i}}{3\,a^2\,f\,\left (3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+3\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.09, size = 146, normalized size = 1.19 \[ \begin {cases} \frac {c \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{6 a^{2} f} + \frac {c \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{2 a^{2} f} + \frac {d x \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{6 a^{2} f} + \frac {d x \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{2 a^{2} f} - \frac {d \log {\left (\tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1 \right )}}{3 a^{2} f^{2}} - \frac {d \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{6 a^{2} f^{2}} & \text {for}\: f \neq 0 \\\frac {c x + \frac {d x^{2}}{2}}{\left (a \cos {\relax (e )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________