Optimal. Leaf size=89 \[ \frac{(c+5 d) (c-d) \tan (e+f x)}{3 f \left (a^2 \sec (e+f x)+a^2\right )}+\frac{d^2 \tanh ^{-1}(\sin (e+f x))}{a^2 f}+\frac{(c-d)^2 \tan (e+f x)}{3 f (a \sec (e+f x)+a)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.153959, antiderivative size = 149, normalized size of antiderivative = 1.67, number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {3987, 89, 78, 63, 217, 203} \[ \frac{(c+5 d) (c-d) \tan (e+f x)}{3 f \left (a^2 \sec (e+f x)+a^2\right )}+\frac{(c-d)^2 \tan (e+f x)}{3 f (a \sec (e+f x)+a)^2}+\frac{2 d^2 \tan (e+f x) \tan ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a (\sec (e+f x)+1)}}\right )}{a f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3987
Rule 89
Rule 78
Rule 63
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{\sec (e+f x) (c+d \sec (e+f x))^2}{(a+a \sec (e+f x))^2} \, dx &=-\frac{\left (a^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{(c+d x)^2}{\sqrt{a-a x} (a+a x)^{5/2}} \, dx,x,\sec (e+f x)\right )}{f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{(c-d)^2 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac{\tan (e+f x) \operatorname{Subst}\left (\int \frac{a^3 \left (c^2+4 c d-2 d^2\right )+3 a^3 d^2 x}{\sqrt{a-a x} (a+a x)^{3/2}} \, dx,x,\sec (e+f x)\right )}{3 a^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{(c-d)^2 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}+\frac{(c-d) (c+5 d) \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}-\frac{\left (d^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-a x} \sqrt{a+a x}} \, dx,x,\sec (e+f x)\right )}{f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{(c-d)^2 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}+\frac{(c-d) (c+5 d) \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}+\frac{\left (2 d^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2 a-x^2}} \, dx,x,\sqrt{a-a \sec (e+f x)}\right )}{a f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{(c-d)^2 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}+\frac{(c-d) (c+5 d) \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}+\frac{\left (2 d^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a+a \sec (e+f x)}}\right )}{a f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{(c-d)^2 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}+\frac{2 d^2 \tan ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a+a \sec (e+f x)}}\right ) \tan (e+f x)}{a f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{(c-d) (c+5 d) \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}\\ \end{align*}
Mathematica [B] time = 0.736878, size = 181, normalized size = 2.03 \[ -\frac{2 \cos \left (\frac{1}{2} (e+f x)\right ) \left (-4 \left (c^2+c d-2 d^2\right ) \sec \left (\frac{e}{2}\right ) \sin \left (\frac{f x}{2}\right ) \cos ^2\left (\frac{1}{2} (e+f x)\right )+(c-d)^2 \tan \left (\frac{e}{2}\right ) \cos \left (\frac{1}{2} (e+f x)\right )+(c-d)^2 \sec \left (\frac{e}{2}\right ) \sin \left (\frac{f x}{2}\right )+6 d^2 \cos ^3\left (\frac{1}{2} (e+f x)\right ) \left (\log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )\right )\right )}{3 a^2 f (\cos (e+f x)+1)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.074, size = 170, normalized size = 1.9 \begin{align*}{\frac{{d}^{2}}{f{a}^{2}}\ln \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) +1 \right ) }+{\frac{cd}{3\,f{a}^{2}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}+{\frac{cd}{f{a}^{2}}\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) }+{\frac{{c}^{2}}{2\,f{a}^{2}}\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) }-{\frac{{d}^{2}}{f{a}^{2}}\ln \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) -1 \right ) }-{\frac{{c}^{2}}{6\,f{a}^{2}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}-{\frac{{d}^{2}}{6\,f{a}^{2}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}-{\frac{3\,{d}^{2}}{2\,f{a}^{2}}\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.00558, size = 263, normalized size = 2.96 \begin{align*} -\frac{d^{2}{\left (\frac{\frac{9 \, \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}}}{a^{2}} - \frac{6 \, \log \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{2}} + \frac{6 \, \log \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{2}}\right )} - \frac{2 \, c d{\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{c^{2}{\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}}}{6 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.481805, size = 383, normalized size = 4.3 \begin{align*} \frac{3 \,{\left (d^{2} \cos \left (f x + e\right )^{2} + 2 \, d^{2} \cos \left (f x + e\right ) + d^{2}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \,{\left (d^{2} \cos \left (f x + e\right )^{2} + 2 \, d^{2} \cos \left (f x + e\right ) + d^{2}\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \,{\left (c^{2} + 4 \, c d - 5 \, d^{2} + 2 \,{\left (c^{2} + c d - 2 \, d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{6 \,{\left (a^{2} f \cos \left (f x + e\right )^{2} + 2 \, a^{2} f \cos \left (f x + e\right ) + a^{2} f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{c^{2} \sec{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec{\left (e + f x \right )} + 1}\, dx + \int \frac{d^{2} \sec ^{3}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec{\left (e + f x \right )} + 1}\, dx + \int \frac{2 c d \sec ^{2}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec{\left (e + f x \right )} + 1}\, dx}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.35398, size = 224, normalized size = 2.52 \begin{align*} \frac{\frac{6 \, d^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1 \right |}\right )}{a^{2}} - \frac{6 \, d^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1 \right |}\right )}{a^{2}} - \frac{a^{4} c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 2 \, a^{4} c d \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + a^{4} d^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 3 \, a^{4} c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 6 \, a^{4} c d \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 9 \, a^{4} d^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{a^{6}}}{6 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]