Optimal. Leaf size=76 \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right )}{a^{5/2} f \sqrt{a+b}}+\frac{(a-b) \sin (e+f x)}{a^2 f}-\frac{\sin ^3(e+f x)}{3 a f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0890151, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {4147, 390, 208} \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right )}{a^{5/2} f \sqrt{a+b}}+\frac{(a-b) \sin (e+f x)}{a^2 f}-\frac{\sin ^3(e+f x)}{3 a f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4147
Rule 390
Rule 208
Rubi steps
\begin{align*} \int \frac{\cos ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{a+b-a x^2} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a-b}{a^2}-\frac{x^2}{a}+\frac{b^2}{a^2 \left (a+b-a x^2\right )}\right ) \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{(a-b) \sin (e+f x)}{a^2 f}-\frac{\sin ^3(e+f x)}{3 a f}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{a+b-a x^2} \, dx,x,\sin (e+f x)\right )}{a^2 f}\\ &=\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right )}{a^{5/2} \sqrt{a+b} f}+\frac{(a-b) \sin (e+f x)}{a^2 f}-\frac{\sin ^3(e+f x)}{3 a f}\\ \end{align*}
Mathematica [A] time = 0.299817, size = 105, normalized size = 1.38 \[ \frac{a^{3/2} \sin (3 (e+f x))+\frac{6 b^2 \left (\log \left (\sqrt{a+b}+\sqrt{a} \sin (e+f x)\right )-\log \left (\sqrt{a+b}-\sqrt{a} \sin (e+f x)\right )\right )}{\sqrt{a+b}}+3 \sqrt{a} (3 a-4 b) \sin (e+f x)}{12 a^{5/2} f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.097, size = 70, normalized size = 0.9 \begin{align*}{\frac{1}{f} \left ( -{\frac{1}{{a}^{2}} \left ({\frac{a \left ( \sin \left ( fx+e \right ) \right ) ^{3}}{3}}-\sin \left ( fx+e \right ) a+\sin \left ( fx+e \right ) b \right ) }+{\frac{{b}^{2}}{{a}^{2}}{\it Artanh} \left ({\sin \left ( fx+e \right ) a{\frac{1}{\sqrt{ \left ( a+b \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) a}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.550273, size = 517, normalized size = 6.8 \begin{align*} \left [\frac{3 \, \sqrt{a^{2} + a b} b^{2} \log \left (-\frac{a \cos \left (f x + e\right )^{2} - 2 \, \sqrt{a^{2} + a b} \sin \left (f x + e\right ) - 2 \, a - b}{a \cos \left (f x + e\right )^{2} + b}\right ) + 2 \,{\left (2 \, a^{3} - a^{2} b - 3 \, a b^{2} +{\left (a^{3} + a^{2} b\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}{6 \,{\left (a^{4} + a^{3} b\right )} f}, -\frac{3 \, \sqrt{-a^{2} - a b} b^{2} \arctan \left (\frac{\sqrt{-a^{2} - a b} \sin \left (f x + e\right )}{a + b}\right ) -{\left (2 \, a^{3} - a^{2} b - 3 \, a b^{2} +{\left (a^{3} + a^{2} b\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}{3 \,{\left (a^{4} + a^{3} b\right )} f}\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 = 1.26462, size = 120, normalized size = 1.58 \begin{align*} -\frac{\frac{3 \, b^{2} \arctan \left (\frac{a \sin \left (f x + e\right )}{\sqrt{-a^{2} - a b}}\right )}{\sqrt{-a^{2} - a b} a^{2}} + \frac{a^{2} \sin \left (f x + e\right )^{3} - 3 \, a^{2} \sin \left (f x + e\right ) + 3 \, a b \sin \left (f x + e\right )}{a^{3}}}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]