Optimal. Leaf size=121 \[ \frac{2 a \tan ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (d+e x)\right )+c}{\sqrt{a^2-b^2-c^2}}\right )}{e \left (a^2-b^2-c^2\right )^{3/2}}+\frac{c \cos (d+e x)-b \sin (d+e x)}{e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.107953, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3129, 12, 3124, 618, 204} \[ \frac{2 a \tan ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (d+e x)\right )+c}{\sqrt{a^2-b^2-c^2}}\right )}{e \left (a^2-b^2-c^2\right )^{3/2}}+\frac{c \cos (d+e x)-b \sin (d+e x)}{e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3129
Rule 12
Rule 3124
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{(a+b \cos (d+e x)+c \sin (d+e x))^2} \, dx &=\frac{c \cos (d+e x)-b \sin (d+e x)}{\left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))}-\frac{\int \frac{a}{a+b \cos (d+e x)+c \sin (d+e x)} \, dx}{-a^2+b^2+c^2}\\ &=\frac{c \cos (d+e x)-b \sin (d+e x)}{\left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))}+\frac{a \int \frac{1}{a+b \cos (d+e x)+c \sin (d+e x)} \, dx}{a^2-b^2-c^2}\\ &=\frac{c \cos (d+e x)-b \sin (d+e x)}{\left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))}+\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{a+b+2 c x+(a-b) x^2} \, dx,x,\tan \left (\frac{1}{2} (d+e x)\right )\right )}{\left (a^2-b^2-c^2\right ) e}\\ &=\frac{c \cos (d+e x)-b \sin (d+e x)}{\left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))}-\frac{(4 a) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2-b^2-c^2\right )-x^2} \, dx,x,2 c+2 (a-b) \tan \left (\frac{1}{2} (d+e x)\right )\right )}{\left (a^2-b^2-c^2\right ) e}\\ &=\frac{2 a \tan ^{-1}\left (\frac{c+(a-b) \tan \left (\frac{1}{2} (d+e x)\right )}{\sqrt{a^2-b^2-c^2}}\right )}{\left (a^2-b^2-c^2\right )^{3/2} e}+\frac{c \cos (d+e x)-b \sin (d+e x)}{\left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))}\\ \end{align*}
Mathematica [A] time = 0.341416, size = 116, normalized size = 0.96 \[ \frac{\frac{a c+\left (b^2+c^2\right ) \sin (d+e x)}{b \left (-a^2+b^2+c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))}+\frac{2 a \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (d+e x)\right )+c}{\sqrt{-a^2+b^2+c^2}}\right )}{\left (-a^2+b^2+c^2\right )^{3/2}}}{e} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.142, size = 424, normalized size = 3.5 \begin{align*} -2\,{\frac{a\tan \left ( d/2+1/2\,ex \right ) b}{e \left ( a \left ( \tan \left ( d/2+1/2\,ex \right ) \right ) ^{2}-b \left ( \tan \left ( d/2+1/2\,ex \right ) \right ) ^{2}+2\,c\tan \left ( d/2+1/2\,ex \right ) +a+b \right ) \left ({a}^{3}-{a}^{2}b-a{b}^{2}-a{c}^{2}+{b}^{3}+b{c}^{2} \right ) }}+2\,{\frac{\tan \left ( d/2+1/2\,ex \right ){b}^{2}}{e \left ( a \left ( \tan \left ( d/2+1/2\,ex \right ) \right ) ^{2}-b \left ( \tan \left ( d/2+1/2\,ex \right ) \right ) ^{2}+2\,c\tan \left ( d/2+1/2\,ex \right ) +a+b \right ) \left ({a}^{3}-{a}^{2}b-a{b}^{2}-a{c}^{2}+{b}^{3}+b{c}^{2} \right ) }}+2\,{\frac{\tan \left ( d/2+1/2\,ex \right ){c}^{2}}{e \left ( a \left ( \tan \left ( d/2+1/2\,ex \right ) \right ) ^{2}-b \left ( \tan \left ( d/2+1/2\,ex \right ) \right ) ^{2}+2\,c\tan \left ( d/2+1/2\,ex \right ) +a+b \right ) \left ({a}^{3}-{a}^{2}b-a{b}^{2}-a{c}^{2}+{b}^{3}+b{c}^{2} \right ) }}+2\,{\frac{ac}{e \left ( a \left ( \tan \left ( d/2+1/2\,ex \right ) \right ) ^{2}-b \left ( \tan \left ( d/2+1/2\,ex \right ) \right ) ^{2}+2\,c\tan \left ( d/2+1/2\,ex \right ) +a+b \right ) \left ({a}^{3}-{a}^{2}b-a{b}^{2}-a{c}^{2}+{b}^{3}+b{c}^{2} \right ) }}+2\,{\frac{a}{e \left ({a}^{2}-{b}^{2}-{c}^{2} \right ) ^{3/2}}\arctan \left ( 1/2\,{\frac{2\, \left ( a-b \right ) \tan \left ( d/2+1/2\,ex \right ) +2\,c}{\sqrt{{a}^{2}-{b}^{2}-{c}^{2}}}} \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 [B] time = 2.98829, size = 1796, normalized size = 14.84 \begin{align*} \text{result too large to display} \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 = 1.14243, size = 300, normalized size = 2.48 \begin{align*} -2 \,{\left (\frac{{\left (\pi \left \lfloor \frac{x e + d}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac{a \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) - b \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + c}{\sqrt{a^{2} - b^{2} - c^{2}}}\right )\right )} a}{{\left (a^{2} - b^{2} - c^{2}\right )}^{\frac{3}{2}}} + \frac{a b \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) - b^{2} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) - c^{2} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) - a c}{{\left (a^{3} - a^{2} b - a b^{2} + b^{3} - a c^{2} + b c^{2}\right )}{\left (a \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{2} - b \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{2} + 2 \, c \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + a + b\right )}}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]