Optimal. Leaf size=141 \[ -\frac{b}{3 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{a (b \cos (c+d x)-a \sin (c+d x))}{2 d \left (a^2+b^2\right )^2 (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{a \tanh ^{-1}\left (\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{5/2}} \]
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Rubi [A] time = 0.110976, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {3158, 12, 3076, 3074, 206} \[ -\frac{b}{3 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{a (b \cos (c+d x)-a \sin (c+d x))}{2 d \left (a^2+b^2\right )^2 (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{a \tanh ^{-1}\left (\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3158
Rule 12
Rule 3076
Rule 3074
Rule 206
Rubi steps
\begin{align*} \int \frac{\cos (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx &=-\frac{b}{3 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{\int \frac{3 a}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx}{3 \left (a^2+b^2\right )}\\ &=-\frac{b}{3 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{a \int \frac{1}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx}{a^2+b^2}\\ &=-\frac{b}{3 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{a (b \cos (c+d x)-a \sin (c+d x))}{2 \left (a^2+b^2\right )^2 d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{a \int \frac{1}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{2 \left (a^2+b^2\right )^2}\\ &=-\frac{b}{3 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{a (b \cos (c+d x)-a \sin (c+d x))}{2 \left (a^2+b^2\right )^2 d (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{a \operatorname{Subst}\left (\int \frac{1}{a^2+b^2-x^2} \, dx,x,b \cos (c+d x)-a \sin (c+d x)\right )}{2 \left (a^2+b^2\right )^2 d}\\ &=-\frac{a \tanh ^{-1}\left (\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{5/2} d}-\frac{b}{3 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{a (b \cos (c+d x)-a \sin (c+d x))}{2 \left (a^2+b^2\right )^2 d (a \cos (c+d x)+b \sin (c+d x))^2}\\ \end{align*}
Mathematica [A] time = 0.699489, size = 128, normalized size = 0.91 \[ \frac{\frac{3 \left (a^3-a b^2\right ) \sin (2 (c+d x))-4 b \left (a^2+b^2\right )-6 a^2 b \cos (2 (c+d x))}{2 \left (a^2+b^2\right )^2 (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{6 a \tanh ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )-b}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}}{6 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.233, size = 383, normalized size = 2.7 \begin{align*}{\frac{1}{d} \left ( -2\,{\frac{1}{ \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a-2\,\tan \left ( 1/2\,dx+c/2 \right ) b-a \right ) ^{3}} \left ( -1/2\,{\frac{ \left ({a}^{4}+4\,{a}^{2}{b}^{2}+2\,{b}^{4} \right ) \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}}{a \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) }}-1/2\,{\frac{b \left ({a}^{4}-8\,{a}^{2}{b}^{2}-4\,{b}^{4} \right ) \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}}{{a}^{2} \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) }}+1/3\,{\frac{{b}^{2} \left ( 15\,{a}^{4}-4\,{a}^{2}{b}^{2}-4\,{b}^{4} \right ) \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}}{{a}^{3} \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) }}+{\frac{b \left ( 2\,{a}^{4}-5\,{a}^{2}{b}^{2}-2\,{b}^{4} \right ) \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{{a}^{2} \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) }}+1/2\,{\frac{ \left ({a}^{4}-6\,{a}^{2}{b}^{2}-2\,{b}^{4} \right ) \tan \left ( 1/2\,dx+c/2 \right ) }{a \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) }}-1/6\,{\frac{b \left ( 5\,{a}^{2}+2\,{b}^{2} \right ) }{{a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4}}} \right ) }+{\frac{a}{{a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4}}{\it Artanh} \left ({\frac{1}{2} \left ( 2\,a\tan \left ( 1/2\,dx+c/2 \right ) -2\,b \right ){\frac{1}{\sqrt{{a}^{2}+{b}^{2}}}}} \right ){\frac{1}{\sqrt{{a}^{2}+{b}^{2}}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [B] time = 0.566791, size = 945, normalized size = 6.7 \begin{align*} \frac{2 \, a^{4} b - 2 \, a^{2} b^{3} - 4 \, b^{5} - 12 \,{\left (a^{4} b + a^{2} b^{3}\right )} \cos \left (d x + c\right )^{2} + 6 \,{\left (a^{5} - a b^{4}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 3 \,{\left (3 \, a^{2} b^{2} \cos \left (d x + c\right ) +{\left (a^{4} - 3 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3} +{\left (a b^{3} +{\left (3 \, a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt{a^{2} + b^{2}} \log \left (-\frac{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt{a^{2} + b^{2}}{\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right )}{12 \,{\left ({\left (a^{9} - 6 \, a^{5} b^{4} - 8 \, a^{3} b^{6} - 3 \, a b^{8}\right )} d \cos \left (d x + c\right )^{3} + 3 \,{\left (a^{7} b^{2} + 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} + a b^{8}\right )} d \cos \left (d x + c\right ) +{\left ({\left (3 \, a^{8} b + 8 \, a^{6} b^{3} + 6 \, a^{4} b^{5} - b^{9}\right )} d \cos \left (d x + c\right )^{2} +{\left (a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}\right )} d\right )} \sin \left (d x + c\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [B] time = 1.30217, size = 575, normalized size = 4.08 \begin{align*} -\frac{\frac{3 \, a \log \left (\frac{{\left | 2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt{a^{2} + b^{2}}} - \frac{2 \,{\left (3 \, a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 12 \, a^{4} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, a^{2} b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, a^{5} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 24 \, a^{3} b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 12 \, a b^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 30 \, a^{4} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 8 \, a^{2} b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 8 \, b^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 12 \, a^{5} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 30 \, a^{3} b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 12 \, a b^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3 \, a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 18 \, a^{4} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, a^{2} b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 5 \, a^{5} b + 2 \, a^{3} b^{3}\right )}}{{\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )}{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 2 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - a\right )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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