Optimal. Leaf size=156 \[ -\frac{3 (b \cos (c+d x)-a \sin (c+d x))}{8 d \left (a^2+b^2\right )^2 (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{b \cos (c+d x)-a \sin (c+d x)}{4 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{3 \tanh ^{-1}\left (\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right )}{8 d \left (a^2+b^2\right )^{5/2}} \]
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Rubi [A] time = 0.0860091, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {3076, 3074, 206} \[ -\frac{3 (b \cos (c+d x)-a \sin (c+d x))}{8 d \left (a^2+b^2\right )^2 (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{b \cos (c+d x)-a \sin (c+d x)}{4 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{3 \tanh ^{-1}\left (\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right )}{8 d \left (a^2+b^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3076
Rule 3074
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(a \cos (c+d x)+b \sin (c+d x))^5} \, dx &=-\frac{b \cos (c+d x)-a \sin (c+d x)}{4 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{3 \int \frac{1}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx}{4 \left (a^2+b^2\right )}\\ &=-\frac{b \cos (c+d x)-a \sin (c+d x)}{4 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{3 (b \cos (c+d x)-a \sin (c+d x))}{8 \left (a^2+b^2\right )^2 d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{3 \int \frac{1}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{8 \left (a^2+b^2\right )^2}\\ &=-\frac{b \cos (c+d x)-a \sin (c+d x)}{4 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{3 (b \cos (c+d x)-a \sin (c+d x))}{8 \left (a^2+b^2\right )^2 d (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{3 \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 )}{8 \left (a^2+b^2\right )^2 d}\\ &=-\frac{3 \tanh ^{-1}\left (\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right )}{8 \left (a^2+b^2\right )^{5/2} d}-\frac{b \cos (c+d x)-a \sin (c+d x)}{4 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{3 (b \cos (c+d x)-a \sin (c+d x))}{8 \left (a^2+b^2\right )^2 d (a \cos (c+d x)+b \sin (c+d x))^2}\\ \end{align*}
Mathematica [A] time = 1.12765, size = 157, normalized size = 1.01 \[ \frac{\frac{-11 b \left (a^2+b^2\right ) \cos (c+d x)+\left (3 b^3-9 a^2 b\right ) \cos (3 (c+d x))+2 a \sin (c+d x) \left (3 \left (a^2-3 b^2\right ) \cos (2 (c+d x))+7 a^2+b^2\right )}{4 \left (a^2+b^2\right )^2 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{6 \tanh ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )-b}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}}{8 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.222, size = 514, normalized size = 3.3 \begin{align*}{\frac{1}{d} \left ( -2\,{\frac{1}{ \left ( a \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-2\,b\tan \left ( 1/2\,dx+c/2 \right ) -a \right ) ^{4}} \left ( -1/8\,{\frac{ \left ( 5\,{a}^{4}+16\,{a}^{2}{b}^{2}+8\,{b}^{4} \right ) \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{7}}{a \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) }}+3/8\,{\frac{b \left ({a}^{4}+16\,{a}^{2}{b}^{2}+8\,{b}^{4} \right ) \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}}{{a}^{2} \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) }}-1/8\,{\frac{ \left ( 3\,{a}^{6}-36\,{a}^{4}{b}^{2}+56\,{a}^{2}{b}^{4}+32\,{b}^{6} \right ) \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}}{{a}^{3} \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) }}+1/8\,{\frac{b \left ( 15\,{a}^{6}-114\,{a}^{4}{b}^{2}-8\,{a}^{2}{b}^{4}+16\,{b}^{6} \right ) \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}}{{a}^{4} \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) }}-1/8\,{\frac{ \left ( 3\,{a}^{6}+84\,{a}^{4}{b}^{2}-56\,{a}^{2}{b}^{4}-32\,{b}^{6} \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 ) }}-1/8\,{\frac{b \left ( 23\,{a}^{4}-64\,{a}^{2}{b}^{2}-24\,{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 ) }}-1/8\,{\frac{ \left ( 5\,{a}^{4}-24\,{a}^{2}{b}^{2}-8\,{b}^{4} \right ) \tan \left ( 1/2\,dx+c/2 \right ) }{a \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) }}+1/8\,{\frac{b \left ( 5\,{a}^{2}+2\,{b}^{2} \right ) }{{a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4}}} \right ) }+{\frac{3}{4\,{a}^{4}+8\,{a}^{2}{b}^{2}+4\,{b}^{4}}{\it Artanh} \left ({\frac{1}{2} \left ( 2\,\tan \left ( 1/2\,dx+c/2 \right ) a-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 = 2.53538, size = 1216, normalized size = 7.79 \begin{align*} -\frac{6 \,{\left (3 \, a^{4} b + 2 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{3} - 3 \,{\left ({\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + b^{4} + 2 \,{\left (3 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} + 4 \,{\left (a b^{3} \cos \left (d x + c\right ) +{\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3}\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 ) - 2 \,{\left (4 \, a^{4} b - a^{2} b^{3} - 5 \, b^{5}\right )} \cos \left (d x + c\right ) - 2 \,{\left (2 \, a^{5} + 7 \, a^{3} b^{2} + 5 \, a b^{4} + 3 \,{\left (a^{5} - 2 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \,{\left ({\left (a^{10} - 3 \, a^{8} b^{2} - 14 \, a^{6} b^{4} - 14 \, a^{4} b^{6} - 3 \, a^{2} b^{8} + b^{10}\right )} d \cos \left (d x + c\right )^{4} + 2 \,{\left (3 \, a^{8} b^{2} + 8 \, a^{6} b^{4} + 6 \, a^{4} b^{6} - b^{10}\right )} d \cos \left (d x + c\right )^{2} +{\left (a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}\right )} d + 4 \,{\left ({\left (a^{9} b + 2 \, a^{7} b^{3} - 2 \, a^{3} b^{7} - a b^{9}\right )} d \cos \left (d x + c\right )^{3} +{\left (a^{7} b^{3} + 3 \, a^{5} b^{5} + 3 \, a^{3} b^{7} + a b^{9}\right )} d \cos \left (d x + c\right )\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 = 1.34513, size = 794, normalized size = 5.09 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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