Optimal. Leaf size=195 \[ \frac{2 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^5 b d}+\frac{b \left (b^2-2 a^2\right ) \cot (c+d x)}{a^4 d}-\frac{\left (-20 a^2 b^2+15 a^4+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^5 d}+\frac{\left (7 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac{b \cot ^3(c+d x)}{3 a^2 d}-\frac{\cot ^3(c+d x) \csc (c+d x)}{4 a d}-\frac{x}{b} \]
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Rubi [A] time = 0.289505, antiderivative size = 275, normalized size of antiderivative = 1.41, number of steps used = 15, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {2897, 3770, 3767, 8, 3768, 2660, 618, 204} \[ \frac{2 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^5 b d}-\frac{b \left (3 a^2-b^2\right ) \cot (c+d x)}{a^4 d}+\frac{\left (3 a^2-b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\left (-3 a^2 b^2+3 a^4+b^4\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{\left (3 a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac{b \cot ^3(c+d x)}{3 a^2 d}+\frac{b \cot (c+d x)}{a^2 d}-\frac{3 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac{3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac{x}{b} \]
Antiderivative was successfully verified.
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Rule 2897
Rule 3770
Rule 3767
Rule 8
Rule 3768
Rule 2660
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \cot ^5(c+d x)}{a+b \sin (c+d x)} \, dx &=\int \left (-\frac{1}{b}+\frac{\left (3 a^4-3 a^2 b^2+b^4\right ) \csc (c+d x)}{a^5}+\frac{\left (3 a^2 b-b^3\right ) \csc ^2(c+d x)}{a^4}+\frac{\left (-3 a^2+b^2\right ) \csc ^3(c+d x)}{a^3}-\frac{b \csc ^4(c+d x)}{a^2}+\frac{\csc ^5(c+d x)}{a}+\frac{\left (a^2-b^2\right )^3}{a^5 b (a+b \sin (c+d x))}\right ) \, dx\\ &=-\frac{x}{b}+\frac{\int \csc ^5(c+d x) \, dx}{a}-\frac{b \int \csc ^4(c+d x) \, dx}{a^2}+\frac{\left (a^2-b^2\right )^3 \int \frac{1}{a+b \sin (c+d x)} \, dx}{a^5 b}-\frac{\left (3 a^2-b^2\right ) \int \csc ^3(c+d x) \, dx}{a^3}+\frac{\left (b \left (3 a^2-b^2\right )\right ) \int \csc ^2(c+d x) \, dx}{a^4}+\frac{\left (3 a^4-3 a^2 b^2+b^4\right ) \int \csc (c+d x) \, dx}{a^5}\\ &=-\frac{x}{b}-\frac{\left (3 a^4-3 a^2 b^2+b^4\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{\left (3 a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac{3 \int \csc ^3(c+d x) \, dx}{4 a}-\frac{\left (3 a^2-b^2\right ) \int \csc (c+d x) \, dx}{2 a^3}+\frac{b \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^2 d}+\frac{\left (2 \left (a^2-b^2\right )^3\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^5 b d}-\frac{\left (b \left (3 a^2-b^2\right )\right ) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^4 d}\\ &=-\frac{x}{b}+\frac{\left (3 a^2-b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\left (3 a^4-3 a^2 b^2+b^4\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{b \cot (c+d x)}{a^2 d}-\frac{b \left (3 a^2-b^2\right ) \cot (c+d x)}{a^4 d}+\frac{b \cot ^3(c+d x)}{3 a^2 d}-\frac{3 \cot (c+d x) \csc (c+d x)}{8 a d}+\frac{\left (3 a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac{3 \int \csc (c+d x) \, dx}{8 a}-\frac{\left (4 \left (a^2-b^2\right )^3\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^5 b d}\\ &=-\frac{x}{b}+\frac{2 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{a^5 b d}-\frac{3 \tanh ^{-1}(\cos (c+d x))}{8 a d}+\frac{\left (3 a^2-b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\left (3 a^4-3 a^2 b^2+b^4\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{b \cot (c+d x)}{a^2 d}-\frac{b \left (3 a^2-b^2\right ) \cot (c+d x)}{a^4 d}+\frac{b \cot ^3(c+d x)}{3 a^2 d}-\frac{3 \cot (c+d x) \csc (c+d x)}{8 a d}+\frac{\left (3 a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}\\ \end{align*}
Mathematica [B] time = 6.18269, size = 448, normalized size = 2.3 \[ \frac{\left (9 a^2-4 b^2\right ) \csc ^2\left (\frac{1}{2} (c+d x)\right )}{32 a^3 d}+\frac{\left (4 b^2-9 a^2\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )}{32 a^3 d}+\frac{\left (-20 a^2 b^2+15 a^4+8 b^4\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{8 a^5 d}+\frac{\left (20 a^2 b^2-15 a^4-8 b^4\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{8 a^5 d}+\frac{\csc \left (\frac{1}{2} (c+d x)\right ) \left (3 b^3 \cos \left (\frac{1}{2} (c+d x)\right )-7 a^2 b \cos \left (\frac{1}{2} (c+d x)\right )\right )}{6 a^4 d}+\frac{\sec \left (\frac{1}{2} (c+d x)\right ) \left (7 a^2 b \sin \left (\frac{1}{2} (c+d x)\right )-3 b^3 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{6 a^4 d}+\frac{2 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{\sec \left (\frac{1}{2} (c+d x)\right ) \left (a \sin \left (\frac{1}{2} (c+d x)\right )+b \cos \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{a^2-b^2}}\right )}{a^5 b d}+\frac{b \cot \left (\frac{1}{2} (c+d x)\right ) \csc ^2\left (\frac{1}{2} (c+d x)\right )}{24 a^2 d}-\frac{b \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )}{24 a^2 d}-\frac{\csc ^4\left (\frac{1}{2} (c+d x)\right )}{64 a d}+\frac{\sec ^4\left (\frac{1}{2} (c+d x)\right )}{64 a d}-\frac{c+d x}{b d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.125, size = 523, normalized size = 2.7 \begin{align*} \text{result too large to display} \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 = 4.56521, size = 2427, normalized size = 12.45 \begin{align*} \text{result too large to display} \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 = 1.26005, size = 535, normalized size = 2.74 \begin{align*} -\frac{\frac{192 \,{\left (d x + c\right )}}{b} - \frac{3 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 8 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 48 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 24 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 216 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 96 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{4}} - \frac{24 \,{\left (15 \, a^{4} - 20 \, a^{2} b^{2} + 8 \, b^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{5}} - \frac{384 \,{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )}{\left (\pi \left \lfloor \frac{d x + c}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (a\right ) + \arctan \left (\frac{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + b}{\sqrt{a^{2} - b^{2}}}\right )\right )}}{\sqrt{a^{2} - b^{2}} a^{5} b} + \frac{750 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 1000 \, a^{2} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 400 \, b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 216 \, a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 96 \, a b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 48 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 24 \, a^{2} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 8 \, a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, a^{4}}{a^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4}}}{192 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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