Optimal. Leaf size=55 \[ \frac {\cot ^5(c+d x)}{5 a d}+\frac {\cot ^3(c+d x)}{3 a d}-\frac {\csc ^5(c+d x)}{5 a d} \]
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Rubi [A] time = 0.14, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3872, 2839, 2606, 30, 2607, 14} \[ \frac {\cot ^5(c+d x)}{5 a d}+\frac {\cot ^3(c+d x)}{3 a d}-\frac {\csc ^5(c+d x)}{5 a d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 2606
Rule 2607
Rule 2839
Rule 3872
Rubi steps
\begin {align*} \int \frac {\csc ^4(c+d x)}{a+a \sec (c+d x)} \, dx &=-\int \frac {\cot (c+d x) \csc ^3(c+d x)}{-a-a \cos (c+d x)} \, dx\\ &=-\frac {\int \cot ^2(c+d x) \csc ^4(c+d x) \, dx}{a}+\frac {\int \cot (c+d x) \csc ^5(c+d x) \, dx}{a}\\ &=-\frac {\operatorname {Subst}\left (\int x^4 \, dx,x,\csc (c+d x)\right )}{a d}-\frac {\operatorname {Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=-\frac {\csc ^5(c+d x)}{5 a d}-\frac {\operatorname {Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=\frac {\cot ^3(c+d x)}{3 a d}+\frac {\cot ^5(c+d x)}{5 a d}-\frac {\csc ^5(c+d x)}{5 a d}\\ \end {align*}
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Mathematica [B] time = 0.54, size = 116, normalized size = 2.11 \[ -\frac {\csc (c) (-54 \sin (c+d x)-18 \sin (2 (c+d x))+18 \sin (3 (c+d x))+9 \sin (4 (c+d x))-32 \sin (c+2 d x)+32 \sin (2 c+3 d x)+16 \sin (3 c+4 d x)+240 \sin (c)-96 \sin (d x)) \csc ^3(c+d x) \sec (c+d x)}{960 a d (\sec (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 89, normalized size = 1.62 \[ -\frac {2 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\right ) - 3}{15 \, {\left (a d \cos \left (d x + c\right )^{3} + a d \cos \left (d x + c\right )^{2} - a d \cos \left (d x + c\right ) - a d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 74, normalized size = 1.35 \[ -\frac {\frac {5 \, {\left (6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} + \frac {3 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 10 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}{a^{5}}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.57, size = 62, normalized size = 1.13 \[ \frac {-\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{16 d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 96, normalized size = 1.75 \[ -\frac {\frac {\frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a} + \frac {5 \, {\left (\frac {6 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{3}}{a \sin \left (d x + c\right )^{3}}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 60, normalized size = 1.09 \[ -\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+30\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+5}{240\,a\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\csc ^{4}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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