Optimal. Leaf size=102 \[ -\frac {\cos ^5(c+d x)}{5 a^3 d}+\frac {3 \cos ^4(c+d x)}{4 a^3 d}-\frac {4 \cos ^3(c+d x)}{3 a^3 d}+\frac {2 \cos ^2(c+d x)}{a^3 d}-\frac {4 \cos (c+d x)}{a^3 d}+\frac {4 \log (\cos (c+d x)+1)}{a^3 d} \]
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Rubi [A] time = 0.18, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3872, 2836, 12, 88} \[ -\frac {\cos ^5(c+d x)}{5 a^3 d}+\frac {3 \cos ^4(c+d x)}{4 a^3 d}-\frac {4 \cos ^3(c+d x)}{3 a^3 d}+\frac {2 \cos ^2(c+d x)}{a^3 d}-\frac {4 \cos (c+d x)}{a^3 d}+\frac {4 \log (\cos (c+d x)+1)}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rule 3872
Rubi steps
\begin {align*} \int \frac {\sin ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\int \frac {\cos ^3(c+d x) \sin ^5(c+d x)}{(-a-a \cos (c+d x))^3} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {(-a-x)^2 x^3}{a^3 (-a+x)} \, dx,x,-a \cos (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(-a-x)^2 x^3}{-a+x} \, dx,x,-a \cos (c+d x)\right )}{a^8 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (4 a^4-\frac {4 a^5}{a-x}+4 a^3 x+4 a^2 x^2+3 a x^3+x^4\right ) \, dx,x,-a \cos (c+d x)\right )}{a^8 d}\\ &=-\frac {4 \cos (c+d x)}{a^3 d}+\frac {2 \cos ^2(c+d x)}{a^3 d}-\frac {4 \cos ^3(c+d x)}{3 a^3 d}+\frac {3 \cos ^4(c+d x)}{4 a^3 d}-\frac {\cos ^5(c+d x)}{5 a^3 d}+\frac {4 \log (1+\cos (c+d x))}{a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.96, size = 73, normalized size = 0.72 \[ \frac {-4920 \cos (c+d x)+1320 \cos (2 (c+d x))-380 \cos (3 (c+d x))+90 \cos (4 (c+d x))-12 \cos (5 (c+d x))+7680 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+3857}{960 a^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 70, normalized size = 0.69 \[ -\frac {12 \, \cos \left (d x + c\right )^{5} - 45 \, \cos \left (d x + c\right )^{4} + 80 \, \cos \left (d x + c\right )^{3} - 120 \, \cos \left (d x + c\right )^{2} + 240 \, \cos \left (d x + c\right ) - 240 \, \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{60 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 172, normalized size = 1.69 \[ -\frac {\frac {60 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{3}} + \frac {\frac {85 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {20 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {200 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {205 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {137 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 29}{a^{3} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{5}}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.72, size = 114, normalized size = 1.12 \[ -\frac {1}{5 d \,a^{3} \sec \left (d x +c \right )^{5}}+\frac {3}{4 d \,a^{3} \sec \left (d x +c \right )^{4}}-\frac {4}{3 d \,a^{3} \sec \left (d x +c \right )^{3}}+\frac {2}{d \,a^{3} \sec \left (d x +c \right )^{2}}-\frac {4}{d \,a^{3} \sec \left (d x +c \right )}-\frac {4 \ln \left (\sec \left (d x +c \right )\right )}{d \,a^{3}}+\frac {4 \ln \left (1+\sec \left (d x +c \right )\right )}{d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 73, normalized size = 0.72 \[ -\frac {\frac {12 \, \cos \left (d x + c\right )^{5} - 45 \, \cos \left (d x + c\right )^{4} + 80 \, \cos \left (d x + c\right )^{3} - 120 \, \cos \left (d x + c\right )^{2} + 240 \, \cos \left (d x + c\right )}{a^{3}} - \frac {240 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.90, size = 82, normalized size = 0.80 \[ \frac {\frac {4\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{a^3}-\frac {4\,\cos \left (c+d\,x\right )}{a^3}+\frac {2\,{\cos \left (c+d\,x\right )}^2}{a^3}-\frac {4\,{\cos \left (c+d\,x\right )}^3}{3\,a^3}+\frac {3\,{\cos \left (c+d\,x\right )}^4}{4\,a^3}-\frac {{\cos \left (c+d\,x\right )}^5}{5\,a^3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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