Optimal. Leaf size=112 \[ -\frac {a^2 \cos ^5(c+d x)}{5 d}-\frac {a^2 \cos ^4(c+d x)}{2 d}+\frac {a^2 \cos ^3(c+d x)}{3 d}+\frac {2 a^2 \cos ^2(c+d x)}{d}+\frac {a^2 \cos (c+d x)}{d}+\frac {a^2 \sec (c+d x)}{d}-\frac {2 a^2 \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.16, antiderivative size = 112, 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} \[ -\frac {a^2 \cos ^5(c+d x)}{5 d}-\frac {a^2 \cos ^4(c+d x)}{2 d}+\frac {a^2 \cos ^3(c+d x)}{3 d}+\frac {2 a^2 \cos ^2(c+d x)}{d}+\frac {a^2 \cos (c+d x)}{d}+\frac {a^2 \sec (c+d x)}{d}-\frac {2 a^2 \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rule 3872
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^2 \sin ^5(c+d x) \, dx &=\int (-a-a \cos (c+d x))^2 \sin ^3(c+d x) \tan ^2(c+d x) \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {a^2 (-a-x)^2 (-a+x)^4}{x^2} \, dx,x,-a \cos (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(-a-x)^2 (-a+x)^4}{x^2} \, dx,x,-a \cos (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-a^4+\frac {a^6}{x^2}-\frac {2 a^5}{x}+4 a^3 x-a^2 x^2-2 a x^3+x^4\right ) \, dx,x,-a \cos (c+d x)\right )}{a^3 d}\\ &=\frac {a^2 \cos (c+d x)}{d}+\frac {2 a^2 \cos ^2(c+d x)}{d}+\frac {a^2 \cos ^3(c+d x)}{3 d}-\frac {a^2 \cos ^4(c+d x)}{2 d}-\frac {a^2 \cos ^5(c+d x)}{5 d}-\frac {2 a^2 \log (\cos (c+d x))}{d}+\frac {a^2 \sec (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 87, normalized size = 0.78 \[ -\frac {a^2 \sec (c+d x) (-275 \cos (2 (c+d x))-165 \cos (3 (c+d x))-2 \cos (4 (c+d x))+15 \cos (5 (c+d x))+3 \cos (6 (c+d x))+30 \cos (c+d x) (32 \log (\cos (c+d x))-3)-750)}{480 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 115, normalized size = 1.03 \[ -\frac {48 \, a^{2} \cos \left (d x + c\right )^{6} + 120 \, a^{2} \cos \left (d x + c\right )^{5} - 80 \, a^{2} \cos \left (d x + c\right )^{4} - 480 \, a^{2} \cos \left (d x + c\right )^{3} - 240 \, a^{2} \cos \left (d x + c\right )^{2} + 480 \, a^{2} \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) + 195 \, a^{2} \cos \left (d x + c\right ) - 240 \, a^{2}}{240 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.38, size = 270, normalized size = 2.41 \[ \frac {60 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 60 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {60 \, {\left (2 \, a^{2} + \frac {a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}{\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1} + \frac {69 \, a^{2} - \frac {525 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {1650 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1610 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {745 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {137 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{5}}}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.74, size = 130, normalized size = 1.16 \[ \frac {32 a^{2} \cos \left (d x +c \right )}{15 d}+\frac {4 a^{2} \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{5 d}+\frac {16 a^{2} \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{15 d}-\frac {a^{2} \left (\sin ^{4}\left (d x +c \right )\right )}{2 d}-\frac {a^{2} \left (\sin ^{2}\left (d x +c \right )\right )}{d}-\frac {2 a^{2} \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {a^{2} \left (\sin ^{6}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 94, normalized size = 0.84 \[ -\frac {6 \, a^{2} \cos \left (d x + c\right )^{5} + 15 \, a^{2} \cos \left (d x + c\right )^{4} - 10 \, a^{2} \cos \left (d x + c\right )^{3} - 60 \, a^{2} \cos \left (d x + c\right )^{2} - 30 \, a^{2} \cos \left (d x + c\right ) + 60 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) - \frac {30 \, a^{2}}{\cos \left (d x + c\right )}}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.89, size = 91, normalized size = 0.81 \[ \frac {a^2\,\cos \left (c+d\,x\right )+\frac {a^2}{\cos \left (c+d\,x\right )}+2\,a^2\,{\cos \left (c+d\,x\right )}^2+\frac {a^2\,{\cos \left (c+d\,x\right )}^3}{3}-\frac {a^2\,{\cos \left (c+d\,x\right )}^4}{2}-\frac {a^2\,{\cos \left (c+d\,x\right )}^5}{5}-2\,a^2\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
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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