Optimal. Leaf size=194 \[ \frac {b \cos ^4(c+d x)}{2 a^3 d}-\frac {\cos ^5(c+d x)}{5 a^2 d}+\frac {b^2 \left (a^2-b^2\right )^2}{a^7 d (a \cos (c+d x)+b)}-\frac {2 b \left (a^2-b^2\right ) \cos ^2(c+d x)}{a^5 d}+\frac {\left (2 a^2-3 b^2\right ) \cos ^3(c+d x)}{3 a^4 d}+\frac {2 b \left (a^4-4 a^2 b^2+3 b^4\right ) \log (a \cos (c+d x)+b)}{a^7 d}-\frac {\left (a^4-6 a^2 b^2+5 b^4\right ) \cos (c+d x)}{a^6 d} \]
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Rubi [A] time = 0.30, antiderivative size = 194, 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, 2837, 12, 948} \[ \frac {\left (2 a^2-3 b^2\right ) \cos ^3(c+d x)}{3 a^4 d}-\frac {2 b \left (a^2-b^2\right ) \cos ^2(c+d x)}{a^5 d}-\frac {\left (-6 a^2 b^2+a^4+5 b^4\right ) \cos (c+d x)}{a^6 d}+\frac {b^2 \left (a^2-b^2\right )^2}{a^7 d (a \cos (c+d x)+b)}+\frac {2 b \left (-4 a^2 b^2+a^4+3 b^4\right ) \log (a \cos (c+d x)+b)}{a^7 d}+\frac {b \cos ^4(c+d x)}{2 a^3 d}-\frac {\cos ^5(c+d x)}{5 a^2 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 948
Rule 2837
Rule 3872
Rubi steps
\begin {align*} \int \frac {\sin ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx &=\int \frac {\cos ^2(c+d x) \sin ^5(c+d x)}{(-b-a \cos (c+d x))^2} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (a^2-x^2\right )^2}{a^2 (-b+x)^2} \, dx,x,-a \cos (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (a^2-x^2\right )^2}{(-b+x)^2} \, dx,x,-a \cos (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^4 \left (1+\frac {-6 a^2 b^2+5 b^4}{a^4}\right )+\frac {b^2 \left (a^2-b^2\right )^2}{(b-x)^2}-\frac {2 b \left (a^4-4 a^2 b^2+3 b^4\right )}{b-x}+4 b \left (-a^2+b^2\right ) x-\left (2 a^2-3 b^2\right ) x^2+2 b x^3+x^4\right ) \, dx,x,-a \cos (c+d x)\right )}{a^7 d}\\ &=-\frac {\left (a^4-6 a^2 b^2+5 b^4\right ) \cos (c+d x)}{a^6 d}-\frac {2 b \left (a^2-b^2\right ) \cos ^2(c+d x)}{a^5 d}+\frac {\left (2 a^2-3 b^2\right ) \cos ^3(c+d x)}{3 a^4 d}+\frac {b \cos ^4(c+d x)}{2 a^3 d}-\frac {\cos ^5(c+d x)}{5 a^2 d}+\frac {b^2 \left (a^2-b^2\right )^2}{a^7 d (b+a \cos (c+d x))}+\frac {2 b \left (a^4-4 a^2 b^2+3 b^4\right ) \log (b+a \cos (c+d x))}{a^7 d}\\ \end {align*}
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Mathematica [A] time = 1.64, size = 280, normalized size = 1.44 \[ \frac {22 a^6 \cos (4 (c+d x))-3 a^6 \cos (6 (c+d x))-150 a^6-115 a^5 b \cos (3 (c+d x))+9 a^5 b \cos (5 (c+d x))-30 a^4 b^2 \cos (4 (c+d x))+960 a^4 b^2 \log (a \cos (c+d x)+b)+1740 a^4 b^2+120 a^3 b^3 \cos (3 (c+d x))-3840 a^2 b^4 \log (a \cos (c+d x)+b)-2160 a^2 b^4+120 a b \cos (c+d x) \left (-4 a^4+23 a^2 b^2+8 \left (a^4-4 a^2 b^2+3 b^4\right ) \log (a \cos (c+d x)+b)-20 b^4\right )-5 \left (25 a^6-168 a^4 b^2+144 a^2 b^4\right ) \cos (2 (c+d x))+2880 b^6 \log (a \cos (c+d x)+b)+480 b^6}{480 a^7 d (a \cos (c+d x)+b)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 240, normalized size = 1.24 \[ -\frac {48 \, a^{6} \cos \left (d x + c\right )^{6} - 72 \, a^{5} b \cos \left (d x + c\right )^{5} - 435 \, a^{4} b^{2} + 720 \, a^{2} b^{4} - 240 \, b^{6} - 40 \, {\left (4 \, a^{6} - 3 \, a^{4} b^{2}\right )} \cos \left (d x + c\right )^{4} + 80 \, {\left (4 \, a^{5} b - 3 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{3} + 240 \, {\left (a^{6} - 4 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (3 \, a^{5} b - 80 \, a^{3} b^{3} + 80 \, a b^{5}\right )} \cos \left (d x + c\right ) - 480 \, {\left (a^{4} b^{2} - 4 \, a^{2} b^{4} + 3 \, b^{6} + {\left (a^{5} b - 4 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{240 \, {\left (a^{8} d \cos \left (d x + c\right ) + a^{7} b d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.37, size = 1102, normalized size = 5.68 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.52, size = 285, normalized size = 1.47 \[ -\frac {\cos ^{5}\left (d x +c \right )}{5 a^{2} d}+\frac {b \left (\cos ^{4}\left (d x +c \right )\right )}{2 a^{3} d}+\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{3 a^{2} d}-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) b^{2}}{d \,a^{4}}-\frac {2 b \left (\cos ^{2}\left (d x +c \right )\right )}{a^{3} d}+\frac {2 \left (\cos ^{2}\left (d x +c \right )\right ) b^{3}}{d \,a^{5}}-\frac {\cos \left (d x +c \right )}{a^{2} d}+\frac {6 \cos \left (d x +c \right ) b^{2}}{d \,a^{4}}-\frac {5 \cos \left (d x +c \right ) b^{4}}{d \,a^{6}}+\frac {2 b \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{3} d}-\frac {8 b^{3} \ln \left (b +a \cos \left (d x +c \right )\right )}{d \,a^{5}}+\frac {6 b^{5} \ln \left (b +a \cos \left (d x +c \right )\right )}{d \,a^{7}}+\frac {b^{2}}{a^{3} d \left (b +a \cos \left (d x +c \right )\right )}-\frac {2 b^{4}}{d \,a^{5} \left (b +a \cos \left (d x +c \right )\right )}+\frac {b^{6}}{d \,a^{7} \left (b +a \cos \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 184, normalized size = 0.95 \[ \frac {\frac {30 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )}}{a^{8} \cos \left (d x + c\right ) + a^{7} b} - \frac {6 \, a^{4} \cos \left (d x + c\right )^{5} - 15 \, a^{3} b \cos \left (d x + c\right )^{4} - 10 \, {\left (2 \, a^{4} - 3 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3} + 60 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{2} + 30 \, {\left (a^{4} - 6 \, a^{2} b^{2} + 5 \, b^{4}\right )} \cos \left (d x + c\right )}{a^{6}} + \frac {60 \, {\left (a^{4} b - 4 \, a^{2} b^{3} + 3 \, b^{5}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{7}}}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 253, normalized size = 1.30 \[ \frac {{\cos \left (c+d\,x\right )}^3\,\left (\frac {2}{3\,a^2}-\frac {b^2}{a^4}\right )}{d}-\frac {{\cos \left (c+d\,x\right )}^2\,\left (\frac {b^3}{a^5}+\frac {b\,\left (\frac {2}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{d}-\frac {\cos \left (c+d\,x\right )\,\left (\frac {1}{a^2}+\frac {b^2\,\left (\frac {2}{a^2}-\frac {3\,b^2}{a^4}\right )}{a^2}-\frac {2\,b\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {2}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{a}\right )}{d}-\frac {{\cos \left (c+d\,x\right )}^5}{5\,a^2\,d}+\frac {b\,{\cos \left (c+d\,x\right )}^4}{2\,a^3\,d}+\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (2\,a^4\,b-8\,a^2\,b^3+6\,b^5\right )}{a^7\,d}+\frac {a^4\,b^2-2\,a^2\,b^4+b^6}{a\,d\,\left (\cos \left (c+d\,x\right )\,a^7+b\,a^6\right )} \]
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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