Optimal. Leaf size=269 \[ -\frac {3 \left (a^2+4 a b+5 b^2\right ) \log (1-\sin (c+d x))}{16 d (a+b)^4}+\frac {3 \left (a^2-4 a b+5 b^2\right ) \log (\sin (c+d x)+1)}{16 d (a-b)^4}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))}+\frac {\sec ^2(c+d x) \left (3 a \left (a^2-3 b^2\right ) \sin (c+d x)+b \left (a^2+5 b^2\right )\right )}{8 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))}-\frac {6 a b^5 \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^4}-\frac {3 b \left (a^4-4 a^2 b^2-5 b^4\right )}{8 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))} \]
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Rubi [A] time = 0.32, antiderivative size = 269, 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 = {2668, 741, 823, 801} \[ -\frac {3 b \left (-4 a^2 b^2+a^4-5 b^4\right )}{8 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}-\frac {6 a b^5 \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^4}-\frac {3 \left (a^2+4 a b+5 b^2\right ) \log (1-\sin (c+d x))}{16 d (a+b)^4}+\frac {3 \left (a^2-4 a b+5 b^2\right ) \log (\sin (c+d x)+1)}{16 d (a-b)^4}+\frac {\sec ^2(c+d x) \left (3 a \left (a^2-3 b^2\right ) \sin (c+d x)+b \left (a^2+5 b^2\right )\right )}{8 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))} \]
Antiderivative was successfully verified.
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Rule 741
Rule 801
Rule 823
Rule 2668
Rubi steps
\begin {align*} \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\frac {b^5 \operatorname {Subst}\left (\int \frac {1}{(a+x)^2 \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {b^3 \operatorname {Subst}\left (\int \frac {3 a^2-5 b^2+4 a x}{(a+x)^2 \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 \left (a^2-b^2\right ) d}\\ &=-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\sec ^2(c+d x) \left (b \left (a^2+5 b^2\right )+3 a \left (a^2-3 b^2\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}-\frac {b \operatorname {Subst}\left (\int \frac {-3 \left (a^4-2 a^2 b^2+5 b^4\right )-6 a \left (a^2-3 b^2\right ) x}{(a+x)^2 \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d}\\ &=-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\sec ^2(c+d x) \left (b \left (a^2+5 b^2\right )+3 a \left (a^2-3 b^2\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}-\frac {b \operatorname {Subst}\left (\int \left (-\frac {3 (a-b)^2 \left (a^2+4 a b+5 b^2\right )}{2 b (a+b)^2 (b-x)}-\frac {3 \left (a^4-4 a^2 b^2-5 b^4\right )}{\left (a^2-b^2\right ) (a+x)^2}+\frac {48 a b^4}{\left (a^2-b^2\right )^2 (a+x)}-\frac {3 (a+b)^2 \left (a^2-4 a b+5 b^2\right )}{2 (a-b)^2 b (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d}\\ &=-\frac {3 \left (a^2+4 a b+5 b^2\right ) \log (1-\sin (c+d x))}{16 (a+b)^4 d}+\frac {3 \left (a^2-4 a b+5 b^2\right ) \log (1+\sin (c+d x))}{16 (a-b)^4 d}-\frac {6 a b^5 \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^4 d}-\frac {3 b \left (a^4-4 a^2 b^2-5 b^4\right )}{8 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\sec ^2(c+d x) \left (b \left (a^2+5 b^2\right )+3 a \left (a^2-3 b^2\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 6.11, size = 406, normalized size = 1.51 \[ \frac {b^5 \left (\frac {\sec ^4(c+d x) \left (b^2-a b \sin (c+d x)\right )}{4 b^6 \left (b^2-a^2\right ) (a+b \sin (c+d x))}-\frac {\frac {\left (6 a^2 \left (a^2-3 b^2\right )-3 \left (a^4-2 a^2 b^2+5 b^4\right )\right ) \left (\frac {1}{\left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac {\log (1-\sin (c+d x))}{2 b (a+b)^2}+\frac {\log (\sin (c+d x)+1)}{2 b (a-b)^2}-\frac {2 a \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}\right )-6 a \left (a^2-3 b^2\right ) \left (-\frac {\log (a+b \sin (c+d x))}{a^2-b^2}-\frac {\log (1-\sin (c+d x))}{2 b (a+b)}+\frac {\log (\sin (c+d x)+1)}{2 b (a-b)}\right )}{2 b^2 \left (b^2-a^2\right )}-\frac {\sec ^2(c+d x) \left (-b \left (4 a b^2-a \left (3 a^2-5 b^2\right )\right ) \sin (c+d x)+4 a^2 b^2-b^2 \left (3 a^2-5 b^2\right )\right )}{2 b^4 \left (b^2-a^2\right ) (a+b \sin (c+d x))}}{4 b^2 \left (b^2-a^2\right )}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.77, size = 527, normalized size = 1.96 \[ -\frac {4 \, a^{6} b - 12 \, a^{4} b^{3} + 12 \, a^{2} b^{5} - 4 \, b^{7} + 6 \, {\left (a^{6} b - 5 \, a^{4} b^{3} - a^{2} b^{5} + 5 \, b^{7}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{6} b + 3 \, a^{4} b^{3} - 9 \, a^{2} b^{5} + 5 \, b^{7}\right )} \cos \left (d x + c\right )^{2} + 96 \, {\left (a b^{6} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + a^{2} b^{5} \cos \left (d x + c\right )^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 3 \, {\left ({\left (a^{6} b - 5 \, a^{4} b^{3} + 15 \, a^{2} b^{5} + 16 \, a b^{6} + 5 \, b^{7}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + {\left (a^{7} - 5 \, a^{5} b^{2} + 15 \, a^{3} b^{4} + 16 \, a^{2} b^{5} + 5 \, a b^{6}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left ({\left (a^{6} b - 5 \, a^{4} b^{3} + 15 \, a^{2} b^{5} - 16 \, a b^{6} + 5 \, b^{7}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + {\left (a^{7} - 5 \, a^{5} b^{2} + 15 \, a^{3} b^{4} - 16 \, a^{2} b^{5} + 5 \, a b^{6}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (2 \, a^{7} - 6 \, a^{5} b^{2} + 6 \, a^{3} b^{4} - 2 \, a b^{6} + 3 \, {\left (a^{7} - 5 \, a^{5} b^{2} + 7 \, a^{3} b^{4} - 3 \, a b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, {\left ({\left (a^{8} b - 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} - 4 \, a^{2} b^{7} + b^{9}\right )} d \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + {\left (a^{9} - 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} - 4 \, a^{3} b^{6} + a b^{8}\right )} d \cos \left (d x + c\right )^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 460, normalized size = 1.71 \[ -\frac {\frac {96 \, a b^{6} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{8} b - 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} - 4 \, a^{2} b^{7} + b^{9}} - \frac {3 \, {\left (a^{2} - 4 \, a b + 5 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} + \frac {3 \, {\left (a^{2} + 4 \, a b + 5 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} - \frac {16 \, {\left (6 \, a b^{6} \sin \left (d x + c\right ) + 7 \, a^{2} b^{5} - b^{7}\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}} + \frac {2 \, {\left (36 \, a b^{5} \sin \left (d x + c\right )^{4} + 3 \, a^{6} \sin \left (d x + c\right )^{3} - 15 \, a^{4} b^{2} \sin \left (d x + c\right )^{3} + 5 \, a^{2} b^{4} \sin \left (d x + c\right )^{3} + 7 \, b^{6} \sin \left (d x + c\right )^{3} + 16 \, a^{3} b^{3} \sin \left (d x + c\right )^{2} - 88 \, a b^{5} \sin \left (d x + c\right )^{2} - 5 \, a^{6} \sin \left (d x + c\right ) + 17 \, a^{4} b^{2} \sin \left (d x + c\right ) - 3 \, a^{2} b^{4} \sin \left (d x + c\right ) - 9 \, b^{6} \sin \left (d x + c\right ) + 4 \, a^{5} b - 24 \, a^{3} b^{3} + 56 \, a b^{5}\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} {\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 331, normalized size = 1.23 \[ \frac {1}{16 d \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {7 b}{16 d \left (a +b \right )^{3} \left (\sin \left (d x +c \right )-1\right )}-\frac {3 a}{16 d \left (a +b \right )^{3} \left (\sin \left (d x +c \right )-1\right )}-\frac {3 \ln \left (\sin \left (d x +c \right )-1\right ) a^{2}}{16 d \left (a +b \right )^{4}}-\frac {3 \ln \left (\sin \left (d x +c \right )-1\right ) a b}{4 d \left (a +b \right )^{4}}-\frac {15 \ln \left (\sin \left (d x +c \right )-1\right ) b^{2}}{16 d \left (a +b \right )^{4}}+\frac {b^{5}}{d \left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )}-\frac {6 b^{5} a \ln \left (a +b \sin \left (d x +c \right )\right )}{d \left (a +b \right )^{4} \left (a -b \right )^{4}}-\frac {1}{16 d \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {7 b}{16 d \left (a -b \right )^{3} \left (1+\sin \left (d x +c \right )\right )}-\frac {3 a}{16 d \left (a -b \right )^{3} \left (1+\sin \left (d x +c \right )\right )}+\frac {3 \ln \left (1+\sin \left (d x +c \right )\right ) a^{2}}{16 d \left (a -b \right )^{4}}-\frac {3 \ln \left (1+\sin \left (d x +c \right )\right ) a b}{4 d \left (a -b \right )^{4}}+\frac {15 \ln \left (1+\sin \left (d x +c \right )\right ) b^{2}}{16 d \left (a -b \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 505, normalized size = 1.88 \[ -\frac {\frac {96 \, a b^{5} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} - \frac {3 \, {\left (a^{2} - 4 \, a b + 5 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} + \frac {3 \, {\left (a^{2} + 4 \, a b + 5 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {2 \, {\left (4 \, a^{4} b - 20 \, a^{2} b^{3} - 8 \, b^{5} + 3 \, {\left (a^{4} b - 4 \, a^{2} b^{3} - 5 \, b^{5}\right )} \sin \left (d x + c\right )^{4} + 3 \, {\left (a^{5} - 4 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \sin \left (d x + c\right )^{3} - {\left (5 \, a^{4} b - 28 \, a^{2} b^{3} - 25 \, b^{5}\right )} \sin \left (d x + c\right )^{2} - {\left (5 \, a^{5} - 16 \, a^{3} b^{2} + 11 \, a b^{4}\right )} \sin \left (d x + c\right )\right )}}{a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6} + {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \sin \left (d x + c\right )^{5} + {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \sin \left (d x + c\right )^{4} - 2 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \sin \left (d x + c\right )^{3} - 2 \, {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \sin \left (d x + c\right )^{2} + {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \sin \left (d x + c\right )}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.94, size = 449, normalized size = 1.67 \[ \frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (\frac {3\,b^2}{8\,{\left (a-b\right )}^4}-\frac {3\,b}{8\,{\left (a-b\right )}^3}+\frac {3}{16\,{\left (a-b\right )}^2}\right )}{d}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (\frac {3\,b}{8\,{\left (a+b\right )}^3}+\frac {3}{16\,{\left (a+b\right )}^2}+\frac {3\,b^2}{8\,{\left (a+b\right )}^4}\right )}{d}+\frac {\frac {-a^4\,b+5\,a^2\,b^3+2\,b^5}{2\,\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {3\,{\sin \left (c+d\,x\right )}^3\,\left (3\,a\,b^2-a^3\right )}{8\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {3\,{\sin \left (c+d\,x\right )}^4\,\left (-a^4\,b+4\,a^2\,b^3+5\,b^5\right )}{8\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}-\frac {\sin \left (c+d\,x\right )\,\left (11\,a\,b^2-5\,a^3\right )}{8\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\sin \left (c+d\,x\right )}^2\,\left (-5\,a^4\,b+28\,a^2\,b^3+25\,b^5\right )}{8\,\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{d\,\left (b\,{\sin \left (c+d\,x\right )}^5+a\,{\sin \left (c+d\,x\right )}^4-2\,b\,{\sin \left (c+d\,x\right )}^3-2\,a\,{\sin \left (c+d\,x\right )}^2+b\,\sin \left (c+d\,x\right )+a\right )}-\frac {6\,a\,b^5\,\ln \left (a+b\,\sin \left (c+d\,x\right )\right )}{d\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )} \]
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 \[ \int \frac {\sec ^{5}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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