Optimal. Leaf size=328 \[ -\frac {3 \left (a^2+5 a b+8 b^2\right ) \log (1-\sin (c+d x))}{16 d (a+b)^5}+\frac {3 \left (a^2-5 a b+8 b^2\right ) \log (\sin (c+d x)+1)}{16 d (a-b)^5}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}+\frac {\sec ^2(c+d x) \left (a \left (3 a^2-11 b^2\right ) \sin (c+d x)+2 b \left (a^2+3 b^2\right )\right )}{8 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}-\frac {3 b^5 \left (7 a^2+b^2\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^5}-\frac {3 a b \left (a^4-6 a^2 b^2-27 b^4\right )}{8 d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))}-\frac {3 b \left (a^4-5 a^2 b^2-4 b^4\right )}{8 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^2} \]
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Rubi [A] time = 0.42, antiderivative size = 328, 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} \[ -\frac {3 a b \left (-6 a^2 b^2+a^4-27 b^4\right )}{8 d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))}-\frac {3 b \left (-5 a^2 b^2+a^4-4 b^4\right )}{8 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^2}-\frac {3 b^5 \left (7 a^2+b^2\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^5}-\frac {3 \left (a^2+5 a b+8 b^2\right ) \log (1-\sin (c+d x))}{16 d (a+b)^5}+\frac {3 \left (a^2-5 a b+8 b^2\right ) \log (\sin (c+d x)+1)}{16 d (a-b)^5}+\frac {\sec ^2(c+d x) \left (a \left (3 a^2-11 b^2\right ) \sin (c+d x)+2 b \left (a^2+3 b^2\right )\right )}{8 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2} \]
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))^3} \, dx &=\frac {b^5 \operatorname {Subst}\left (\int \frac {1}{(a+x)^3 \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))^2}+\frac {b^3 \operatorname {Subst}\left (\int \frac {3 \left (a^2-2 b^2\right )+5 a x}{(a+x)^3 \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))^2}+\frac {\sec ^2(c+d x) \left (2 b \left (a^2+3 b^2\right )+a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}-\frac {b \operatorname {Subst}\left (\int \frac {-3 \left (a^4-a^2 b^2+8 b^4\right )-3 a \left (3 a^2-11 b^2\right ) x}{(a+x)^3 \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))^2}+\frac {\sec ^2(c+d x) \left (2 b \left (a^2+3 b^2\right )+a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}-\frac {b \operatorname {Subst}\left (\int \left (-\frac {3 (a-b)^2 \left (a^2+5 a b+8 b^2\right )}{2 b (a+b)^3 (b-x)}-\frac {6 \left (a^4-5 a^2 b^2-4 b^4\right )}{\left (a^2-b^2\right ) (a+x)^3}-\frac {3 \left (a^5-6 a^3 b^2-27 a b^4\right )}{\left (a^2-b^2\right )^2 (a+x)^2}+\frac {24 \left (7 a^2 b^4+b^6\right )}{\left (a^2-b^2\right )^3 (a+x)}-\frac {3 (a+b)^2 \left (a^2-5 a b+8 b^2\right )}{2 (a-b)^3 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+5 a b+8 b^2\right ) \log (1-\sin (c+d x))}{16 (a+b)^5 d}+\frac {3 \left (a^2-5 a b+8 b^2\right ) \log (1+\sin (c+d x))}{16 (a-b)^5 d}-\frac {3 b^5 \left (7 a^2+b^2\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^5 d}-\frac {3 b \left (a^4-5 a^2 b^2-4 b^4\right )}{8 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^2}-\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))^2}-\frac {3 a b \left (a^4-6 a^2 b^2-27 b^4\right )}{8 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))}+\frac {\sec ^2(c+d x) \left (2 b \left (a^2+3 b^2\right )+a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}\\ \end {align*}
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Mathematica [A] time = 2.74, size = 388, normalized size = 1.18 \[ \frac {\frac {\sec ^2(c+d x) \left (a \left (3 a^2-11 b^2\right ) \sin (c+d x)+2 b \left (a^2+3 b^2\right )\right )}{\left (b^2-a^2\right ) (a+b \sin (c+d x))^2}-\frac {b \left (3 \left (a^4-5 a^2 b^2-4 b^4\right ) \left (\frac {1}{\left (a^2-b^2\right ) (a+b \sin (c+d x))^2}-\frac {2 \left (3 a^2+b^2\right ) \log (a+b \sin (c+d x))}{(a-b)^3 (a+b)^3}+\frac {4 a}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}-\frac {\log (1-\sin (c+d x))}{b (a+b)^3}+\frac {\log (\sin (c+d x)+1)}{b (a-b)^3}\right )-3 a \left (3 a^2-11 b^2\right ) \left (\frac {1}{\left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac {\log (1-\sin (c+d x))}{2 b (a+b)^2}+\frac {\log (\sin (c+d x)+1)}{2 b (a-b)^2}-\frac {2 a \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}\right )\right )}{b^2-a^2}+\frac {2 \sec ^4(c+d x) (b-a \sin (c+d x))}{(a+b \sin (c+d x))^2}}{8 d \left (b^2-a^2\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.44, size = 895, normalized size = 2.73 \[ \frac {4 \, a^{8} b - 16 \, a^{6} b^{3} + 24 \, a^{4} b^{5} - 16 \, a^{2} b^{7} + 4 \, b^{9} + 12 \, {\left (a^{8} b - 7 \, a^{6} b^{3} - 7 \, a^{4} b^{5} + 15 \, a^{2} b^{7} - 2 \, b^{9}\right )} \cos \left (d x + c\right )^{4} - 4 \, {\left (a^{8} b - 6 \, a^{4} b^{5} + 8 \, a^{2} b^{7} - 3 \, b^{9}\right )} \cos \left (d x + c\right )^{2} - 48 \, {\left ({\left (7 \, a^{2} b^{7} + b^{9}\right )} \cos \left (d x + c\right )^{6} - 2 \, {\left (7 \, a^{3} b^{6} + a b^{8}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - {\left (7 \, a^{4} b^{5} + 8 \, a^{2} b^{7} + b^{9}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + 3 \, {\left ({\left (a^{7} b^{2} - 7 \, a^{5} b^{4} + 35 \, a^{3} b^{6} + 56 \, a^{2} b^{7} + 35 \, a b^{8} + 8 \, b^{9}\right )} \cos \left (d x + c\right )^{6} - 2 \, {\left (a^{8} b - 7 \, a^{6} b^{3} + 35 \, a^{4} b^{5} + 56 \, a^{3} b^{6} + 35 \, a^{2} b^{7} + 8 \, a b^{8}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - {\left (a^{9} - 6 \, a^{7} b^{2} + 28 \, a^{5} b^{4} + 56 \, a^{4} b^{5} + 70 \, a^{3} b^{6} + 64 \, a^{2} b^{7} + 35 \, a b^{8} + 8 \, b^{9}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left ({\left (a^{7} b^{2} - 7 \, a^{5} b^{4} + 35 \, a^{3} b^{6} - 56 \, a^{2} b^{7} + 35 \, a b^{8} - 8 \, b^{9}\right )} \cos \left (d x + c\right )^{6} - 2 \, {\left (a^{8} b - 7 \, a^{6} b^{3} + 35 \, a^{4} b^{5} - 56 \, a^{3} b^{6} + 35 \, a^{2} b^{7} - 8 \, a b^{8}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - {\left (a^{9} - 6 \, a^{7} b^{2} + 28 \, a^{5} b^{4} - 56 \, a^{4} b^{5} + 70 \, a^{3} b^{6} - 64 \, a^{2} b^{7} + 35 \, a b^{8} - 8 \, b^{9}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (2 \, a^{9} - 8 \, a^{7} b^{2} + 12 \, a^{5} b^{4} - 8 \, a^{3} b^{6} + 2 \, a b^{8} - 3 \, {\left (a^{7} b^{2} - 7 \, a^{5} b^{4} - 21 \, a^{3} b^{6} + 27 \, a b^{8}\right )} \cos \left (d x + c\right )^{4} + {\left (3 \, a^{9} - 20 \, a^{7} b^{2} + 42 \, a^{5} b^{4} - 36 \, a^{3} b^{6} + 11 \, a b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, {\left ({\left (a^{10} b^{2} - 5 \, a^{8} b^{4} + 10 \, a^{6} b^{6} - 10 \, a^{4} b^{8} + 5 \, a^{2} b^{10} - b^{12}\right )} d \cos \left (d x + c\right )^{6} - 2 \, {\left (a^{11} b - 5 \, a^{9} b^{3} + 10 \, a^{7} b^{5} - 10 \, a^{5} b^{7} + 5 \, a^{3} b^{9} - a b^{11}\right )} d \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - {\left (a^{12} - 4 \, a^{10} b^{2} + 5 \, a^{8} b^{4} - 5 \, a^{4} b^{8} + 4 \, a^{2} b^{10} - b^{12}\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 = 0.73, size = 575, normalized size = 1.75 \[ -\frac {\frac {48 \, {\left (7 \, a^{2} b^{6} + b^{8}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{10} b - 5 \, a^{8} b^{3} + 10 \, a^{6} b^{5} - 10 \, a^{4} b^{7} + 5 \, a^{2} b^{9} - b^{11}} - \frac {3 \, {\left (a^{2} - 5 \, a b + 8 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}} + \frac {3 \, {\left (a^{2} + 5 \, a b + 8 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}} + \frac {2 \, {\left (3 \, a^{5} b^{2} \sin \left (d x + c\right )^{5} - 18 \, a^{3} b^{4} \sin \left (d x + c\right )^{5} - 81 \, a b^{6} \sin \left (d x + c\right )^{5} + 6 \, a^{6} b \sin \left (d x + c\right )^{4} - 36 \, a^{4} b^{3} \sin \left (d x + c\right )^{4} - 78 \, a^{2} b^{5} \sin \left (d x + c\right )^{4} + 12 \, b^{7} \sin \left (d x + c\right )^{4} + 3 \, a^{7} \sin \left (d x + c\right )^{3} - 23 \, a^{5} b^{2} \sin \left (d x + c\right )^{3} + 61 \, a^{3} b^{4} \sin \left (d x + c\right )^{3} + 151 \, a b^{6} \sin \left (d x + c\right )^{3} - 10 \, a^{6} b \sin \left (d x + c\right )^{2} + 74 \, a^{4} b^{3} \sin \left (d x + c\right )^{2} + 146 \, a^{2} b^{5} \sin \left (d x + c\right )^{2} - 18 \, b^{7} \sin \left (d x + c\right )^{2} - 5 \, a^{7} \sin \left (d x + c\right ) + 26 \, a^{5} b^{2} \sin \left (d x + c\right ) - 49 \, a^{3} b^{4} \sin \left (d x + c\right ) - 68 \, a b^{6} \sin \left (d x + c\right ) + 6 \, a^{6} b - 44 \, a^{4} b^{3} - 62 \, a^{2} b^{5} + 4 \, b^{7}\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} {\left (b \sin \left (d x + c\right )^{3} + a \sin \left (d x + c\right )^{2} - b \sin \left (d x + c\right ) - a\right )}^{2}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 398, normalized size = 1.21 \[ \frac {1}{16 d \left (a +b \right )^{3} \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {3 a}{16 d \left (a +b \right )^{4} \left (\sin \left (d x +c \right )-1\right )}-\frac {9 b}{16 d \left (a +b \right )^{4} \left (\sin \left (d x +c \right )-1\right )}-\frac {3 \ln \left (\sin \left (d x +c \right )-1\right ) a^{2}}{16 d \left (a +b \right )^{5}}-\frac {15 \ln \left (\sin \left (d x +c \right )-1\right ) a b}{16 d \left (a +b \right )^{5}}-\frac {3 \ln \left (\sin \left (d x +c \right )-1\right ) b^{2}}{2 d \left (a +b \right )^{5}}+\frac {b^{5}}{2 d \left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )^{2}}+\frac {6 b^{5} a}{d \left (a +b \right )^{4} \left (a -b \right )^{4} \left (a +b \sin \left (d x +c \right )\right )}-\frac {21 b^{5} \ln \left (a +b \sin \left (d x +c \right )\right ) a^{2}}{d \left (a +b \right )^{5} \left (a -b \right )^{5}}-\frac {3 b^{7} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \left (a +b \right )^{5} \left (a -b \right )^{5}}-\frac {1}{16 d \left (a -b \right )^{3} \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {3 a}{16 d \left (a -b \right )^{4} \left (1+\sin \left (d x +c \right )\right )}+\frac {9 b}{16 d \left (a -b \right )^{4} \left (1+\sin \left (d x +c \right )\right )}+\frac {3 \ln \left (1+\sin \left (d x +c \right )\right ) a^{2}}{16 d \left (a -b \right )^{5}}-\frac {15 \ln \left (1+\sin \left (d x +c \right )\right ) a b}{16 d \left (a -b \right )^{5}}+\frac {3 \ln \left (1+\sin \left (d x +c \right )\right ) b^{2}}{2 d \left (a -b \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.37, size = 725, normalized size = 2.21 \[ -\frac {\frac {48 \, {\left (7 \, a^{2} b^{5} + b^{7}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{10} - 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} - 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} - b^{10}} - \frac {3 \, {\left (a^{2} - 5 \, a b + 8 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}} + \frac {3 \, {\left (a^{2} + 5 \, a b + 8 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}} + \frac {2 \, {\left (6 \, a^{6} b - 44 \, a^{4} b^{3} - 62 \, a^{2} b^{5} + 4 \, b^{7} + 3 \, {\left (a^{5} b^{2} - 6 \, a^{3} b^{4} - 27 \, a b^{6}\right )} \sin \left (d x + c\right )^{5} + 6 \, {\left (a^{6} b - 6 \, a^{4} b^{3} - 13 \, a^{2} b^{5} + 2 \, b^{7}\right )} \sin \left (d x + c\right )^{4} + {\left (3 \, a^{7} - 23 \, a^{5} b^{2} + 61 \, a^{3} b^{4} + 151 \, a b^{6}\right )} \sin \left (d x + c\right )^{3} - 2 \, {\left (5 \, a^{6} b - 37 \, a^{4} b^{3} - 73 \, a^{2} b^{5} + 9 \, b^{7}\right )} \sin \left (d x + c\right )^{2} - {\left (5 \, a^{7} - 26 \, a^{5} b^{2} + 49 \, a^{3} b^{4} + 68 \, a b^{6}\right )} \sin \left (d x + c\right )\right )}}{a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8} + {\left (a^{8} b^{2} - 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} - 4 \, a^{2} b^{8} + b^{10}\right )} \sin \left (d x + c\right )^{6} + 2 \, {\left (a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right )} \sin \left (d x + c\right )^{5} + {\left (a^{10} - 6 \, a^{8} b^{2} + 14 \, a^{6} b^{4} - 16 \, a^{4} b^{6} + 9 \, a^{2} b^{8} - 2 \, b^{10}\right )} \sin \left (d x + c\right )^{4} - 4 \, {\left (a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right )} \sin \left (d x + c\right )^{3} - {\left (2 \, a^{10} - 9 \, a^{8} b^{2} + 16 \, a^{6} b^{4} - 14 \, a^{4} b^{6} + 6 \, a^{2} b^{8} - b^{10}\right )} \sin \left (d x + c\right )^{2} + 2 \, {\left (a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\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 = 6.56, size = 688, normalized size = 2.10 \[ \frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (\frac {3\,b^2}{4\,{\left (a-b\right )}^5}-\frac {9\,b}{16\,{\left (a-b\right )}^4}+\frac {3}{16\,{\left (a-b\right )}^3}\right )}{d}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (\frac {9\,b}{16\,{\left (a+b\right )}^4}+\frac {3}{16\,{\left (a+b\right )}^3}+\frac {3\,b^2}{4\,{\left (a+b\right )}^5}\right )}{d}-\frac {\frac {3\,a^6\,b-22\,a^4\,b^3-31\,a^2\,b^5+2\,b^7}{4\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}-\frac {\sin \left (c+d\,x\right )\,\left (5\,a^7-26\,a^5\,b^2+49\,a^3\,b^4+68\,a\,b^6\right )}{8\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}-\frac {3\,{\sin \left (c+d\,x\right )}^5\,\left (-a^5\,b^2+6\,a^3\,b^4+27\,a\,b^6\right )}{8\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}+\frac {{\sin \left (c+d\,x\right )}^3\,\left (3\,a^7-23\,a^5\,b^2+61\,a^3\,b^4+151\,a\,b^6\right )}{8\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}+\frac {3\,{\sin \left (c+d\,x\right )}^4\,\left (a^6\,b-6\,a^4\,b^3-13\,a^2\,b^5+2\,b^7\right )}{4\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}-\frac {{\sin \left (c+d\,x\right )}^2\,\left (5\,a^6\,b-37\,a^4\,b^3-73\,a^2\,b^5+9\,b^7\right )}{4\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}}{d\,\left ({\sin \left (c+d\,x\right )}^4\,\left (a^2-2\,b^2\right )+a^2-{\sin \left (c+d\,x\right )}^2\,\left (2\,a^2-b^2\right )+b^2\,{\sin \left (c+d\,x\right )}^6+2\,a\,b\,\sin \left (c+d\,x\right )-4\,a\,b\,{\sin \left (c+d\,x\right )}^3+2\,a\,b\,{\sin \left (c+d\,x\right )}^5\right )}-\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (21\,a^2\,b^5+3\,b^7\right )}{d\,\left (a^{10}-5\,a^8\,b^2+10\,a^6\,b^4-10\,a^4\,b^6+5\,a^2\,b^8-b^{10}\right )} \]
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 \[ \int \frac {\sec ^{5}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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