Optimal. Leaf size=226 \[ -\frac {a b \left (a^2+11 b^2\right )}{2 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}-\frac {b \left (a^2+2 b^2\right )}{2 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}-\frac {\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}+\frac {2 b^3 \left (5 a^2+b^2\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^4}-\frac {(a+4 b) \log (1-\sin (c+d x))}{4 d (a+b)^4}+\frac {(a-4 b) \log (\sin (c+d x)+1)}{4 d (a-b)^4} \]
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Rubi [A] time = 0.28, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2668, 741, 801} \[ -\frac {a b \left (a^2+11 b^2\right )}{2 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}-\frac {b \left (a^2+2 b^2\right )}{2 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}+\frac {2 b^3 \left (5 a^2+b^2\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^4}-\frac {\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}-\frac {(a+4 b) \log (1-\sin (c+d x))}{4 d (a+b)^4}+\frac {(a-4 b) \log (\sin (c+d x)+1)}{4 d (a-b)^4} \]
Antiderivative was successfully verified.
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Rule 741
Rule 801
Rule 2668
Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx &=\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{(a+x)^3 \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac {\sec ^2(c+d x) (b-a \sin (c+d x))}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}+\frac {b \operatorname {Subst}\left (\int \frac {a^2-4 b^2+3 a x}{(a+x)^3 \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{2 \left (a^2-b^2\right ) d}\\ &=-\frac {\sec ^2(c+d x) (b-a \sin (c+d x))}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}+\frac {b \operatorname {Subst}\left (\int \left (\frac {(a-b) (a+4 b)}{2 b (a+b)^3 (b-x)}+\frac {2 \left (a^2+2 b^2\right )}{(a-b) (a+b) (a+x)^3}+\frac {a \left (a^2+11 b^2\right )}{(a-b)^2 (a+b)^2 (a+x)^2}+\frac {4 \left (5 a^2 b^2+b^4\right )}{(a-b)^3 (a+b)^3 (a+x)}+\frac {(a-4 b) (a+b)}{2 (a-b)^3 b (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{2 \left (a^2-b^2\right ) d}\\ &=-\frac {(a+4 b) \log (1-\sin (c+d x))}{4 (a+b)^4 d}+\frac {(a-4 b) \log (1+\sin (c+d x))}{4 (a-b)^4 d}+\frac {2 b^3 \left (5 a^2+b^2\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^4 d}-\frac {b \left (a^2+2 b^2\right )}{2 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}-\frac {\sec ^2(c+d x) (b-a \sin (c+d x))}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}-\frac {a b \left (a^2+11 b^2\right )}{2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 3.95, size = 283, normalized size = 1.25 \[ \frac {b \left (a^2+2 b^2\right ) \left (\frac {1}{\left (a^2-b^2\right ) (a+b \sin (c+d x))^2}-\frac {2 \left (3 a^2+b^2\right ) \log (a+b \sin (c+d x))}{(a-b)^3 (a+b)^3}+\frac {4 a}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}-\frac {\log (1-\sin (c+d x))}{b (a+b)^3}+\frac {\log (\sin (c+d x)+1)}{b (a-b)^3}\right )+\frac {3}{2} a \left (\frac {2 b}{\left (b^2-a^2\right ) (a+b \sin (c+d x))}+\frac {4 a b \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}+\frac {\log (1-\sin (c+d x))}{(a+b)^2}-\frac {\log (\sin (c+d x)+1)}{(a-b)^2}\right )+\frac {\sec ^2(c+d x) (b-a \sin (c+d x))}{(a+b \sin (c+d x))^2}}{2 d \left (b^2-a^2\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 707, normalized size = 3.13 \[ \frac {2 \, a^{6} b - 6 \, a^{4} b^{3} + 6 \, a^{2} b^{5} - 2 \, b^{7} + 4 \, {\left (a^{6} b + 5 \, a^{4} b^{3} - 7 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left ({\left (5 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (5 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - {\left (5 \, a^{4} b^{3} + 6 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + {\left ({\left (a^{5} b^{2} - 10 \, a^{3} b^{4} - 20 \, a^{2} b^{5} - 15 \, a b^{6} - 4 \, b^{7}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{6} b - 10 \, a^{4} b^{3} - 20 \, a^{3} b^{4} - 15 \, a^{2} b^{5} - 4 \, a b^{6}\right )} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - {\left (a^{7} - 9 \, a^{5} b^{2} - 20 \, a^{4} b^{3} - 25 \, a^{3} b^{4} - 24 \, a^{2} b^{5} - 15 \, a b^{6} - 4 \, b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left ({\left (a^{5} b^{2} - 10 \, a^{3} b^{4} + 20 \, a^{2} b^{5} - 15 \, a b^{6} + 4 \, b^{7}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{6} b - 10 \, a^{4} b^{3} + 20 \, a^{3} b^{4} - 15 \, a^{2} b^{5} + 4 \, a b^{6}\right )} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - {\left (a^{7} - 9 \, a^{5} b^{2} + 20 \, a^{4} b^{3} - 25 \, a^{3} b^{4} + 24 \, a^{2} b^{5} - 15 \, a b^{6} + 4 \, b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6} - {\left (a^{5} b^{2} + 10 \, a^{3} b^{4} - 11 \, a b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{8} b^{2} - 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} - 4 \, a^{2} b^{8} + b^{10}\right )} d \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right )} d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - {\left (a^{10} - 3 \, a^{8} b^{2} + 2 \, a^{6} b^{4} + 2 \, a^{4} b^{6} - 3 \, a^{2} b^{8} + b^{10}\right )} d \cos \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.38, size = 413, normalized size = 1.83 \[ \frac {\frac {8 \, {\left (5 \, a^{2} b^{4} + b^{6}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{8} b - 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} - 4 \, a^{2} b^{7} + b^{9}} + \frac {{\left (a - 4 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} - \frac {{\left (a + 4 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {2 \, {\left (10 \, a^{2} b^{3} \sin \left (d x + c\right )^{2} + 2 \, b^{5} \sin \left (d x + c\right )^{2} - a^{5} \sin \left (d x + c\right ) - 2 \, a^{3} b^{2} \sin \left (d x + c\right ) + 3 \, a b^{4} \sin \left (d x + c\right ) + 3 \, a^{4} b - 12 \, a^{2} b^{3} - 3 \, b^{5}\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} {\left (\sin \left (d x + c\right )^{2} - 1\right )}} - \frac {2 \, {\left (30 \, a^{2} b^{5} \sin \left (d x + c\right )^{2} + 6 \, b^{7} \sin \left (d x + c\right )^{2} + 68 \, a^{3} b^{4} \sin \left (d x + c\right ) + 4 \, a b^{6} \sin \left (d x + c\right ) + 39 \, a^{4} b^{3} - 4 \, a^{2} b^{5} + b^{7}\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{2}}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 258, normalized size = 1.14 \[ -\frac {1}{4 d \left (a +b \right )^{3} \left (\sin \left (d x +c \right )-1\right )}-\frac {\ln \left (\sin \left (d x +c \right )-1\right ) a}{4 d \left (a +b \right )^{4}}-\frac {\ln \left (\sin \left (d x +c \right )-1\right ) b}{d \left (a +b \right )^{4}}-\frac {b^{3}}{2 d \left (a +b \right )^{2} \left (a -b \right )^{2} \left (a +b \sin \left (d x +c \right )\right )^{2}}-\frac {4 a \,b^{3}}{d \left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )}+\frac {10 b^{3} \ln \left (a +b \sin \left (d x +c \right )\right ) a^{2}}{d \left (a +b \right )^{4} \left (a -b \right )^{4}}+\frac {2 b^{5} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \left (a +b \right )^{4} \left (a -b \right )^{4}}-\frac {1}{4 d \left (a -b \right )^{3} \left (1+\sin \left (d x +c \right )\right )}+\frac {\ln \left (1+\sin \left (d x +c \right )\right ) a}{4 d \left (a -b \right )^{4}}-\frac {\ln \left (1+\sin \left (d x +c \right )\right ) b}{d \left (a -b \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 438, normalized size = 1.94 \[ \frac {\frac {8 \, {\left (5 \, a^{2} b^{3} + b^{5}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} + \frac {{\left (a - 4 \, b\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} - \frac {{\left (a + 4 \, b\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} - \frac {2 \, {\left (3 \, a^{4} b + 10 \, a^{2} b^{3} - b^{5} - {\left (a^{3} b^{2} + 11 \, a b^{4}\right )} \sin \left (d x + c\right )^{3} - 2 \, {\left (a^{4} b + 6 \, a^{2} b^{3} - b^{5}\right )} \sin \left (d x + c\right )^{2} - {\left (a^{5} - 3 \, a^{3} b^{2} - 10 \, a b^{4}\right )} \sin \left (d x + c\right )\right )}}{a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6} - {\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )} \sin \left (d x + c\right )^{4} - 2 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} \sin \left (d x + c\right )^{3} - {\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \sin \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} \sin \left (d x + c\right )}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.78, size = 388, normalized size = 1.72 \[ \frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (\frac {3\,b}{4\,{\left (a+b\right )}^4}+\frac {1}{4\,{\left (a+b\right )}^3}+\frac {3\,b}{4\,{\left (a-b\right )}^4}-\frac {1}{4\,{\left (a-b\right )}^3}\right )}{d}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (\frac {3\,b}{4\,{\left (a+b\right )}^4}+\frac {1}{4\,{\left (a+b\right )}^3}\right )}{d}+\frac {\frac {3\,a^4\,b+10\,a^2\,b^3-b^5}{2\,\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\sin \left (c+d\,x\right )}^3\,\left (a^3\,b^2+11\,a\,b^4\right )}{2\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}-\frac {{\sin \left (c+d\,x\right )}^2\,\left (a^4\,b+6\,a^2\,b^3-b^5\right )}{\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {a\,\sin \left (c+d\,x\right )\,\left (-a^4+3\,a^2\,b^2+10\,b^4\right )}{2\,\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{d\,\left ({\sin \left (c+d\,x\right )}^2\,\left (a^2-b^2\right )-a^2+b^2\,{\sin \left (c+d\,x\right )}^4-2\,a\,b\,\sin \left (c+d\,x\right )+2\,a\,b\,{\sin \left (c+d\,x\right )}^3\right )}+\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (a-4\,b\right )}{4\,d\,{\left (a-b\right )}^4} \]
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 \[ \int \frac {\sec ^{3}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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