Optimal. Leaf size=145 \[ \frac {2 a b}{d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))}+\frac {b}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}-\frac {b \left (3 a^2+b^2\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^3}-\frac {\log (1-\sin (c+d x))}{2 d (a+b)^3}+\frac {\log (\sin (c+d x)+1)}{2 d (a-b)^3} \]
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Rubi [A] time = 0.15, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2668, 710, 801} \[ \frac {2 a b}{d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))}+\frac {b}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}-\frac {b \left (3 a^2+b^2\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^3}-\frac {\log (1-\sin (c+d x))}{2 d (a+b)^3}+\frac {\log (\sin (c+d x)+1)}{2 d (a-b)^3} \]
Antiderivative was successfully verified.
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Rule 710
Rule 801
Rule 2668
Rubi steps
\begin {align*} \int \frac {\sec (c+d x)}{(a+b \sin (c+d x))^3} \, dx &=\frac {b \operatorname {Subst}\left (\int \frac {1}{(a+x)^3 \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {b}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}+\frac {b \operatorname {Subst}\left (\int \frac {a-x}{(a+x)^2 \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{\left (a^2-b^2\right ) d}\\ &=\frac {b}{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}{2 b (a+b)^2 (b-x)}-\frac {2 a}{(a-b) (a+b) (a+x)^2}+\frac {-3 a^2-b^2}{(a-b)^2 (a+b)^2 (a+x)}+\frac {a+b}{2 (a-b)^2 b (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{\left (a^2-b^2\right ) d}\\ &=-\frac {\log (1-\sin (c+d x))}{2 (a+b)^3 d}+\frac {\log (1+\sin (c+d x))}{2 (a-b)^3 d}-\frac {b \left (3 a^2+b^2\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^3 d}+\frac {b}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}+\frac {2 a b}{\left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.59, size = 135, normalized size = 0.93 \[ \frac {b \left (\frac {1}{\left (a^2-b^2\right ) (a+b \sin (c+d x))^2}-\frac {2 \left (3 a^2+b^2\right ) \log (a+b \sin (c+d x))}{(a-b)^3 (a+b)^3}+\frac {4 a}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}-\frac {\log (1-\sin (c+d x))}{b (a+b)^3}+\frac {\log (\sin (c+d x)+1)}{b (a-b)^3}\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 462, normalized size = 3.19 \[ -\frac {5 \, a^{4} b - 6 \, a^{2} b^{3} + b^{5} - 2 \, {\left (3 \, a^{4} b + 4 \, a^{2} b^{3} + b^{5} - {\left (3 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + {\left (a^{5} + 3 \, a^{4} b + 4 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5} - {\left (a^{3} b^{2} + 3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (a^{5} - 3 \, a^{4} b + 4 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5} - {\left (a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 4 \, {\left (a^{3} b^{2} - a b^{4}\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )} d \cos \left (d x + c\right )^{2} - 2 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} d \sin \left (d x + c\right ) - {\left (a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8}\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.69, size = 242, normalized size = 1.67 \[ -\frac {\frac {2 \, {\left (3 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}} - \frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac {\log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac {9 \, a^{2} b^{3} \sin \left (d x + c\right )^{2} + 3 \, b^{5} \sin \left (d x + c\right )^{2} + 22 \, a^{3} b^{2} \sin \left (d x + c\right ) + 2 \, a b^{4} \sin \left (d x + c\right ) + 14 \, a^{4} b - 3 \, a^{2} b^{3} + b^{5}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 166, normalized size = 1.14 \[ -\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{2 d \left (a +b \right )^{3}}+\frac {b}{2 d \left (a +b \right ) \left (a -b \right ) \left (a +b \sin \left (d x +c \right )\right )^{2}}+\frac {2 a b}{d \left (a +b \right )^{2} \left (a -b \right )^{2} \left (a +b \sin \left (d x +c \right )\right )}-\frac {3 b \ln \left (a +b \sin \left (d x +c \right )\right ) a^{2}}{d \left (a +b \right )^{3} \left (a -b \right )^{3}}-\frac {b^{3} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \left (a +b \right )^{3} \left (a -b \right )^{3}}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{2 \left (a -b \right )^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 223, normalized size = 1.54 \[ -\frac {\frac {2 \, {\left (3 \, a^{2} b + b^{3}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} - \frac {4 \, a b^{2} \sin \left (d x + c\right ) + 5 \, a^{2} b - b^{3}}{a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4} + {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \sin \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \sin \left (d x + c\right )} - \frac {\log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac {\log \left (\sin \left (d x + c\right ) - 1\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.40, size = 169, normalized size = 1.17 \[ \frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (\frac {1}{2\,{\left (a+b\right )}^3}-\frac {1}{2\,{\left (a-b\right )}^3}\right )}{d}+\frac {\frac {5\,a^2\,b-b^3}{2\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {2\,a\,b^2\,\sin \left (c+d\,x\right )}{a^4-2\,a^2\,b^2+b^4}}{d\,\left (a^2+2\,a\,b\,\sin \left (c+d\,x\right )+b^2\,{\sin \left (c+d\,x\right )}^2\right )}+\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )}{2\,d\,{\left (a-b\right )}^3}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )}{2\,d\,{\left (a+b\right )}^3} \]
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 \[ \int \frac {\sec {\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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