Optimal. Leaf size=269 \[ -\frac{3 b \left (-4 a^2 b^2+a^4-5 b^4\right )}{8 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}-\frac{6 a b^5 \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^4}-\frac{3 \left (a^2+4 a b+5 b^2\right ) \log (1-\sin (c+d x))}{16 d (a+b)^4}+\frac{3 \left (a^2-4 a b+5 b^2\right ) \log (\sin (c+d x)+1)}{16 d (a-b)^4}+\frac{\sec ^2(c+d x) \left (3 a \left (a^2-3 b^2\right ) \sin (c+d x)+b \left (a^2+5 b^2\right )\right )}{8 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))} \]
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Rubi [A] time = 0.321401, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, 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 2668
Rule 741
Rule 823
Rule 801
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*}
Mathematica [A] time = 6.11299, size = 406, normalized size = 1.51 \[ \frac{b^5 \left (\frac{\sec ^4(c+d x) \left (b^2-a b \sin (c+d x)\right )}{4 b^6 \left (b^2-a^2\right ) (a+b \sin (c+d x))}-\frac{\frac{\left (6 a^2 \left (a^2-3 b^2\right )-3 \left (-2 a^2 b^2+a^4+5 b^4\right )\right ) \left (\frac{1}{\left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac{\log (1-\sin (c+d x))}{2 b (a+b)^2}+\frac{\log (\sin (c+d x)+1)}{2 b (a-b)^2}-\frac{2 a \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}\right )-6 a \left (a^2-3 b^2\right ) \left (-\frac{\log (a+b \sin (c+d x))}{a^2-b^2}-\frac{\log (1-\sin (c+d x))}{2 b (a+b)}+\frac{\log (\sin (c+d x)+1)}{2 b (a-b)}\right )}{2 b^2 \left (b^2-a^2\right )}-\frac{\sec ^2(c+d x) \left (-b \left (4 a b^2-a \left (3 a^2-5 b^2\right )\right ) \sin (c+d x)+4 a^2 b^2-b^2 \left (3 a^2-5 b^2\right )\right )}{2 b^4 \left (b^2-a^2\right ) (a+b \sin (c+d x))}}{4 b^2 \left (b^2-a^2\right )}\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.118, size = 331, normalized size = 1.2 \begin{align*}{\frac{{b}^{5}}{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3} \left ( a+b\sin \left ( dx+c \right ) \right ) }}-6\,{\frac{{b}^{5}a\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4}}}+{\frac{1}{16\,d \left ( a+b \right ) ^{2} \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}-{\frac{7\,b}{16\,d \left ( a+b \right ) ^{3} \left ( \sin \left ( dx+c \right ) -1 \right ) }}-{\frac{3\,a}{16\,d \left ( a+b \right ) ^{3} \left ( \sin \left ( dx+c \right ) -1 \right ) }}-{\frac{3\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ){a}^{2}}{16\,d \left ( a+b \right ) ^{4}}}-{\frac{3\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ) ab}{4\,d \left ( a+b \right ) ^{4}}}-{\frac{15\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ){b}^{2}}{16\,d \left ( a+b \right ) ^{4}}}-{\frac{1}{16\,d \left ( a-b \right ) ^{2} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{7\,b}{16\,d \left ( a-b \right ) ^{3} \left ( 1+\sin \left ( dx+c \right ) \right ) }}-{\frac{3\,a}{16\,d \left ( a-b \right ) ^{3} \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\frac{3\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ){a}^{2}}{16\,d \left ( a-b \right ) ^{4}}}-{\frac{3\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ) ab}{4\,d \left ( a-b \right ) ^{4}}}+{\frac{15\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ){b}^{2}}{16\,d \left ( a-b \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03183, size = 682, normalized size = 2.54 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 5.57136, size = 1195, normalized size = 4.44 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18301, size = 621, normalized size = 2.31 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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