Optimal. Leaf size=328 \[ -\frac{3 a b \left (-6 a^2 b^2+a^4-27 b^4\right )}{8 d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))}-\frac{3 b \left (-5 a^2 b^2+a^4-4 b^4\right )}{8 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^2}-\frac{3 b^5 \left (7 a^2+b^2\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^5}-\frac{3 \left (a^2+5 a b+8 b^2\right ) \log (1-\sin (c+d x))}{16 d (a+b)^5}+\frac{3 \left (a^2-5 a b+8 b^2\right ) \log (\sin (c+d x)+1)}{16 d (a-b)^5}+\frac{\sec ^2(c+d x) \left (a \left (3 a^2-11 b^2\right ) \sin (c+d x)+2 b \left (a^2+3 b^2\right )\right )}{8 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2} \]
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Rubi [A] time = 0.420186, antiderivative size = 328, 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} \[ -\frac{3 a b \left (-6 a^2 b^2+a^4-27 b^4\right )}{8 d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))}-\frac{3 b \left (-5 a^2 b^2+a^4-4 b^4\right )}{8 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^2}-\frac{3 b^5 \left (7 a^2+b^2\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^5}-\frac{3 \left (a^2+5 a b+8 b^2\right ) \log (1-\sin (c+d x))}{16 d (a+b)^5}+\frac{3 \left (a^2-5 a b+8 b^2\right ) \log (\sin (c+d x)+1)}{16 d (a-b)^5}+\frac{\sec ^2(c+d x) \left (a \left (3 a^2-11 b^2\right ) \sin (c+d x)+2 b \left (a^2+3 b^2\right )\right )}{8 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 741
Rule 823
Rule 801
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*}
Mathematica [A] time = 2.6111, size = 388, normalized size = 1.18 \[ \frac{-\frac{b \left (3 \left (-5 a^2 b^2+a^4-4 b^4\right ) \left (\frac{1}{\left (a^2-b^2\right ) (a+b \sin (c+d x))^2}-\frac{2 \left (3 a^2+b^2\right ) \log (a+b \sin (c+d x))}{(a-b)^3 (a+b)^3}+\frac{4 a}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}-\frac{\log (1-\sin (c+d x))}{b (a+b)^3}+\frac{\log (\sin (c+d x)+1)}{b (a-b)^3}\right )-3 a \left (3 a^2-11 b^2\right ) \left (\frac{1}{\left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac{\log (1-\sin (c+d x))}{2 b (a+b)^2}+\frac{\log (\sin (c+d x)+1)}{2 b (a-b)^2}-\frac{2 a \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}\right )\right )}{b^2-a^2}+\frac{\sec ^2(c+d x) \left (a \left (3 a^2-11 b^2\right ) \sin (c+d x)+2 b \left (a^2+3 b^2\right )\right )}{\left (b^2-a^2\right ) (a+b \sin (c+d x))^2}+\frac{2 \sec ^4(c+d x) (b-a \sin (c+d x))}{(a+b \sin (c+d x))^2}}{8 d \left (b^2-a^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.142, size = 398, normalized size = 1.2 \begin{align*}{\frac{{b}^{5}}{2\,d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{2}}}+6\,{\frac{{b}^{5}a}{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4} \left ( a+b\sin \left ( dx+c \right ) \right ) }}-21\,{\frac{{b}^{5}\ln \left ( a+b\sin \left ( dx+c \right ) \right ){a}^{2}}{d \left ( a+b \right ) ^{5} \left ( a-b \right ) ^{5}}}-3\,{\frac{{b}^{7}\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{5} \left ( a-b \right ) ^{5}}}+{\frac{1}{16\,d \left ( a+b \right ) ^{3} \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}-{\frac{3\,a}{16\,d \left ( a+b \right ) ^{4} \left ( \sin \left ( dx+c \right ) -1 \right ) }}-{\frac{9\,b}{16\,d \left ( a+b \right ) ^{4} \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 ) ^{5}}}-{\frac{15\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ) ab}{16\,d \left ( a+b \right ) ^{5}}}-{\frac{3\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ){b}^{2}}{2\,d \left ( a+b \right ) ^{5}}}-{\frac{1}{16\,d \left ( a-b \right ) ^{3} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\,a}{16\,d \left ( a-b \right ) ^{4} \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\frac{9\,b}{16\,d \left ( a-b \right ) ^{4} \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 ) ^{5}}}-{\frac{15\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ) ab}{16\,d \left ( a-b \right ) ^{5}}}+{\frac{3\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ){b}^{2}}{2\,d \left ( a-b \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04404, size = 979, normalized size = 2.98 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 9.12643, size = 1995, normalized size = 6.08 \begin{align*} \text{result too large to display} \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 = 1.20583, size = 776, normalized size = 2.37 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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