Optimal. Leaf size=190 \[ -\frac{a^4 b \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^3}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left (a^2-b^2\right )}+\frac{\sec ^2(c+d x) \left (4 b \left (2 a^2-b^2\right )-a \left (5 a^2-b^2\right ) \sin (c+d x)\right )}{8 d \left (a^2-b^2\right )^2}-\frac{a (3 a+b) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac{a (3 a-b) \log (\sin (c+d x)+1)}{16 d (a-b)^3} \]
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Rubi [A] time = 0.431329, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {2837, 12, 1647, 801} \[ -\frac{a^4 b \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^3}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left (a^2-b^2\right )}+\frac{\sec ^2(c+d x) \left (4 b \left (2 a^2-b^2\right )-a \left (5 a^2-b^2\right ) \sin (c+d x)\right )}{8 d \left (a^2-b^2\right )^2}-\frac{a (3 a+b) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac{a (3 a-b) \log (\sin (c+d x)+1)}{16 d (a-b)^3} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 12
Rule 1647
Rule 801
Rubi steps
\begin{align*} \int \frac{\sec (c+d x) \tan ^4(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{b^5 \operatorname{Subst}\left (\int \frac{x^4}{b^4 (a+x) \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{b \operatorname{Subst}\left (\int \frac{x^4}{(a+x) \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac{\sec ^4(c+d x) \left (\frac{b}{a^2-b^2}-\frac{a \sin (c+d x)}{a^2-b^2}\right )}{4 d}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{a^2 b^4}{a^2-b^2}+\frac{3 a b^4 x}{a^2-b^2}-4 b^2 x^2}{(a+x) \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 b d}\\ &=-\frac{\sec ^4(c+d x) \left (\frac{b}{a^2-b^2}-\frac{a \sin (c+d x)}{a^2-b^2}\right )}{4 d}+\frac{\sec ^2(c+d x) \left (4 b \left (2 a^2-b^2\right )-a \left (5 a^2-b^2\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d}+\frac{\operatorname{Subst}\left (\int \frac{\frac{a^2 b^4 \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^2}-\frac{a b^4 \left (5 a^2-b^2\right ) x}{\left (a^2-b^2\right )^2}}{(a+x) \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{8 b^3 d}\\ &=-\frac{\sec ^4(c+d x) \left (\frac{b}{a^2-b^2}-\frac{a \sin (c+d x)}{a^2-b^2}\right )}{4 d}+\frac{\sec ^2(c+d x) \left (4 b \left (2 a^2-b^2\right )-a \left (5 a^2-b^2\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d}+\frac{\operatorname{Subst}\left (\int \left (\frac{a b^3 (3 a+b)}{2 (a+b)^3 (b-x)}-\frac{8 a^4 b^4}{(a-b)^3 (a+b)^3 (a+x)}+\frac{a (3 a-b) b^3}{2 (a-b)^3 (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{8 b^3 d}\\ &=-\frac{a (3 a+b) \log (1-\sin (c+d x))}{16 (a+b)^3 d}+\frac{a (3 a-b) \log (1+\sin (c+d x))}{16 (a-b)^3 d}-\frac{a^4 b \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^3 d}-\frac{\sec ^4(c+d x) \left (\frac{b}{a^2-b^2}-\frac{a \sin (c+d x)}{a^2-b^2}\right )}{4 d}+\frac{\sec ^2(c+d x) \left (4 b \left (2 a^2-b^2\right )-a \left (5 a^2-b^2\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d}\\ \end{align*}
Mathematica [A] time = 1.59681, size = 169, normalized size = 0.89 \[ \frac{-\frac{16 a^4 b \log (a+b \sin (c+d x))}{(a-b)^3 (a+b)^3}+\frac{5 a+3 b}{(a+b)^2 (\sin (c+d x)-1)}+\frac{5 a-3 b}{(a-b)^2 (\sin (c+d x)+1)}+\frac{1}{(a+b) (\sin (c+d x)-1)^2}-\frac{1}{(a-b) (\sin (c+d x)+1)^2}-\frac{a (3 a+b) \log (1-\sin (c+d x))}{(a+b)^3}+\frac{a (3 a-b) \log (\sin (c+d x)+1)}{(a-b)^3}}{16 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.086, size = 260, normalized size = 1.4 \begin{align*} -{\frac{{a}^{4}b\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3}}}+{\frac{1}{2\,d \left ( 8\,a+8\,b \right ) \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}+{\frac{5\,a}{16\,d \left ( a+b \right ) ^{2} \left ( \sin \left ( dx+c \right ) -1 \right ) }}+{\frac{3\,b}{16\,d \left ( a+b \right ) ^{2} \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 ) ^{3}}}-{\frac{\ln \left ( \sin \left ( dx+c \right ) -1 \right ) ab}{16\,d \left ( a+b \right ) ^{3}}}-{\frac{1}{2\,d \left ( 8\,a-8\,b \right ) \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{5\,a}{16\,d \left ( a-b \right ) ^{2} \left ( 1+\sin \left ( dx+c \right ) \right ) }}-{\frac{3\,b}{16\,d \left ( a-b \right ) ^{2} \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 ) ^{3}}}-{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) ab}{16\,d \left ( a-b \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00613, size = 373, normalized size = 1.96 \begin{align*} -\frac{\frac{16 \, a^{4} b \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{{\left (3 \, a^{2} - a b\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac{{\left (3 \, a^{2} + a b\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac{2 \,{\left ({\left (5 \, a^{3} - a b^{2}\right )} \sin \left (d x + c\right )^{3} + 6 \, a^{2} b - 2 \, b^{3} - 4 \,{\left (2 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )^{2} -{\left (3 \, a^{3} + a b^{2}\right )} \sin \left (d x + c\right )\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{4} + a^{4} - 2 \, a^{2} b^{2} + b^{4} - 2 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{2}}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.35374, size = 589, normalized size = 3.1 \begin{align*} -\frac{16 \, a^{4} b \cos \left (d x + c\right )^{4} \log \left (b \sin \left (d x + c\right ) + a\right ) -{\left (3 \, a^{5} + 8 \, a^{4} b + 6 \, a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) +{\left (3 \, a^{5} - 8 \, a^{4} b + 6 \, a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 4 \, a^{4} b - 8 \, a^{2} b^{3} + 4 \, b^{5} - 8 \,{\left (2 \, a^{4} b - 3 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2} - 2 \,{\left (2 \, a^{5} - 4 \, a^{3} b^{2} + 2 \, a b^{4} -{\left (5 \, a^{5} - 6 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \,{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} d \cos \left (d x + c\right )^{4}} \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 = 1.27666, size = 450, normalized size = 2.37 \begin{align*} -\frac{\frac{16 \, a^{4} b^{2} \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{{\left (3 \, a^{2} - a b\right )} \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{{\left (3 \, a^{2} + a b\right )} \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{2 \,{\left (6 \, a^{4} b \sin \left (d x + c\right )^{4} - 5 \, a^{5} \sin \left (d x + c\right )^{3} + 6 \, a^{3} b^{2} \sin \left (d x + c\right )^{3} - a b^{4} \sin \left (d x + c\right )^{3} - 4 \, a^{4} b \sin \left (d x + c\right )^{2} - 12 \, a^{2} b^{3} \sin \left (d x + c\right )^{2} + 4 \, b^{5} \sin \left (d x + c\right )^{2} + 3 \, a^{5} \sin \left (d x + c\right ) - 2 \, a^{3} b^{2} \sin \left (d x + c\right ) - a b^{4} \sin \left (d x + c\right ) + 8 \, a^{2} b^{3} - 2 \, b^{5}\right )}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\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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