Optimal. Leaf size=211 \[ -\frac {6 a b^2 \left (a^2+b^2\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^4}-\frac {\csc ^2(c+d x) \left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \cos (c+d x)\right )}{2 d \left (a^2-b^2\right )^3}+\frac {b^4}{2 a d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}-\frac {4 a b^3}{d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}-\frac {3 b \log (1-\cos (c+d x))}{4 d (a+b)^4}+\frac {3 b \log (\cos (c+d x)+1)}{4 d (a-b)^4} \]
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Rubi [A] time = 0.67, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {4397, 2837, 12, 1647, 1629} \[ \frac {b^4}{2 a d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}-\frac {4 a b^3}{d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}-\frac {6 a b^2 \left (a^2+b^2\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^4}-\frac {\csc ^2(c+d x) \left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \cos (c+d x)\right )}{2 d \left (a^2-b^2\right )^3}-\frac {3 b \log (1-\cos (c+d x))}{4 d (a+b)^4}+\frac {3 b \log (\cos (c+d x)+1)}{4 d (a-b)^4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 1629
Rule 1647
Rule 2837
Rule 4397
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx &=\int \frac {\cos (c+d x) \cot ^3(c+d x)}{(b+a \cos (c+d x))^3} \, dx\\ &=-\frac {a^3 \operatorname {Subst}\left (\int \frac {x^4}{a^4 (b+x)^3 \left (a^2-x^2\right )^2} \, dx,x,a \cos (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {x^4}{(b+x)^3 \left (a^2-x^2\right )^2} \, dx,x,a \cos (c+d x)\right )}{a d}\\ &=-\frac {\left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^3 d}-\frac {\operatorname {Subst}\left (\int \frac {\frac {a^4 b^4 \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3}+\frac {a^4 b^3 \left (7 a^2-3 b^2\right ) x}{\left (a^2-b^2\right )^3}+\frac {a^2 b^2 \left (3 a^4-9 a^2 b^2+2 b^4\right ) x^2}{\left (a^2-b^2\right )^3}-\frac {a^4 b \left (3 a^2+b^2\right ) x^3}{\left (a^2-b^2\right )^3}}{(b+x)^3 \left (a^2-x^2\right )} \, dx,x,a \cos (c+d x)\right )}{2 a^3 d}\\ &=-\frac {\left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^3 d}-\frac {\operatorname {Subst}\left (\int \left (-\frac {3 a^3 b}{2 (a+b)^4 (a-x)}-\frac {3 a^3 b}{2 (a-b)^4 (a+x)}+\frac {2 a^2 b^4}{\left (a^2-b^2\right )^2 (b+x)^3}-\frac {8 a^4 b^3}{\left (a^2-b^2\right )^3 (b+x)^2}+\frac {12 a^4 b^2 \left (a^2+b^2\right )}{\left (a^2-b^2\right )^4 (b+x)}\right ) \, dx,x,a \cos (c+d x)\right )}{2 a^3 d}\\ &=\frac {b^4}{2 a \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))^2}-\frac {4 a b^3}{\left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}-\frac {\left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^3 d}-\frac {3 b \log (1-\cos (c+d x))}{4 (a+b)^4 d}+\frac {3 b \log (1+\cos (c+d x))}{4 (a-b)^4 d}-\frac {6 a b^2 \left (a^2+b^2\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^4 d}\\ \end {align*}
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Mathematica [A] time = 5.33, size = 184, normalized size = 0.87 \[ \frac {-\frac {48 a b^2 \left (a^2+b^2\right ) \log (a \cos (c+d x)+b)}{\left (a^2-b^2\right )^4}+\frac {4 b^4}{a (a-b)^2 (a+b)^2 (a \cos (c+d x)+b)^2}+\frac {32 a b^3}{(b-a)^3 (a+b)^3 (a \cos (c+d x)+b)}-\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right )}{(a+b)^3}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right )}{(b-a)^3}-\frac {12 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{(a+b)^4}+\frac {12 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{(a-b)^4}}{8 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 994, normalized size = 4.71 \[ \frac {2 \, a^{6} b^{2} + 18 \, a^{4} b^{4} - 18 \, a^{2} b^{6} - 2 \, b^{8} - 6 \, {\left (a^{7} b + 2 \, a^{5} b^{3} - 3 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (a^{8} - 4 \, a^{6} b^{2} - 6 \, a^{4} b^{4} + 8 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (2 \, a^{7} b + 9 \, a^{5} b^{3} - 12 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (d x + c\right ) + 24 \, {\left (a^{4} b^{4} + a^{2} b^{6} - {\left (a^{6} b^{2} + a^{4} b^{4}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{5} b^{3} + a^{3} b^{5}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{6} b^{2} - a^{2} b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{3} + a^{3} b^{5}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right ) - 3 \, {\left (a^{5} b^{3} + 4 \, a^{4} b^{4} + 6 \, a^{3} b^{5} + 4 \, a^{2} b^{6} + a b^{7} - {\left (a^{7} b + 4 \, a^{6} b^{2} + 6 \, a^{5} b^{3} + 4 \, a^{4} b^{4} + a^{3} b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{6} b^{2} + 4 \, a^{5} b^{3} + 6 \, a^{4} b^{4} + 4 \, a^{3} b^{5} + a^{2} b^{6}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{7} b + 4 \, a^{6} b^{2} + 5 \, a^{5} b^{3} - 5 \, a^{3} b^{5} - 4 \, a^{2} b^{6} - a b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{6} b^{2} + 4 \, a^{5} b^{3} + 6 \, a^{4} b^{4} + 4 \, a^{3} b^{5} + a^{2} b^{6}\right )} \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3 \, {\left (a^{5} b^{3} - 4 \, a^{4} b^{4} + 6 \, a^{3} b^{5} - 4 \, a^{2} b^{6} + a b^{7} - {\left (a^{7} b - 4 \, a^{6} b^{2} + 6 \, a^{5} b^{3} - 4 \, a^{4} b^{4} + a^{3} b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{6} b^{2} - 4 \, a^{5} b^{3} + 6 \, a^{4} b^{4} - 4 \, a^{3} b^{5} + a^{2} b^{6}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{7} b - 4 \, a^{6} b^{2} + 5 \, a^{5} b^{3} - 5 \, a^{3} b^{5} + 4 \, a^{2} b^{6} - a b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{6} b^{2} - 4 \, a^{5} b^{3} + 6 \, a^{4} b^{4} - 4 \, a^{3} b^{5} + a^{2} b^{6}\right )} \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{4 \, {\left ({\left (a^{11} - 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} - 4 \, a^{5} b^{6} + a^{3} b^{8}\right )} d \cos \left (d x + c\right )^{4} + 2 \, {\left (a^{10} b - 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} - 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} d \cos \left (d x + c\right )^{3} - {\left (a^{11} - 5 \, a^{9} b^{2} + 10 \, a^{7} b^{4} - 10 \, a^{5} b^{6} + 5 \, a^{3} b^{8} - a b^{10}\right )} d \cos \left (d x + c\right )^{2} - 2 \, {\left (a^{10} b - 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} - 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} d \cos \left (d x + c\right ) - {\left (a^{9} b^{2} - 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} - 4 \, a^{3} b^{8} + a b^{10}\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.44, size = 690, normalized size = 3.27 \[ -\frac {\frac {6 \, b \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \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 {48 \, {\left (a^{3} b^{2} + a b^{4}\right )} \log \left ({\left | -a - b - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} \right |}\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} - \frac {{\left (a + b + \frac {6 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} {\left (\cos \left (d x + c\right ) - 1\right )}} - \frac {\cos \left (d x + c\right ) - 1}{{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} - \frac {8 \, {\left (9 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 2 \, a^{3} b^{4} + 8 \, a^{2} b^{5} + 5 \, a b^{6} - 2 \, b^{7} + \frac {18 \, a^{5} b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {8 \, a^{4} b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {2 \, a^{3} b^{4} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {6 \, a^{2} b^{5} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {16 \, a b^{6} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {2 \, b^{7} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {9 \, a^{5} b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {18 \, a^{4} b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {18 \, a^{3} b^{4} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {18 \, a^{2} b^{5} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {9 \, a b^{6} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} {\left (a + b + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 220, normalized size = 1.04 \[ \frac {b^{4}}{2 d \left (a +b \right )^{2} \left (a -b \right )^{2} a \left (b +a \cos \left (d x +c \right )\right )^{2}}-\frac {4 a \,b^{3}}{d \left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +a \cos \left (d x +c \right )\right )}-\frac {6 a^{3} b^{2} \ln \left (b +a \cos \left (d x +c \right )\right )}{d \left (a +b \right )^{4} \left (a -b \right )^{4}}-\frac {6 b^{4} a \ln \left (b +a \cos \left (d x +c \right )\right )}{d \left (a +b \right )^{4} \left (a -b \right )^{4}}+\frac {1}{4 d \left (a +b \right )^{3} \left (\cos \left (d x +c \right )-1\right )}-\frac {3 \ln \left (\cos \left (d x +c \right )-1\right ) b}{4 d \left (a +b \right )^{4}}-\frac {1}{4 d \left (a -b \right )^{3} \left (1+\cos \left (d x +c \right )\right )}+\frac {3 b \ln \left (1+\cos \left (d x +c \right )\right )}{4 \left (a -b \right )^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.62, size = 593, normalized size = 2.81 \[ -\frac {\frac {48 \, {\left (a^{3} b^{2} + a b^{4}\right )} \log \left (a + b - \frac {{\left (a - b\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} + \frac {12 \, b \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {a^{6} - 2 \, a^{5} b - a^{4} b^{2} + 4 \, a^{3} b^{3} - a^{2} b^{4} - 2 \, a b^{5} + b^{6} - \frac {2 \, {\left (a^{6} - 4 \, a^{5} b + 5 \, a^{4} b^{2} - 32 \, a^{3} b^{3} - 37 \, a^{2} b^{4} - 4 \, a b^{5} - 9 \, b^{6}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {{\left (a^{6} - 6 \, a^{5} b + 15 \, a^{4} b^{2} - 84 \, a^{3} b^{3} + 63 \, a^{2} b^{4} - 6 \, a b^{5} + 17 \, b^{6}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{\frac {{\left (a^{9} + a^{8} b - 4 \, a^{7} b^{2} - 4 \, a^{6} b^{3} + 6 \, a^{5} b^{4} + 6 \, a^{4} b^{5} - 4 \, a^{3} b^{6} - 4 \, a^{2} b^{7} + a b^{8} + b^{9}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {2 \, {\left (a^{9} - a^{8} b - 4 \, a^{7} b^{2} + 4 \, a^{6} b^{3} + 6 \, a^{5} b^{4} - 6 \, a^{4} b^{5} - 4 \, a^{3} b^{6} + 4 \, a^{2} b^{7} + a b^{8} - b^{9}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {{\left (a^{9} - 3 \, a^{8} b + 8 \, a^{6} b^{3} - 6 \, a^{5} b^{4} - 6 \, a^{4} b^{5} + 8 \, a^{3} b^{6} - 3 \, a b^{8} + b^{9}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\sin \left (d x + c\right )^{2}}{{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.11, size = 490, normalized size = 2.32 \[ \frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (a^5-5\,a^4\,b+10\,a^3\,b^2-42\,a^2\,b^3+5\,a\,b^4-9\,b^5\right )}{\left (a-b\right )\,\left (a^2+2\,a\,b+b^2\right )}-\frac {a^3-3\,a^2\,b+3\,a\,b^2-b^3}{2\,\left (a+b\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (-a^5+5\,a^4\,b-10\,a^3\,b^2+74\,a^2\,b^3+11\,a\,b^4+17\,b^5\right )}{2\,\left (a+b\right )\,\left (a^2+2\,a\,b+b^2\right )}}{d\,\left (\left (4\,a^5-20\,a^4\,b+40\,a^3\,b^2-40\,a^2\,b^3+20\,a\,b^4-4\,b^5\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (-8\,a^5+24\,a^4\,b-16\,a^3\,b^2-16\,a^2\,b^3+24\,a\,b^4-8\,b^5\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (4\,a^5-4\,a^4\,b-8\,a^3\,b^2+8\,a^2\,b^3+4\,a\,b^4-4\,b^5\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d\,{\left (a-b\right )}^3}-\frac {\ln \left (a+b-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )\,\left (6\,a^3\,b^2+6\,a\,b^4\right )}{d\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}-\frac {3\,b\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d\,\left (2\,a^4+8\,a^3\,b+12\,a^2\,b^2+8\,a\,b^3+2\,b^4\right )} \]
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 \[ \int \frac {\cos {\left (c + d x \right )}}{\left (a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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