Optimal. Leaf size=168 \[ \frac {a b^2}{d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)}+\frac {2 a b \left (a^2+b^2\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^3}+\frac {\csc ^2(c+d x) \left (2 a b-\left (a^2+b^2\right ) \cos (c+d x)\right )}{2 d \left (a^2-b^2\right )^2}+\frac {(a-b) \log (1-\cos (c+d x))}{4 d (a+b)^3}-\frac {(a+b) \log (\cos (c+d x)+1)}{4 d (a-b)^3} \]
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Rubi [A] time = 0.43, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3872, 2837, 12, 1647, 1629} \[ \frac {a b^2}{d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)}+\frac {2 a b \left (a^2+b^2\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^3}+\frac {\csc ^2(c+d x) \left (2 a b-\left (a^2+b^2\right ) \cos (c+d x)\right )}{2 d \left (a^2-b^2\right )^2}+\frac {(a-b) \log (1-\cos (c+d x))}{4 d (a+b)^3}-\frac {(a+b) \log (\cos (c+d x)+1)}{4 d (a-b)^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 1629
Rule 1647
Rule 2837
Rule 3872
Rubi steps
\begin {align*} \int \frac {\csc ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx &=\int \frac {\cot ^2(c+d x) \csc (c+d x)}{(-b-a \cos (c+d x))^2} \, dx\\ &=\frac {a^3 \operatorname {Subst}\left (\int \frac {x^2}{a^2 (-b+x)^2 \left (a^2-x^2\right )^2} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {a \operatorname {Subst}\left (\int \frac {x^2}{(-b+x)^2 \left (a^2-x^2\right )^2} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {\left (2 a b-\left (a^2+b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^2 d}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {a^2 b^2 \left (a^2+b^2\right )}{\left (a^2-b^2\right )^2}+\frac {2 a^2 b x}{a^2-b^2}+\frac {a^2 \left (a^2+b^2\right ) x^2}{\left (a^2-b^2\right )^2}}{(-b+x)^2 \left (a^2-x^2\right )} \, dx,x,-a \cos (c+d x)\right )}{2 a d}\\ &=\frac {\left (2 a b-\left (a^2+b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^2 d}+\frac {\operatorname {Subst}\left (\int \left (\frac {a (a+b)}{2 (a-b)^3 (a-x)}+\frac {2 a^2 b^2}{(a-b)^2 (a+b)^2 (b-x)^2}-\frac {4 a^2 b \left (a^2+b^2\right )}{(a-b)^3 (a+b)^3 (b-x)}+\frac {a (a-b)}{2 (a+b)^3 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{2 a d}\\ &=\frac {a b^2}{\left (a^2-b^2\right )^2 d (b+a \cos (c+d x))}+\frac {\left (2 a b-\left (a^2+b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^2 d}+\frac {(a-b) \log (1-\cos (c+d x))}{4 (a+b)^3 d}-\frac {(a+b) \log (1+\cos (c+d x))}{4 (a-b)^3 d}+\frac {2 a b \left (a^2+b^2\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^3 d}\\ \end {align*}
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Mathematica [A] time = 1.31, size = 224, normalized size = 1.33 \[ \frac {\sec ^2(c+d x) (a \cos (c+d x)+b) \left (\frac {16 a b \left (a^2+b^2\right ) (a \cos (c+d x)+b) \log (a \cos (c+d x)+b)}{\left (a^2-b^2\right )^3}+\frac {8 a b^2}{(a-b)^2 (a+b)^2}+\frac {4 (a+b) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b)}{(b-a)^3}-\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{(a+b)^2}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{(a-b)^2}+\frac {4 (a-b) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b)}{(a+b)^3}\right )}{8 d (a+b \sec (c+d x))^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 630, normalized size = 3.75 \[ -\frac {8 \, a^{3} b^{2} - 8 \, a b^{4} - 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right ) + 8 \, {\left (a^{3} b^{2} + a b^{4} - {\left (a^{4} b + a^{2} b^{3}\right )} \cos \left (d x + c\right )^{3} - {\left (a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{4} b + a^{2} b^{3}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right ) - {\left (a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5} - {\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} \cos \left (d x + c\right )^{3} - {\left (a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5} - {\left (a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right )} \cos \left (d x + c\right )^{3} - {\left (a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\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^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} d \cos \left (d x + c\right )^{3} + {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} d \cos \left (d x + c\right ) - {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 456, normalized size = 2.71 \[ \frac {\frac {2 \, {\left (a - b\right )} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {16 \, {\left (a^{3} b + a b^{3}\right )} \log \left ({\left | -a - b - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} \right |}\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac {a^{3} - a^{2} b - a b^{2} + b^{3} - \frac {8 \, a^{2} b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {8 \, a b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {3 \, a^{2} b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {3 \, a b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (\frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}} - \frac {\cos \left (d x + c\right ) - 1}{{\left (a^{2} - 2 \, a b + b^{2}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.65, size = 224, normalized size = 1.33 \[ \frac {b^{2} a}{d \left (a +b \right )^{2} \left (a -b \right )^{2} \left (b +a \cos \left (d x +c \right )\right )}+\frac {2 a^{3} b \ln \left (b +a \cos \left (d x +c \right )\right )}{d \left (a +b \right )^{3} \left (a -b \right )^{3}}+\frac {2 a \,b^{3} \ln \left (b +a \cos \left (d x +c \right )\right )}{d \left (a +b \right )^{3} \left (a -b \right )^{3}}+\frac {1}{4 d \left (a +b \right )^{2} \left (-1+\cos \left (d x +c \right )\right )}+\frac {\ln \left (-1+\cos \left (d x +c \right )\right ) a}{4 d \left (a +b \right )^{3}}-\frac {\ln \left (-1+\cos \left (d x +c \right )\right ) b}{4 d \left (a +b \right )^{3}}+\frac {1}{4 d \left (a -b \right )^{2} \left (1+\cos \left (d x +c \right )\right )}-\frac {\ln \left (1+\cos \left (d x +c \right )\right ) a}{4 d \left (a -b \right )^{3}}-\frac {\ln \left (1+\cos \left (d x +c \right )\right ) b}{4 d \left (a -b \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.80, size = 274, normalized size = 1.63 \[ \frac {\frac {8 \, {\left (a^{3} b + a b^{3}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} - \frac {{\left (a + b\right )} \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac {{\left (a - b\right )} \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {2 \, {\left (4 \, a b^{2} - {\left (a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{2} b - b^{3}\right )} \cos \left (d x + c\right )\right )}}{a^{4} b - 2 \, a^{2} b^{3} + b^{5} - {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )^{3} - {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.47, size = 228, normalized size = 1.36 \[ \frac {\frac {2\,a\,b^2}{{\left (a^2-b^2\right )}^2}+\frac {b\,\cos \left (c+d\,x\right )}{2\,\left (a^2-b^2\right )}-\frac {{\cos \left (c+d\,x\right )}^2\,\left (a^3+3\,a\,b^2\right )}{2\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{d\,\left (-a\,{\cos \left (c+d\,x\right )}^3-b\,{\cos \left (c+d\,x\right )}^2+a\,\cos \left (c+d\,x\right )+b\right )}-\frac {\ln \left (\cos \left (c+d\,x\right )-1\right )\,\left (\frac {b}{2\,{\left (a+b\right )}^3}-\frac {1}{4\,{\left (a+b\right )}^2}\right )}{d}+\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (2\,a^3\,b+2\,a\,b^3\right )}{d\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}-\frac {\ln \left (\cos \left (c+d\,x\right )+1\right )\,\left (a+b\right )}{4\,d\,{\left (a-b\right )}^3} \]
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 \[ \int \frac {\csc ^{3}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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