Optimal. Leaf size=229 \[ \frac {b^2 \left (3 a^2+b^2\right )}{d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}+\frac {\csc ^2(c+d x) \left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right )}{2 d \left (a^2-b^2\right )^3}-\frac {b^3}{2 d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}+\frac {b \left (3 a^4+8 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^4}+\frac {(a-2 b) \log (1-\cos (c+d x))}{4 d (a+b)^4}-\frac {(a+2 b) \log (\cos (c+d x)+1)}{4 d (a-b)^4} \]
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Rubi [A] time = 0.51, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3872, 2721, 1647, 1629} \[ -\frac {b^3}{2 d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}+\frac {b^2 \left (3 a^2+b^2\right )}{d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}+\frac {b \left (8 a^2 b^2+3 a^4+b^4\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^4}+\frac {\csc ^2(c+d x) \left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right )}{2 d \left (a^2-b^2\right )^3}+\frac {(a-2 b) \log (1-\cos (c+d x))}{4 d (a+b)^4}-\frac {(a+2 b) \log (\cos (c+d x)+1)}{4 d (a-b)^4} \]
Antiderivative was successfully verified.
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Rule 1629
Rule 1647
Rule 2721
Rule 3872
Rubi steps
\begin {align*} \int \frac {\csc ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx &=-\int \frac {\cot ^3(c+d x)}{(-b-a \cos (c+d x))^3} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^3}{(-b+x)^3 \left (a^2-x^2\right )^2} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {\left (b \left (3 a^2+b^2\right )-a \left (a^2+3 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^3 \left (a^2+3 b^2\right )}{\left (a^2-b^2\right )^3}-\frac {a^2 b^2 \left (3 a^4+3 a^2 b^2-2 b^4\right ) x}{\left (a^2-b^2\right )^3}+\frac {a^4 b \left (3 a^2-7 b^2\right ) x^2}{\left (a^2-b^2\right )^3}+\frac {a^4 \left (a^2+3 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^2 d}\\ &=\frac {\left (b \left (3 a^2+b^2\right )-a \left (a^2+3 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 {a^2 (a+2 b)}{2 (a-b)^4 (a-x)}-\frac {2 a^2 b^3}{\left (a^2-b^2\right )^2 (b-x)^3}+\frac {2 a^2 b^2 \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3 (b-x)^2}-\frac {2 a^2 b \left (3 a^4+8 a^2 b^2+b^4\right )}{\left (a^2-b^2\right )^4 (b-x)}+\frac {a^2 (a-2 b)}{2 (a+b)^4 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{2 a^2 d}\\ &=-\frac {b^3}{2 \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))^2}+\frac {b^2 \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}+\frac {\left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^3 d}+\frac {(a-2 b) \log (1-\cos (c+d x))}{4 (a+b)^4 d}-\frac {(a+2 b) \log (1+\cos (c+d x))}{4 (a-b)^4 d}+\frac {b \left (3 a^4+8 a^2 b^2+b^4\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^4 d}\\ \end {align*}
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Mathematica [C] time = 6.34, size = 332, normalized size = 1.45 \[ -\frac {b^2 \left (3 a^2+b^2\right )}{d (b-a)^3 (a+b)^3 (a \cos (c+d x)+b)}-\frac {2 i \left (3 a^4 b+8 a^2 b^3+b^5\right ) (c+d x)}{d (a-b)^4 (a+b)^4}+\frac {\left (3 a^4 b+8 a^2 b^3+b^5\right ) \log (a \cos (c+d x)+b)}{d \left (b^2-a^2\right )^4}-\frac {b^3}{2 d (b-a)^2 (a+b)^2 (a \cos (c+d x)+b)^2}-\frac {i (-a-2 b) \tan ^{-1}(\tan (c+d x))}{2 d (b-a)^4}-\frac {i (a-2 b) \tan ^{-1}(\tan (c+d x))}{2 d (a+b)^4}-\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d (a+b)^3}-\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d (b-a)^3}+\frac {(a-2 b) \log \left (\sin ^2\left (\frac {1}{2} (c+d x)\right )\right )}{4 d (a+b)^4}+\frac {(-a-2 b) \log \left (\cos ^2\left (\frac {1}{2} (c+d x)\right )\right )}{4 d (b-a)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.89, size = 1071, normalized size = 4.68 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.65, size = 800, normalized size = 3.49 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.68, size = 322, normalized size = 1.41 \[ -\frac {b^{3}}{2 d \left (a +b \right )^{2} \left (a -b \right )^{2} \left (b +a \cos \left (d x +c \right )\right )^{2}}+\frac {3 b \ln \left (b +a \cos \left (d x +c \right )\right ) a^{4}}{d \left (a +b \right )^{4} \left (a -b \right )^{4}}+\frac {8 b^{3} \ln \left (b +a \cos \left (d x +c \right )\right ) a^{2}}{d \left (a +b \right )^{4} \left (a -b \right )^{4}}+\frac {b^{5} \ln \left (b +a \cos \left (d x +c \right )\right )}{d \left (a +b \right )^{4} \left (a -b \right )^{4}}+\frac {3 b^{2} a^{2}}{d \left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +a \cos \left (d x +c \right )\right )}+\frac {b^{4}}{d \left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +a \cos \left (d x +c \right )\right )}+\frac {1}{4 d \left (a +b \right )^{3} \left (-1+\cos \left (d x +c \right )\right )}+\frac {\ln \left (-1+\cos \left (d x +c \right )\right ) a}{4 d \left (a +b \right )^{4}}-\frac {\ln \left (-1+\cos \left (d x +c \right )\right ) b}{2 d \left (a +b \right )^{4}}+\frac {1}{4 d \left (a -b \right )^{3} \left (1+\cos \left (d x +c \right )\right )}-\frac {\ln \left (1+\cos \left (d x +c \right )\right ) a}{4 d \left (a -b \right )^{4}}-\frac {\ln \left (1+\cos \left (d x +c \right )\right ) b}{2 d \left (a -b \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.79, size = 435, normalized size = 1.90 \[ \frac {\frac {4 \, {\left (3 \, a^{4} b + 8 \, a^{2} b^{3} + b^{5}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} - \frac {{\left (a + 2 \, b\right )} \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} + \frac {{\left (a - 2 \, b\right )} \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {2 \, {\left (8 \, a^{2} b^{3} + 4 \, b^{5} - {\left (a^{5} + 9 \, a^{3} b^{2} + 2 \, a b^{4}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{4} b - 10 \, a^{2} b^{3} - 3 \, b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (11 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )\right )}}{a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8} - {\left (a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} \cos \left (d x + c\right )}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.70, size = 378, normalized size = 1.65 \[ \frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (\frac {3\,b}{4\,{\left (a+b\right )}^4}-\frac {1}{4\,{\left (a+b\right )}^3}+\frac {3\,b}{4\,{\left (a-b\right )}^4}+\frac {1}{4\,{\left (a-b\right )}^3}\right )}{d}-\frac {\ln \left (\cos \left (c+d\,x\right )-1\right )\,\left (\frac {3\,b}{4\,{\left (a+b\right )}^4}-\frac {1}{4\,{\left (a+b\right )}^3}\right )}{d}-\frac {\frac {{\cos \left (c+d\,x\right )}^3\,\left (a^5+9\,a^3\,b^2+2\,a\,b^4\right )}{2\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {{\cos \left (c+d\,x\right )}^2\,\left (-a^4\,b+10\,a^2\,b^3+3\,b^5\right )}{2\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}-\frac {2\,b\,\left (2\,a^2\,b^2+b^4\right )}{\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {a\,\cos \left (c+d\,x\right )\,\left (11\,a^2\,b^2+b^4\right )}{2\,\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{d\,\left ({\cos \left (c+d\,x\right )}^2\,\left (a^2-b^2\right )+b^2-a^2\,{\cos \left (c+d\,x\right )}^4+2\,a\,b\,\cos \left (c+d\,x\right )-2\,a\,b\,{\cos \left (c+d\,x\right )}^3\right )}-\frac {\ln \left (\cos \left (c+d\,x\right )+1\right )\,\left (a+2\,b\right )}{4\,d\,{\left (a-b\right )}^4} \]
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 \[ \int \frac {\csc ^{3}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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