Optimal. Leaf size=91 \[ -\frac {2 a A \cot ^3(c+d x)}{3 d}-\frac {2 a A \cot (c+d x)}{d}-\frac {7 a A \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {a A \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {7 a A \cot (c+d x) \csc (c+d x)}{8 d} \]
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Rubi [A] time = 0.10, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {21, 3788, 3767, 4046, 3768, 3770} \[ -\frac {2 a A \cot ^3(c+d x)}{3 d}-\frac {2 a A \cot (c+d x)}{d}-\frac {7 a A \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {a A \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {7 a A \cot (c+d x) \csc (c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Rule 21
Rule 3767
Rule 3768
Rule 3770
Rule 3788
Rule 4046
Rubi steps
\begin {align*} \int \csc ^3(c+d x) (a+a \csc (c+d x)) (A+A \csc (c+d x)) \, dx &=\frac {A \int \csc ^3(c+d x) (a+a \csc (c+d x))^2 \, dx}{a}\\ &=\frac {A \int \csc ^3(c+d x) \left (a^2+a^2 \csc ^2(c+d x)\right ) \, dx}{a}+(2 a A) \int \csc ^4(c+d x) \, dx\\ &=-\frac {a A \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {1}{4} (7 a A) \int \csc ^3(c+d x) \, dx-\frac {(2 a A) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac {2 a A \cot (c+d x)}{d}-\frac {2 a A \cot ^3(c+d x)}{3 d}-\frac {7 a A \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a A \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {1}{8} (7 a A) \int \csc (c+d x) \, dx\\ &=-\frac {7 a A \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {2 a A \cot (c+d x)}{d}-\frac {2 a A \cot ^3(c+d x)}{3 d}-\frac {7 a A \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a A \cot (c+d x) \csc ^3(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 163, normalized size = 1.79 \[ -\frac {4 a A \cot (c+d x)}{3 d}-\frac {a A \csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {7 a A \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {a A \sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {7 a A \sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {7 a A \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}-\frac {7 a A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}-\frac {2 a A \cot (c+d x) \csc ^2(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 161, normalized size = 1.77 \[ \frac {42 \, A a \cos \left (d x + c\right )^{3} - 54 \, A a \cos \left (d x + c\right ) - 21 \, {\left (A a \cos \left (d x + c\right )^{4} - 2 \, A a \cos \left (d x + c\right )^{2} + A a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 21 \, {\left (A a \cos \left (d x + c\right )^{4} - 2 \, A a \cos \left (d x + c\right )^{2} + A a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 32 \, {\left (2 \, A a \cos \left (d x + c\right )^{3} - 3 \, A a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 155, normalized size = 1.70 \[ \frac {3 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 16 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 48 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 168 \, A a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 144 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {350 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 144 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 48 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 16 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, A a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.23, size = 99, normalized size = 1.09 \[ -\frac {a A \cot \left (d x +c \right ) \left (\csc ^{3}\left (d x +c \right )\right )}{4 d}-\frac {7 a A \cot \left (d x +c \right ) \csc \left (d x +c \right )}{8 d}+\frac {7 a A \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8 d}-\frac {4 a A \cot \left (d x +c \right )}{3 d}-\frac {2 a A \cot \left (d x +c \right ) \left (\csc ^{2}\left (d x +c \right )\right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 145, normalized size = 1.59 \[ \frac {3 \, A a {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 12 \, A a {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {32 \, {\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )} A a}{\tan \left (d x + c\right )^{3}}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 242, normalized size = 2.66 \[ \frac {A\,a\,\left (3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+16\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-16\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+48\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+144\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-144\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-48\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+168\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}{192\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4} \]
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 \[ A a \left (\int \csc ^{3}{\left (c + d x \right )}\, dx + \int 2 \csc ^{4}{\left (c + d x \right )}\, dx + \int \csc ^{5}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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