Optimal. Leaf size=174 \[ \frac {\left (a^2-2 b^2\right ) \cos (c+d x)}{d}-\frac {5 \left (3 a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac {\left (9 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {5 a b \cot ^3(c+d x)}{3 d}+\frac {5 a b \cot (c+d x)}{d}+\frac {a b \cos ^2(c+d x) \cot ^3(c+d x)}{d}+5 a b x-\frac {b^2 \cos ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.28, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2911, 2591, 288, 302, 203, 455, 1814, 1153, 206} \[ \frac {\left (a^2-2 b^2\right ) \cos (c+d x)}{d}-\frac {5 \left (3 a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac {\left (9 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {5 a b \cot ^3(c+d x)}{3 d}+\frac {5 a b \cot (c+d x)}{d}+\frac {a b \cos ^2(c+d x) \cot ^3(c+d x)}{d}+5 a b x-\frac {b^2 \cos ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 288
Rule 302
Rule 455
Rule 1153
Rule 1814
Rule 2591
Rule 2911
Rubi steps
\begin {align*} \int \cos (c+d x) \cot ^5(c+d x) (a+b \sin (c+d x))^2 \, dx &=(2 a b) \int \cos ^2(c+d x) \cot ^4(c+d x) \, dx+\int \cos (c+d x) \cot ^5(c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac {\operatorname {Subst}\left (\int \frac {x^6 \left (a^2+b^2-b^2 x^2\right )}{\left (1-x^2\right )^3} \, dx,x,\cos (c+d x)\right )}{d}-\frac {(2 a b) \operatorname {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac {a b \cos ^2(c+d x) \cot ^3(c+d x)}{d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {\operatorname {Subst}\left (\int \frac {a^2+4 a^2 x^2+4 a^2 x^4-4 b^2 x^6}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{4 d}-\frac {(5 a b) \operatorname {Subst}\left (\int \frac {x^4}{1+x^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac {a b \cos ^2(c+d x) \cot ^3(c+d x)}{d}+\frac {\left (9 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {\operatorname {Subst}\left (\int \frac {7 a^2-4 b^2+8 \left (a^2-b^2\right ) x^2-8 b^2 x^4}{1-x^2} \, dx,x,\cos (c+d x)\right )}{8 d}-\frac {(5 a b) \operatorname {Subst}\left (\int \left (-1+x^2+\frac {1}{1+x^2}\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac {5 a b \cot (c+d x)}{d}-\frac {5 a b \cot ^3(c+d x)}{3 d}+\frac {a b \cos ^2(c+d x) \cot ^3(c+d x)}{d}+\frac {\left (9 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {\operatorname {Subst}\left (\int \left (-8 \left (a^2-2 b^2\right )+8 b^2 x^2+\frac {5 \left (3 a^2-4 b^2\right )}{1-x^2}\right ) \, dx,x,\cos (c+d x)\right )}{8 d}-\frac {(5 a b) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=5 a b x+\frac {\left (a^2-2 b^2\right ) \cos (c+d x)}{d}-\frac {b^2 \cos ^3(c+d x)}{3 d}+\frac {5 a b \cot (c+d x)}{d}-\frac {5 a b \cot ^3(c+d x)}{3 d}+\frac {a b \cos ^2(c+d x) \cot ^3(c+d x)}{d}+\frac {\left (9 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {\left (5 \left (3 a^2-4 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{8 d}\\ &=5 a b x-\frac {5 \left (3 a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac {\left (a^2-2 b^2\right ) \cos (c+d x)}{d}-\frac {b^2 \cos ^3(c+d x)}{3 d}+\frac {5 a b \cot (c+d x)}{d}-\frac {5 a b \cot ^3(c+d x)}{3 d}+\frac {a b \cos ^2(c+d x) \cot ^3(c+d x)}{d}+\frac {\left (9 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 6.18, size = 337, normalized size = 1.94 \[ \frac {\left (9 a^2-4 b^2\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {\left (4 b^2-9 a^2\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {5 \left (3 a^2-4 b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}-\frac {5 \left (3 a^2-4 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}-\frac {a^2 \csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a^2 \sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {5 a b (c+d x)}{d}+\frac {a b \sin (2 (c+d x))}{2 d}+\frac {(2 a-3 b) (2 a+3 b) \cos (c+d x)}{4 d}-\frac {7 a b \tan \left (\frac {1}{2} (c+d x)\right )}{3 d}+\frac {7 a b \cot \left (\frac {1}{2} (c+d x)\right )}{3 d}-\frac {a b \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{12 d}+\frac {a b \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{12 d}-\frac {b^2 \cos (3 (c+d x))}{12 d} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.80, size = 309, normalized size = 1.78 \[ -\frac {16 \, b^{2} \cos \left (d x + c\right )^{7} - 240 \, a b d x \cos \left (d x + c\right )^{4} + 480 \, a b d x \cos \left (d x + c\right )^{2} - 16 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{5} - 240 \, a b d x + 50 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - 30 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right ) + 15 \, {\left ({\left (3 \, a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, a^{2} - 4 \, b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 15 \, {\left ({\left (3 \, a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, a^{2} - 4 \, b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 16 \, {\left (3 \, a b \cos \left (d x + c\right )^{5} - 20 \, a b \cos \left (d x + c\right )^{3} + 15 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.33, size = 346, normalized size = 1.99 \[ \frac {3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 16 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 960 \, {\left (d x + c\right )} a b - 432 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 120 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {128 \, {\left (3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 9 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 6 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{2} + 7 \, b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}} - \frac {750 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1000 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 432 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 16 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.54, size = 334, normalized size = 1.92 \[ -\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{4 d \sin \left (d x +c \right )^{4}}+\frac {3 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{2}}+\frac {3 a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{8 d}+\frac {5 a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{8 d}+\frac {15 a^{2} \cos \left (d x +c \right )}{8 d}+\frac {15 a^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8 d}-\frac {2 a b \left (\cos ^{7}\left (d x +c \right )\right )}{3 d \sin \left (d x +c \right )^{3}}+\frac {8 a b \left (\cos ^{7}\left (d x +c \right )\right )}{3 d \sin \left (d x +c \right )}+\frac {8 a b \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {10 a b \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {5 a b \cos \left (d x +c \right ) \sin \left (d x +c \right )}{d}+5 a b x +\frac {5 a b c}{d}-\frac {b^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{2}}-\frac {b^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{2 d}-\frac {5 b^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{6 d}-\frac {5 b^{2} \cos \left (d x +c \right )}{2 d}-\frac {5 b^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 205, normalized size = 1.18 \[ \frac {16 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a b - 4 \, {\left (4 \, \cos \left (d x + c\right )^{3} - \frac {6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} b^{2} - 3 \, a^{2} {\left (\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.74, size = 479, normalized size = 2.75 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {15\,a^2}{8}-\frac {5\,b^2}{2}\right )}{d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {13\,a^2}{4}-2\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (36\,a^2-98\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {173\,a^2}{4}-\frac {242\,b^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {303\,a^2}{4}-134\,b^2\right )-\frac {a^2}{4}+32\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+136\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {320\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}+4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-\frac {4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}}{d\,\left (16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^2}{4}-\frac {b^2}{8}\right )}{d}+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{12\,d}-\frac {10\,a\,b\,\mathrm {atan}\left (\frac {100\,a^2\,b^2}{-\frac {75\,a^3\,b}{2}+100\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^2+50\,a\,b^3}-\frac {50\,a\,b^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{-\frac {75\,a^3\,b}{2}+100\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^2+50\,a\,b^3}+\frac {75\,a^3\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\left (-\frac {75\,a^3\,b}{2}+100\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^2+50\,a\,b^3\right )}\right )}{d}-\frac {9\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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