Optimal. Leaf size=210 \[ \frac {\left (3 a^2+10 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{256 d}+\frac {\left (21 a^2-10 b^2\right ) \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {\left (93 a^2-170 b^2\right ) \cot (c+d x) \csc ^5(c+d x)}{480 d}+\frac {\left (3 a^2-118 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{384 d}+\frac {\left (3 a^2+10 b^2\right ) \cot (c+d x) \csc (c+d x)}{256 d}-\frac {a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d}-\frac {2 a b \cot ^9(c+d x)}{9 d}-\frac {2 a b \cot ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.34, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2911, 2607, 14, 4366, 455, 1814, 1157, 385, 199, 206} \[ \frac {\left (3 a^2+10 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{256 d}+\frac {\left (21 a^2-10 b^2\right ) \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {\left (93 a^2-170 b^2\right ) \cot (c+d x) \csc ^5(c+d x)}{480 d}+\frac {\left (3 a^2-118 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{384 d}+\frac {\left (3 a^2+10 b^2\right ) \cot (c+d x) \csc (c+d x)}{256 d}-\frac {a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d}-\frac {2 a b \cot ^9(c+d x)}{9 d}-\frac {2 a b \cot ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 199
Rule 206
Rule 385
Rule 455
Rule 1157
Rule 1814
Rule 2607
Rule 2911
Rule 4366
Rubi steps
\begin {align*} \int \cot ^6(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx &=(2 a b) \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx+\int \cot ^6(c+d x) \csc ^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 )^6} \, dx,x,\cos (c+d x)\right )}{d}+\frac {(2 a b) \operatorname {Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac {a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d}+\frac {\operatorname {Subst}\left (\int \frac {a^2+10 a^2 x^2+10 a^2 x^4-10 b^2 x^6}{\left (1-x^2\right )^5} \, dx,x,\cos (c+d x)\right )}{10 d}+\frac {(2 a b) \operatorname {Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac {2 a b \cot ^7(c+d x)}{7 d}-\frac {2 a b \cot ^9(c+d x)}{9 d}+\frac {\left (21 a^2-10 b^2\right ) \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d}-\frac {\operatorname {Subst}\left (\int \frac {13 a^2-10 b^2+80 \left (a^2-b^2\right ) x^2-80 b^2 x^4}{\left (1-x^2\right )^4} \, dx,x,\cos (c+d x)\right )}{80 d}\\ &=-\frac {2 a b \cot ^7(c+d x)}{7 d}-\frac {2 a b \cot ^9(c+d x)}{9 d}-\frac {\left (93 a^2-170 b^2\right ) \cot (c+d x) \csc ^5(c+d x)}{480 d}+\frac {\left (21 a^2-10 b^2\right ) \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d}+\frac {\operatorname {Subst}\left (\int \frac {5 \left (3 a^2-22 b^2\right )-480 b^2 x^2}{\left (1-x^2\right )^3} \, dx,x,\cos (c+d x)\right )}{480 d}\\ &=-\frac {2 a b \cot ^7(c+d x)}{7 d}-\frac {2 a b \cot ^9(c+d x)}{9 d}+\frac {\left (3 a^2-118 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{384 d}-\frac {\left (93 a^2-170 b^2\right ) \cot (c+d x) \csc ^5(c+d x)}{480 d}+\frac {\left (21 a^2-10 b^2\right ) \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d}+\frac {\left (3 a^2+10 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{128 d}\\ &=-\frac {2 a b \cot ^7(c+d x)}{7 d}-\frac {2 a b \cot ^9(c+d x)}{9 d}+\frac {\left (3 a^2+10 b^2\right ) \cot (c+d x) \csc (c+d x)}{256 d}+\frac {\left (3 a^2-118 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{384 d}-\frac {\left (93 a^2-170 b^2\right ) \cot (c+d x) \csc ^5(c+d x)}{480 d}+\frac {\left (21 a^2-10 b^2\right ) \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d}+\frac {\left (3 a^2+10 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{256 d}\\ &=\frac {\left (3 a^2+10 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac {2 a b \cot ^7(c+d x)}{7 d}-\frac {2 a b \cot ^9(c+d x)}{9 d}+\frac {\left (3 a^2+10 b^2\right ) \cot (c+d x) \csc (c+d x)}{256 d}+\frac {\left (3 a^2-118 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{384 d}-\frac {\left (93 a^2-170 b^2\right ) \cot (c+d x) \csc ^5(c+d x)}{480 d}+\frac {\left (21 a^2-10 b^2\right ) \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d}\\ \end {align*}
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Mathematica [A] time = 1.47, size = 244, normalized size = 1.16 \[ -\frac {80640 \left (3 a^2+10 b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-80640 \left (3 a^2+10 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+\csc ^{10}(c+d x) \left (630 \left (1879 a^2+290 b^2\right ) \cos (c+d x)+1260 \left (519 a^2-62 b^2\right ) \cos (3 (c+d x))+218484 a^2 \cos (5 (c+d x))+9135 a^2 \cos (7 (c+d x))-945 a^2 \cos (9 (c+d x))+537600 a b \sin (2 (c+d x))+522240 a b \sin (4 (c+d x))+207360 a b \sin (6 (c+d x))+25600 a b \sin (8 (c+d x))-2560 a b \sin (10 (c+d x))-24360 b^2 \cos (5 (c+d x))-77070 b^2 \cos (7 (c+d x))-3150 b^2 \cos (9 (c+d x))\right )}{20643840 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.79, size = 455, normalized size = 2.17 \[ -\frac {630 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{9} - 420 \, {\left (21 \, a^{2} - 58 \, b^{2}\right )} \cos \left (d x + c\right )^{7} - 5376 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{5} + 2940 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - 630 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right ) - 315 \, {\left ({\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{10} - 5 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{8} + 10 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - 10 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 5 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 3 \, a^{2} - 10 \, b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 315 \, {\left ({\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{10} - 5 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{8} + 10 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - 10 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 5 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 3 \, a^{2} - 10 \, b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 5120 \, {\left (2 \, a b \cos \left (d x + c\right )^{9} - 9 \, a b \cos \left (d x + c\right )^{7}\right )} \sin \left (d x + c\right )}{161280 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.40, size = 468, normalized size = 2.23 \[ \frac {126 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 560 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 315 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 630 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 2160 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 630 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 3360 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 2520 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 5040 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 13440 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1260 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 10080 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 30240 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5040 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {44286 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 147620 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 30240 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1260 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 10080 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 13440 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 2520 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 5040 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 630 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 3360 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2160 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 315 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 630 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 560 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 126 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10}}}{1290240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.46, size = 404, normalized size = 1.92 \[ -\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{10 d \sin \left (d x +c \right )^{10}}-\frac {3 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{80 d \sin \left (d x +c \right )^{8}}-\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{160 d \sin \left (d x +c \right )^{6}}+\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{640 d \sin \left (d x +c \right )^{4}}-\frac {3 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{1280 d \sin \left (d x +c \right )^{2}}-\frac {3 a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{1280 d}-\frac {a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{256 d}-\frac {3 a^{2} \cos \left (d x +c \right )}{256 d}-\frac {3 a^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256 d}-\frac {2 a b \left (\cos ^{7}\left (d x +c \right )\right )}{9 d \sin \left (d x +c \right )^{9}}-\frac {4 a b \left (\cos ^{7}\left (d x +c \right )\right )}{63 d \sin \left (d x +c \right )^{7}}-\frac {b^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{8}}-\frac {b^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{48 d \sin \left (d x +c \right )^{6}}+\frac {b^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{192 d \sin \left (d x +c \right )^{4}}-\frac {b^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{128 d \sin \left (d x +c \right )^{2}}-\frac {b^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{128 d}-\frac {5 b^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{384 d}-\frac {5 b^{2} \cos \left (d x +c \right )}{128 d}-\frac {5 b^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 272, normalized size = 1.30 \[ -\frac {63 \, a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 210 \, b^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {5120 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a b}{\tan \left (d x + c\right )^{9}}}{161280 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.93, size = 394, normalized size = 1.88 \[ \frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,a^2}{256}+\frac {5\,b^2}{128}\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (2\,a^2+4\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^2}{4}-\frac {b^2}{2}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^2}{2}+\frac {8\,b^2}{3}\right )+\frac {a^2}{10}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (a^2+8\,b^2\right )-\frac {12\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{7}+\frac {32\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}-24\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{9}\right )}{1024\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^2}{512}+\frac {b^2}{256}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^2}{1024}+\frac {b^2}{128}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a^2}{2048}+\frac {b^2}{384}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {a^2}{4096}-\frac {b^2}{2048}\right )}{d}+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}-\frac {3\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{1792\,d}+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{2304\,d}-\frac {3\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d} \]
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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