Optimal. Leaf size=408 \[ \frac {a \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) x}{16 b^8}-\frac {2 a^2 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^8 d}+\frac {\left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^7 d}-\frac {a \left (8 a^4-18 a^2 b^2+11 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}+\frac {\left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d}-\frac {\left (6 a^4-13 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a b^4 d}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{4 a^2 d}+\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 b^2 d}+\frac {\cos (c+d x) \sin ^6(c+d x)}{7 b d} \]
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Rubi [A]
time = 0.93, antiderivative size = 408, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2975, 3128,
3102, 2814, 2739, 632, 210} \begin {gather*} -\frac {2 a^2 \left (a^2-b^2\right )^{5/2} \text {ArcTan}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^8 d}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {\left (6 a^4-13 a^2 b^2+8 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{24 a b^4 d}-\frac {a \left (8 a^4-18 a^2 b^2+11 b^4\right ) \sin (c+d x) \cos (c+d x)}{16 b^6 d}+\frac {\left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{105 b^5 d}+\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{140 a^2 b^3 d}+\frac {a x \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right )}{16 b^8}+\frac {\left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^7 d}-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac {\sin ^6(c+d x) \cos (c+d x)}{7 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 2814
Rule 2975
Rule 3102
Rule 3128
Rubi steps
\begin {align*} \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{4 a^2 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 b^2 d}+\frac {\cos (c+d x) \sin ^6(c+d x)}{7 b d}+\frac {\int \frac {\sin ^4(c+d x) \left (84 \left (5 a^4-10 a^2 b^2+6 b^4\right )-6 a b \left (2 a^2-7 b^2\right ) \sin (c+d x)-18 \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{504 a^2 b^2}\\ &=\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{4 a^2 d}+\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 b^2 d}+\frac {\cos (c+d x) \sin ^6(c+d x)}{7 b d}+\frac {\int \frac {\sin ^3(c+d x) \left (-72 a \left (28 a^4-60 a^2 b^2+35 b^4\right )+12 a^2 b \left (7 a^2+10 b^2\right ) \sin (c+d x)+420 a \left (6 a^4-13 a^2 b^2+8 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2520 a^2 b^3}\\ &=\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d}-\frac {\left (6 a^4-13 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a b^4 d}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{4 a^2 d}+\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 b^2 d}+\frac {\cos (c+d x) \sin ^6(c+d x)}{7 b d}+\frac {\int \frac {\sin ^2(c+d x) \left (1260 a^2 \left (6 a^4-13 a^2 b^2+8 b^4\right )-36 a^3 b \left (14 a^2-25 b^2\right ) \sin (c+d x)-288 a^2 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{10080 a^2 b^4}\\ &=\frac {\left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d}-\frac {\left (6 a^4-13 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a b^4 d}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{4 a^2 d}+\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 b^2 d}+\frac {\cos (c+d x) \sin ^6(c+d x)}{7 b d}+\frac {\int \frac {\sin (c+d x) \left (-576 a^3 \left (35 a^4-77 a^2 b^2+45 b^4\right )+36 a^2 b \left (70 a^4-133 a^2 b^2+120 b^4\right ) \sin (c+d x)+3780 a^3 \left (8 a^4-18 a^2 b^2+11 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{30240 a^2 b^5}\\ &=-\frac {a \left (8 a^4-18 a^2 b^2+11 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}+\frac {\left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d}-\frac {\left (6 a^4-13 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a b^4 d}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{4 a^2 d}+\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 b^2 d}+\frac {\cos (c+d x) \sin ^6(c+d x)}{7 b d}+\frac {\int \frac {3780 a^4 \left (8 a^4-18 a^2 b^2+11 b^4\right )-36 a^3 b \left (280 a^4-574 a^2 b^2+285 b^4\right ) \sin (c+d x)-576 a^2 \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{60480 a^2 b^6}\\ &=\frac {\left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^7 d}-\frac {a \left (8 a^4-18 a^2 b^2+11 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}+\frac {\left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d}-\frac {\left (6 a^4-13 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a b^4 d}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{4 a^2 d}+\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 b^2 d}+\frac {\cos (c+d x) \sin ^6(c+d x)}{7 b d}+\frac {\int \frac {3780 a^4 b \left (8 a^4-18 a^2 b^2+11 b^4\right )+3780 a^3 \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{60480 a^2 b^7}\\ &=\frac {a \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) x}{16 b^8}+\frac {\left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^7 d}-\frac {a \left (8 a^4-18 a^2 b^2+11 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}+\frac {\left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d}-\frac {\left (6 a^4-13 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a b^4 d}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{4 a^2 d}+\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 b^2 d}+\frac {\cos (c+d x) \sin ^6(c+d x)}{7 b d}-\frac {\left (a^2 \left (a^2-b^2\right )^3\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{b^8}\\ &=\frac {a \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) x}{16 b^8}+\frac {\left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^7 d}-\frac {a \left (8 a^4-18 a^2 b^2+11 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}+\frac {\left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d}-\frac {\left (6 a^4-13 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a b^4 d}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{4 a^2 d}+\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 b^2 d}+\frac {\cos (c+d x) \sin ^6(c+d x)}{7 b d}-\frac {\left (2 a^2 \left (a^2-b^2\right )^3\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^8 d}\\ &=\frac {a \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) x}{16 b^8}+\frac {\left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^7 d}-\frac {a \left (8 a^4-18 a^2 b^2+11 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}+\frac {\left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d}-\frac {\left (6 a^4-13 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a b^4 d}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{4 a^2 d}+\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 b^2 d}+\frac {\cos (c+d x) \sin ^6(c+d x)}{7 b d}+\frac {\left (4 a^2 \left (a^2-b^2\right )^3\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^8 d}\\ &=\frac {a \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) x}{16 b^8}-\frac {2 a^2 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^8 d}+\frac {\left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^7 d}-\frac {a \left (8 a^4-18 a^2 b^2+11 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}+\frac {\left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d}-\frac {\left (6 a^4-13 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a b^4 d}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{4 a^2 d}+\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 b^2 d}+\frac {\cos (c+d x) \sin ^6(c+d x)}{7 b d}\\ \end {align*}
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Mathematica [A]
time = 2.07, size = 324, normalized size = 0.79 \begin {gather*} -\frac {-6720 a^7 c+16800 a^5 b^2 c-12600 a^3 b^4 c+2100 a b^6 c-6720 a^7 d x+16800 a^5 b^2 d x-12600 a^3 b^4 d x+2100 a b^6 d x+13440 a^2 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+105 b \left (-64 a^6+144 a^4 b^2-88 a^2 b^4+5 b^6\right ) \cos (c+d x)+35 \left (16 a^4 b^3-28 a^2 b^5+9 b^7\right ) \cos (3 (c+d x))-84 a^2 b^5 \cos (5 (c+d x))+105 b^7 \cos (5 (c+d x))+15 b^7 \cos (7 (c+d x))+1680 a^5 b^2 \sin (2 (c+d x))-3360 a^3 b^4 \sin (2 (c+d x))+1575 a b^6 \sin (2 (c+d x))-210 a^3 b^4 \sin (4 (c+d x))+315 a b^6 \sin (4 (c+d x))+35 a b^6 \sin (6 (c+d x))}{6720 b^8 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.45, size = 601, normalized size = 1.47 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 619, normalized size = 1.52 \begin {gather*} \left [-\frac {240 \, b^{7} \cos \left (d x + c\right )^{7} - 336 \, a^{2} b^{5} \cos \left (d x + c\right )^{5} + 560 \, {\left (a^{4} b^{3} - a^{2} b^{5}\right )} \cos \left (d x + c\right )^{3} - 105 \, {\left (16 \, a^{7} - 40 \, a^{5} b^{2} + 30 \, a^{3} b^{4} - 5 \, a b^{6}\right )} d x - 840 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - 1680 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} \cos \left (d x + c\right ) + 35 \, {\left (8 \, a b^{6} \cos \left (d x + c\right )^{5} - 2 \, {\left (6 \, a^{3} b^{4} - 5 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, a^{5} b^{2} - 14 \, a^{3} b^{4} + 5 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, b^{8} d}, -\frac {240 \, b^{7} \cos \left (d x + c\right )^{7} - 336 \, a^{2} b^{5} \cos \left (d x + c\right )^{5} + 560 \, {\left (a^{4} b^{3} - a^{2} b^{5}\right )} \cos \left (d x + c\right )^{3} - 105 \, {\left (16 \, a^{7} - 40 \, a^{5} b^{2} + 30 \, a^{3} b^{4} - 5 \, a b^{6}\right )} d x - 1680 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) - 1680 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} \cos \left (d x + c\right ) + 35 \, {\left (8 \, a b^{6} \cos \left (d x + c\right )^{5} - 2 \, {\left (6 \, a^{3} b^{4} - 5 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, a^{5} b^{2} - 14 \, a^{3} b^{4} + 5 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, b^{8} d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 863 vs.
\(2 (383) = 766\).
time = 0.62, size = 863, normalized size = 2.12 \begin {gather*} \frac {\frac {105 \, {\left (16 \, a^{7} - 40 \, a^{5} b^{2} + 30 \, a^{3} b^{4} - 5 \, a b^{6}\right )} {\left (d x + c\right )}}{b^{8}} - \frac {3360 \, {\left (a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} b^{8}} + \frac {2 \, {\left (840 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 1890 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 1155 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 1680 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 5040 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 5040 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 1680 \, b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 3360 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 5880 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 980 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 10080 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 26880 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 20160 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 4200 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 6090 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 2975 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 25200 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 61040 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 40880 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 8400 \, b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 33600 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 76160 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 49280 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 4200 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6090 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2975 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 25200 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 55440 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 33936 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 5040 \, b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3360 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5880 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 980 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10080 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 22400 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12992 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 840 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1890 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1155 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1680 \, a^{6} - 3920 \, a^{4} b^{2} + 2576 \, a^{2} b^{4} - 240 \, b^{6}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{7} b^{7}}}{1680 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 14.46, size = 2500, normalized size = 6.13 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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