Optimal. Leaf size=378 \[ -\frac {4 a C \tanh ^{-1}(\sin (c+d x))}{b^5 d}-\frac {\left (2 A b^8-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 C)\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} b^5 (a+b)^{7/2} d}-\frac {\left (5 A b^4-\left (12 a^4-23 a^2 b^2+6 b^4\right ) C\right ) \tan (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2+a^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {\left (3 A b^4-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {a \left (2 A b^6+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \]
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Rubi [A]
time = 1.29, antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4184, 4183,
4175, 4167, 4083, 3855, 3916, 2738, 214} \begin {gather*} -\frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\left (5 A b^4-C \left (12 a^4-23 a^2 b^2+6 b^4\right )\right ) \tan (c+d x)}{6 b^4 d \left (a^2-b^2\right )^2}+\frac {\left (-4 a^4 C+a^2 b^2 (2 A+9 C)+3 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{6 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac {a \left (4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)+2 A b^6\right ) \tan (c+d x)}{2 b^4 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac {\left (-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 C)+2 A b^8\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^5 d (a-b)^{7/2} (a+b)^{7/2}}-\frac {4 a C \tanh ^{-1}(\sin (c+d x))}{b^5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 2738
Rule 3855
Rule 3916
Rule 4083
Rule 4167
Rule 4175
Rule 4183
Rule 4184
Rubi steps
\begin {align*} \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx &=-\frac {\left (A b^2+a^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\int \frac {\sec ^3(c+d x) \left (3 \left (A b^2+a^2 C\right )-3 a b (A+C) \sec (c+d x)-\left (A b^2+4 a^2 C-3 b^2 C\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx}{3 b \left (a^2-b^2\right )}\\ &=-\frac {\left (A b^2+a^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {\left (3 A b^4-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\int \frac {\sec ^2(c+d x) \left (2 \left (3 A b^4-4 a^4 C+a^2 b^2 (2 A+9 C)\right )-2 a b \left (5 A b^2-\left (a^2-6 b^2\right ) C\right ) \sec (c+d x)-\left (5 A b^4-\left (12 a^4-23 a^2 b^2+6 b^4\right ) C\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{6 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (A b^2+a^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {\left (3 A b^4-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {a \left (2 A b^6+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {\int \frac {\sec (c+d x) \left (-3 b \left (2 A b^6+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right )-a \left (a^2-b^2\right ) \left (5 A b^4+12 a^4 C-25 a^2 b^2 C+18 b^4 C\right ) \sec (c+d x)-b \left (a^2-b^2\right ) \left (5 A b^4-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{6 b^4 \left (a^2-b^2\right )^3}\\ &=-\frac {\left (5 A b^4-\left (12 a^4-23 a^2 b^2+6 b^4\right ) C\right ) \tan (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2+a^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {\left (3 A b^4-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {a \left (2 A b^6+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {\int \frac {\sec (c+d x) \left (-3 b^2 \left (2 A b^6+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right )-24 a b \left (a^2-b^2\right )^3 C \sec (c+d x)\right )}{a+b \sec (c+d x)} \, dx}{6 b^5 \left (a^2-b^2\right )^3}\\ &=-\frac {\left (5 A b^4-\left (12 a^4-23 a^2 b^2+6 b^4\right ) C\right ) \tan (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2+a^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {\left (3 A b^4-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {a \left (2 A b^6+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {(4 a C) \int \sec (c+d x) \, dx}{b^5}-\frac {\left (2 A b^8-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 C)\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 b^5 \left (a^2-b^2\right )^3}\\ &=-\frac {4 a C \tanh ^{-1}(\sin (c+d x))}{b^5 d}-\frac {\left (5 A b^4-\left (12 a^4-23 a^2 b^2+6 b^4\right ) C\right ) \tan (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2+a^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {\left (3 A b^4-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {a \left (2 A b^6+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\left (2 A b^8-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 C)\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 b^6 \left (a^2-b^2\right )^3}\\ &=-\frac {4 a C \tanh ^{-1}(\sin (c+d x))}{b^5 d}-\frac {\left (5 A b^4-\left (12 a^4-23 a^2 b^2+6 b^4\right ) C\right ) \tan (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2+a^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {\left (3 A b^4-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {a \left (2 A b^6+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\left (2 A b^8-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 C)\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^6 \left (a^2-b^2\right )^3 d}\\ &=-\frac {4 a C \tanh ^{-1}(\sin (c+d x))}{b^5 d}-\frac {\left (3 a^2 A b^6+2 A b^8-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+20 a^2 b^6 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} b^5 (a+b)^{7/2} d}-\frac {\left (5 A b^4-\left (12 a^4-23 a^2 b^2+6 b^4\right ) C\right ) \tan (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2+a^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {\left (3 A b^4-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {a \left (2 A b^6+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 4.68, size = 564, normalized size = 1.49 \begin {gather*} \frac {(b+a \cos (c+d x)) \sec ^3(c+d x) \left (A+C \sec ^2(c+d x)\right ) \left (-\frac {48 \left (-2 A b^8+8 a^8 C-28 a^6 b^2 C+35 a^4 b^4 C-a^2 b^6 (3 A+20 C)\right ) \tanh ^{-1}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) \cos (c+d x) (b+a \cos (c+d x))^3}{\left (a^2-b^2\right )^{7/2}}+192 a C \cos (c+d x) (b+a \cos (c+d x))^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-192 a C \cos (c+d x) (b+a \cos (c+d x))^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-\frac {2 b \left (6 a^4 A b^5+54 a^2 A b^7+120 a^8 b C-318 a^6 b^3 C+246 a^4 b^5 C+36 a^2 b^7 C-24 b^9 C+a \left (5 a^4 b^4 (4 A-61 C)+72 b^8 (A-C)+72 a^8 C-28 a^6 b^2 C+a^2 b^6 (13 A+438 C)\right ) \cos (c+d x)+6 a^2 b \left (3 b^6 (3 A-2 C)+20 a^6 C-57 a^4 b^2 C+a^2 b^4 (A+53 C)\right ) \cos (2 (c+d x))+4 a^5 A b^4 \cos (3 (c+d x))+11 a^3 A b^6 \cos (3 (c+d x))+24 a^9 C \cos (3 (c+d x))-68 a^7 b^2 C \cos (3 (c+d x))+65 a^5 b^4 C \cos (3 (c+d x))-6 a^3 b^6 C \cos (3 (c+d x))\right ) \sin (c+d x)}{\left (-a^2+b^2\right )^3}\right )}{24 b^5 d (A+2 C+A \cos (2 (c+d x))) (a+b \sec (c+d x))^4} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.64, size = 532, normalized size = 1.41
method | result | size |
derivativedivides | \(\frac {\frac {\frac {2 \left (-\frac {\left (2 a^{2} A \,b^{4}+3 A a \,b^{5}+6 A \,b^{6}+6 a^{6} C -2 C \,a^{5} b -18 a^{4} b^{2} C +5 C \,a^{3} b^{3}+20 C \,a^{2} b^{4}\right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}+\frac {2 \left (a^{2} A \,b^{4}+9 A \,b^{6}+9 a^{6} C -29 a^{4} b^{2} C +30 C \,a^{2} b^{4}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (2 a^{2} A \,b^{4}-3 A a \,b^{5}+6 A \,b^{6}+6 a^{6} C +2 C \,a^{5} b -18 a^{4} b^{2} C -5 C \,a^{3} b^{3}+20 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}\right )}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{3}}-\frac {\left (3 a^{2} A \,b^{6}+2 A \,b^{8}-8 a^{8} C +28 a^{6} b^{2} C -35 a^{4} b^{4} C +20 C \,a^{2} b^{6}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{b^{5}}-\frac {C}{b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {4 a C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{5}}-\frac {C}{b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {4 a C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{5}}}{d}\) | \(532\) |
default | \(\frac {\frac {\frac {2 \left (-\frac {\left (2 a^{2} A \,b^{4}+3 A a \,b^{5}+6 A \,b^{6}+6 a^{6} C -2 C \,a^{5} b -18 a^{4} b^{2} C +5 C \,a^{3} b^{3}+20 C \,a^{2} b^{4}\right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}+\frac {2 \left (a^{2} A \,b^{4}+9 A \,b^{6}+9 a^{6} C -29 a^{4} b^{2} C +30 C \,a^{2} b^{4}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (2 a^{2} A \,b^{4}-3 A a \,b^{5}+6 A \,b^{6}+6 a^{6} C +2 C \,a^{5} b -18 a^{4} b^{2} C -5 C \,a^{3} b^{3}+20 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}\right )}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{3}}-\frac {\left (3 a^{2} A \,b^{6}+2 A \,b^{8}-8 a^{8} C +28 a^{6} b^{2} C -35 a^{4} b^{4} C +20 C \,a^{2} b^{6}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{b^{5}}-\frac {C}{b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {4 a C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{5}}-\frac {C}{b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {4 a C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{5}}}{d}\) | \(532\) |
risch | \(\text {Expression too large to display}\) | \(2017\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 1183 vs.
\(2 (362) = 724\).
time = 66.35, size = 2424, normalized size = 6.41 \begin {gather*} \text {Too large to display} \end {gather*}
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 \begin {gather*} \int \frac {\left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{4}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{4}}\, dx \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. 878 vs.
\(2 (362) = 724\).
time = 0.53, size = 878, normalized size = 2.32 \begin {gather*} \frac {\frac {3 \, {\left (8 \, C a^{8} - 28 \, C a^{6} b^{2} + 35 \, C a^{4} b^{4} - 3 \, A a^{2} b^{6} - 20 \, C a^{2} b^{6} - 2 \, A b^{8}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{6} b^{5} - 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} - b^{11}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {12 \, C a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{5}} + \frac {12 \, C a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{5}} - \frac {18 \, C a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 42 \, C a^{8} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 24 \, C a^{7} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 117 \, C a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, A a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 24 \, C a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, A a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, C a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, A a^{3} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 60 \, C a^{3} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 27 \, A a^{2} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, A a b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 36 \, C a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 152 \, C a^{7} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, A a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 236 \, C a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 32 \, A a^{3} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, C a^{3} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, A a b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, C a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 42 \, C a^{8} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, C a^{7} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 117 \, C a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, A a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, C a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, A a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 105 \, C a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, A a^{3} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 60 \, C a^{3} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 27 \, A a^{2} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, A a b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{6} b^{4} - 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} - b^{10}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}^{3}} - \frac {6 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} b^{4}}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 17.16, size = 2500, normalized size = 6.61 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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