Optimal. Leaf size=381 \[ \frac {\left (2 A b^2+\left (12 a^2+b^2\right ) C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}-\frac {a \left (6 A b^6+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+12 a^6 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^5 (a+b)^{5/2} d}-\frac {a \left (a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)+12 a^4 C\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}+\frac {\left (a^2 b^2 (A-10 C)-b^4 (4 A-C)+6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2+a^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (3 A b^4-4 a^4 C+7 a^2 b^2 C\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \]
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Rubi [A]
time = 1.16, antiderivative size = 381, 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,
4177, 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)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac {\left (C \left (12 a^2+b^2\right )+2 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}-\frac {a \left (12 a^4 C+a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)\right ) \tan (c+d x)}{2 b^4 d \left (a^2-b^2\right )^2}+\frac {\left (-4 a^4 C+7 a^2 b^2 C+3 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{2 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}+\frac {\left (6 a^4 C+a^2 b^2 (A-10 C)-b^4 (4 A-C)\right ) \tan (c+d x) \sec (c+d x)}{2 b^3 d \left (a^2-b^2\right )^2}-\frac {a \left (12 a^6 C+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+6 A b^6\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)^{5/2} (a+b)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 2738
Rule 3855
Rule 3916
Rule 4083
Rule 4167
Rule 4177
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))^3} \, dx &=-\frac {\left (A b^2+a^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\int \frac {\sec ^3(c+d x) \left (3 \left (A b^2+a^2 C\right )-2 a b (A+C) \sec (c+d x)-2 \left (A b^2+2 a^2 C-b^2 C\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{2 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)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (3 A b^4-4 a^4 C+7 a^2 b^2 C\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\int \frac {\sec ^2(c+d x) \left (2 \left (3 A b^4-4 a^4 C+7 a^2 b^2 C\right )-a b \left (3 A b^2-\left (a^2-4 b^2\right ) C\right ) \sec (c+d x)+2 \left (a^2 b^2 (A-10 C)-b^4 (4 A-C)+6 a^4 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=\frac {\left (a^2 b^2 (A-10 C)-b^4 (4 A-C)+6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \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)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (3 A b^4-4 a^4 C+7 a^2 b^2 C\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\int \frac {\sec (c+d x) \left (2 a \left (a^2 b^2 (A-10 C)-b^4 (4 A-C)+6 a^4 C\right )-2 b \left (2 a^4 C-b^4 (2 A+C)-a^2 b^2 (A+4 C)\right ) \sec (c+d x)-2 a \left (a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)+12 a^4 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{4 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac {a \left (a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)+12 a^4 C\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}+\frac {\left (a^2 b^2 (A-10 C)-b^4 (4 A-C)+6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \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)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (3 A b^4-4 a^4 C+7 a^2 b^2 C\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\int \frac {\sec (c+d x) \left (2 a b \left (a^2 b^2 (A-10 C)-b^4 (4 A-C)+6 a^4 C\right )+2 \left (a^2-b^2\right )^2 \left (2 A b^2+\left (12 a^2+b^2\right ) C\right ) \sec (c+d x)\right )}{a+b \sec (c+d x)} \, dx}{4 b^4 \left (a^2-b^2\right )^2}\\ &=-\frac {a \left (a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)+12 a^4 C\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}+\frac {\left (a^2 b^2 (A-10 C)-b^4 (4 A-C)+6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \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)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (3 A b^4-4 a^4 C+7 a^2 b^2 C\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (a \left (6 A b^6+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+12 a^6 C\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 b^5 \left (a^2-b^2\right )^2}+\frac {\left (2 A b^2+\left (12 a^2+b^2\right ) C\right ) \int \sec (c+d x) \, dx}{2 b^5}\\ &=\frac {\left (2 A b^2+\left (12 a^2+b^2\right ) C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}-\frac {a \left (a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)+12 a^4 C\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}+\frac {\left (a^2 b^2 (A-10 C)-b^4 (4 A-C)+6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \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)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (3 A b^4-4 a^4 C+7 a^2 b^2 C\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (a \left (6 A b^6+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+12 a^6 C\right )\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 b^6 \left (a^2-b^2\right )^2}\\ &=\frac {\left (2 A b^2+\left (12 a^2+b^2\right ) C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}-\frac {a \left (a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)+12 a^4 C\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}+\frac {\left (a^2 b^2 (A-10 C)-b^4 (4 A-C)+6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \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)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (3 A b^4-4 a^4 C+7 a^2 b^2 C\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (a \left (6 A b^6+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+12 a^6 C\right )\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 )^2 d}\\ &=\frac {\left (2 A b^2+\left (12 a^2+b^2\right ) C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}-\frac {a \left (2 a^4 A b^2-5 a^2 A b^4+6 A b^6+12 a^6 C-29 a^4 b^2 C+20 a^2 b^4 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^5 (a+b)^{5/2} d}-\frac {a \left (a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)+12 a^4 C\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}+\frac {\left (a^2 b^2 (A-10 C)-b^4 (4 A-C)+6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \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)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (3 A b^4-4 a^4 C+7 a^2 b^2 C\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 4.46, size = 559, normalized size = 1.47 \begin {gather*} \frac {(b+a \cos (c+d x)) \sec (c+d x) \left (A+C \sec ^2(c+d x)\right ) \left (\frac {4 a \left (6 A b^6+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+12 a^6 C\right ) \tanh ^{-1}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) (b+a \cos (c+d x))^2}{\left (a^2-b^2\right )^{5/2}}-2 \left (2 A b^2+\left (12 a^2+b^2\right ) C\right ) (b+a \cos (c+d x))^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 \left (2 A b^2+\left (12 a^2+b^2\right ) C\right ) (b+a \cos (c+d x))^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {b^2 C (b+a \cos (c+d x))^2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {12 a b C (b+a \cos (c+d x))^2 \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}-\frac {b^2 C (b+a \cos (c+d x))^2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {12 a b C (b+a \cos (c+d x))^2 \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {2 a^2 b^2 \left (A b^2+a^2 C\right ) \sin (c+d x)}{(-a+b) (a+b)}+\frac {2 a^2 b \left (5 A b^4-6 a^4 C+a^2 b^2 (-2 A+9 C)\right ) (b+a \cos (c+d x)) \sin (c+d x)}{(a-b)^2 (a+b)^2}\right )}{2 b^5 d (A+2 C+A \cos (2 (c+d x))) (a+b \sec (c+d x))^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.99, size = 462, normalized size = 1.21
method | result | size |
derivativedivides | \(\frac {\frac {2 a \left (\frac {\frac {\left (2 a^{2} A \,b^{2}-a A \,b^{3}-6 A \,b^{4}+6 a^{4} C -a^{3} b C -10 C \,a^{2} b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {b a \left (2 a^{2} A \,b^{2}+a A \,b^{3}-6 A \,b^{4}+6 a^{4} C +a^{3} b C -10 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}-\frac {\left (2 A \,a^{4} b^{2}-5 a^{2} A \,b^{4}+6 A \,b^{6}+12 a^{6} C -29 a^{4} b^{2} C +20 C \,a^{2} b^{4}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{5}}+\frac {C}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (-2 A \,b^{2}-12 a^{2} C -b^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{5}}+\frac {C \left (6 a +b \right )}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {C}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (2 A \,b^{2}+12 a^{2} C +b^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{5}}+\frac {C \left (6 a +b \right )}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\) | \(462\) |
default | \(\frac {\frac {2 a \left (\frac {\frac {\left (2 a^{2} A \,b^{2}-a A \,b^{3}-6 A \,b^{4}+6 a^{4} C -a^{3} b C -10 C \,a^{2} b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {b a \left (2 a^{2} A \,b^{2}+a A \,b^{3}-6 A \,b^{4}+6 a^{4} C +a^{3} b C -10 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}-\frac {\left (2 A \,a^{4} b^{2}-5 a^{2} A \,b^{4}+6 A \,b^{6}+12 a^{6} C -29 a^{4} b^{2} C +20 C \,a^{2} b^{4}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{5}}+\frac {C}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (-2 A \,b^{2}-12 a^{2} C -b^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{5}}+\frac {C \left (6 a +b \right )}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {C}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (2 A \,b^{2}+12 a^{2} C +b^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{5}}+\frac {C \left (6 a +b \right )}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\) | \(462\) |
risch | \(\text {Expression too large to display}\) | \(2014\) |
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. 1014 vs.
\(2 (362) = 724\).
time = 56.15, size = 2084, normalized size = 5.47 \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 )^{3}}\, 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. 1187 vs.
\(2 (362) = 724\).
time = 0.59, size = 1187, normalized size = 3.12 \begin {gather*} -\frac {\frac {2 \, {\left (12 \, C a^{7} + 2 \, A a^{5} b^{2} - 29 \, C a^{5} b^{2} - 5 \, A a^{3} b^{4} + 20 \, C a^{3} b^{4} + 6 \, A a b^{6}\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^{4} b^{5} - 2 \, a^{2} b^{7} + b^{9}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {2 \, {\left (12 \, C a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 18 \, C a^{6} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 2 \, A a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 17 \, C a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 3 \, A a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 33 \, C a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 5 \, A a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 2 \, C a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 6 \, A a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 13 \, C a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 4 \, C a b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + C b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 36 \, C a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, C a^{6} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, A a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 67 \, C a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, A a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 29 \, C a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, A a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 26 \, C a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, A a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 5 \, C a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4 \, C a b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, C a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, C a^{6} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 67 \, C a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, A a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 29 \, C a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, A a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 26 \, C a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, A a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, C a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, C a b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, C b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 18 \, C a^{6} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, A a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 17 \, C a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, A a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 33 \, C a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, A a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, C a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, A a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 13 \, C a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, C a b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )}^{2}} - \frac {{\left (12 \, C a^{2} + 2 \, A b^{2} + C b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{5}} + \frac {{\left (12 \, C a^{2} + 2 \, A b^{2} + C b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{5}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 18.47, size = 2500, normalized size = 6.56 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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