Optimal. Leaf size=232 \[ \frac {\left (8 A b^4+4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) x}{8 a^5}-\frac {2 b^3 \left (A b^2+a^2 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 \sqrt {a-b} \sqrt {a+b} d}-\frac {b \left (3 A b^2+a^2 (2 A+3 C)\right ) \sin (c+d x)}{3 a^4 d}+\frac {\left (4 A b^2+a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {A b \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d} \]
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Rubi [A]
time = 0.68, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4190, 4189,
4004, 3916, 2738, 214} \begin {gather*} -\frac {A b \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d}-\frac {2 b^3 \left (a^2 C+A b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d \sqrt {a-b} \sqrt {a+b}}-\frac {b \left (a^2 (2 A+3 C)+3 A b^2\right ) \sin (c+d x)}{3 a^4 d}+\frac {\left (a^2 (3 A+4 C)+4 A b^2\right ) \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac {x \left (a^4 (3 A+4 C)+4 a^2 b^2 (A+2 C)+8 A b^4\right )}{8 a^5}+\frac {A \sin (c+d x) \cos ^3(c+d x)}{4 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 2738
Rule 3916
Rule 4004
Rule 4189
Rule 4190
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx &=\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\int \frac {\cos ^3(c+d x) \left (4 A b-a (3 A+4 C) \sec (c+d x)-3 A b \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{4 a}\\ &=-\frac {A b \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d}+\frac {\int \frac {\cos ^2(c+d x) \left (3 \left (4 A b^2+a^2 (3 A+4 C)\right )+a A b \sec (c+d x)-8 A b^2 \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{12 a^2}\\ &=\frac {\left (4 A b^2+a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {A b \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\int \frac {\cos (c+d x) \left (8 b \left (3 A b^2+a^2 (2 A+3 C)\right )+a \left (4 A b^2-3 a^2 (3 A+4 C)\right ) \sec (c+d x)-3 b \left (4 A b^2+a^2 (3 A+4 C)\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{24 a^3}\\ &=-\frac {b \left (3 A b^2+a^2 (2 A+3 C)\right ) \sin (c+d x)}{3 a^4 d}+\frac {\left (4 A b^2+a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {A b \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d}+\frac {\int \frac {3 \left (8 A b^4+4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right )+3 a b \left (4 A b^2+a^2 (3 A+4 C)\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{24 a^4}\\ &=\frac {\left (8 A b^4+4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) x}{8 a^5}-\frac {b \left (3 A b^2+a^2 (2 A+3 C)\right ) \sin (c+d x)}{3 a^4 d}+\frac {\left (4 A b^2+a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {A b \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\left (b^3 \left (A b^2+a^2 C\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{a^5}\\ &=\frac {\left (8 A b^4+4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) x}{8 a^5}-\frac {b \left (3 A b^2+a^2 (2 A+3 C)\right ) \sin (c+d x)}{3 a^4 d}+\frac {\left (4 A b^2+a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {A b \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\left (b^2 \left (A b^2+a^2 C\right )\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{a^5}\\ &=\frac {\left (8 A b^4+4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) x}{8 a^5}-\frac {b \left (3 A b^2+a^2 (2 A+3 C)\right ) \sin (c+d x)}{3 a^4 d}+\frac {\left (4 A b^2+a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {A b \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\left (2 b^2 \left (A b^2+a^2 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 )}{a^5 d}\\ &=\frac {\left (8 A b^4+4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) x}{8 a^5}-\frac {2 b^3 \left (A b^2+a^2 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 \sqrt {a-b} \sqrt {a+b} d}-\frac {b \left (3 A b^2+a^2 (2 A+3 C)\right ) \sin (c+d x)}{3 a^4 d}+\frac {\left (4 A b^2+a^2 (3 A+4 C)\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {A b \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d}\\ \end {align*}
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Mathematica [A]
time = 0.70, size = 191, normalized size = 0.82 \begin {gather*} \frac {12 \left (8 A b^4+4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) (c+d x)+\frac {192 b^3 \left (A b^2+a^2 C\right ) \tanh ^{-1}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-24 a b \left (4 A b^2+a^2 (3 A+4 C)\right ) \sin (c+d x)+24 a^2 \left (A b^2+a^2 (A+C)\right ) \sin (2 (c+d x))-8 a^3 A b \sin (3 (c+d x))+3 a^4 A \sin (4 (c+d x))}{96 a^5 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 353, normalized size = 1.52
method | result | size |
derivativedivides | \(\frac {-\frac {2 b^{3} \left (A \,b^{2}+a^{2} C \right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5} \sqrt {\left (a +b \right ) \left (a -b \right )}}+\frac {\frac {2 \left (\left (-\frac {5}{8} A \,a^{4}-A \,a^{3} b -\frac {1}{2} a^{2} A \,b^{2}-a A \,b^{3}-\frac {1}{2} a^{4} C -a^{3} b C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {3}{8} A \,a^{4}-\frac {5}{3} A \,a^{3} b -3 a A \,b^{3}-3 a^{3} b C -\frac {1}{2} a^{2} A \,b^{2}-\frac {1}{2} a^{4} C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {3}{8} A \,a^{4}+\frac {1}{2} a^{2} A \,b^{2}+\frac {1}{2} a^{4} C -\frac {5}{3} A \,a^{3} b -3 a A \,b^{3}-3 a^{3} b C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{8} A \,a^{4}+\frac {1}{2} a^{2} A \,b^{2}+\frac {1}{2} a^{4} C -A \,a^{3} b -a A \,b^{3}-a^{3} b C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {\left (3 A \,a^{4}+4 a^{2} A \,b^{2}+8 A \,b^{4}+4 a^{4} C +8 C \,a^{2} b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{a^{5}}}{d}\) | \(353\) |
default | \(\frac {-\frac {2 b^{3} \left (A \,b^{2}+a^{2} C \right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5} \sqrt {\left (a +b \right ) \left (a -b \right )}}+\frac {\frac {2 \left (\left (-\frac {5}{8} A \,a^{4}-A \,a^{3} b -\frac {1}{2} a^{2} A \,b^{2}-a A \,b^{3}-\frac {1}{2} a^{4} C -a^{3} b C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {3}{8} A \,a^{4}-\frac {5}{3} A \,a^{3} b -3 a A \,b^{3}-3 a^{3} b C -\frac {1}{2} a^{2} A \,b^{2}-\frac {1}{2} a^{4} C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {3}{8} A \,a^{4}+\frac {1}{2} a^{2} A \,b^{2}+\frac {1}{2} a^{4} C -\frac {5}{3} A \,a^{3} b -3 a A \,b^{3}-3 a^{3} b C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{8} A \,a^{4}+\frac {1}{2} a^{2} A \,b^{2}+\frac {1}{2} a^{4} C -A \,a^{3} b -a A \,b^{3}-a^{3} b C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {\left (3 A \,a^{4}+4 a^{2} A \,b^{2}+8 A \,b^{4}+4 a^{4} C +8 C \,a^{2} b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{a^{5}}}{d}\) | \(353\) |
risch | \(\frac {3 A x}{8 a}+\frac {x A \,b^{2}}{2 a^{3}}+\frac {x A \,b^{4}}{a^{5}}+\frac {x C}{2 a}+\frac {x C \,b^{2}}{a^{3}}-\frac {i b \,{\mathrm e}^{-i \left (d x +c \right )} C}{2 a^{2} d}+\frac {i b^{3} {\mathrm e}^{i \left (d x +c \right )} A}{2 a^{4} d}+\frac {i b \,{\mathrm e}^{i \left (d x +c \right )} C}{2 a^{2} d}-\frac {3 i A b \,{\mathrm e}^{-i \left (d x +c \right )}}{8 a^{2} d}+\frac {3 i A b \,{\mathrm e}^{i \left (d x +c \right )}}{8 a^{2} d}-\frac {i b^{3} {\mathrm e}^{-i \left (d x +c \right )} A}{2 a^{4} d}+\frac {b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A}{\sqrt {a^{2}-b^{2}}\, d \,a^{5}}+\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) C}{\sqrt {a^{2}-b^{2}}\, d \,a^{3}}-\frac {b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A}{\sqrt {a^{2}-b^{2}}\, d \,a^{5}}-\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) C}{\sqrt {a^{2}-b^{2}}\, d \,a^{3}}+\frac {A \sin \left (4 d x +4 c \right )}{32 a d}-\frac {A b \sin \left (3 d x +3 c \right )}{12 a^{2} d}+\frac {A \sin \left (2 d x +2 c \right )}{4 a d}+\frac {\sin \left (2 d x +2 c \right ) A \,b^{2}}{4 a^{3} d}+\frac {\sin \left (2 d x +2 c \right ) C}{4 a d}\) | \(560\) |
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 = 2.67, size = 599, normalized size = 2.58 \begin {gather*} \left [\frac {3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{6} + {\left (A + 4 \, C\right )} a^{4} b^{2} + 4 \, {\left (A - 2 \, C\right )} a^{2} b^{4} - 8 \, A b^{6}\right )} d x + 12 \, {\left (C a^{2} b^{3} + A b^{5}\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) - {\left (8 \, {\left (2 \, A + 3 \, C\right )} a^{5} b + 8 \, {\left (A - 3 \, C\right )} a^{3} b^{3} - 24 \, A a b^{5} - 6 \, {\left (A a^{6} - A a^{4} b^{2}\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (A a^{5} b - A a^{3} b^{3}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{6} + {\left (A - 4 \, C\right )} a^{4} b^{2} - 4 \, A a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left (a^{7} - a^{5} b^{2}\right )} d}, \frac {3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{6} + {\left (A + 4 \, C\right )} a^{4} b^{2} + 4 \, {\left (A - 2 \, C\right )} a^{2} b^{4} - 8 \, A b^{6}\right )} d x - 24 \, {\left (C a^{2} b^{3} + A b^{5}\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) - {\left (8 \, {\left (2 \, A + 3 \, C\right )} a^{5} b + 8 \, {\left (A - 3 \, C\right )} a^{3} b^{3} - 24 \, A a b^{5} - 6 \, {\left (A a^{6} - A a^{4} b^{2}\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (A a^{5} b - A a^{3} b^{3}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{6} + {\left (A - 4 \, C\right )} a^{4} b^{2} - 4 \, A a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left (a^{7} - a^{5} b^{2}\right )} d}\right ] \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 ) \cos ^{4}{\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, 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. 574 vs.
\(2 (213) = 426\).
time = 0.50, size = 574, normalized size = 2.47 \begin {gather*} \frac {\frac {3 \, {\left (3 \, A a^{4} + 4 \, C a^{4} + 4 \, A a^{2} b^{2} + 8 \, C a^{2} b^{2} + 8 \, A b^{4}\right )} {\left (d x + c\right )}}{a^{5}} - \frac {48 \, {\left (C a^{2} b^{3} + A b^{5}\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 )}}{\sqrt {-a^{2} + b^{2}} a^{5}} - \frac {2 \, {\left (15 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 9 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 72 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 72 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a^{4}}}{24 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.30, size = 2500, normalized size = 10.78 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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