Optimal. Leaf size=215 \[ \frac {\left (2 a^2 b B+b^3 B-2 a^3 C-a b^2 (2 A+C)\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}+\frac {2 a^2 \left (A b^2-a (b B-a C)\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^4 \sqrt {a+b} d}+\frac {\left (3 A b^2-3 a b B+3 a^2 C+2 b^2 C\right ) \tan (c+d x)}{3 b^3 d}+\frac {(b B-a C) \sec (c+d x) \tan (c+d x)}{2 b^2 d}+\frac {C \sec ^2(c+d x) \tan (c+d x)}{3 b d} \]
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Rubi [A]
time = 0.53, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {4187, 4177,
4167, 4083, 3855, 3916, 2738, 214} \begin {gather*} \frac {2 a^2 \left (A b^2-a (b B-a C)\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^4 d \sqrt {a-b} \sqrt {a+b}}+\frac {\tan (c+d x) \left (3 a^2 C-3 a b B+3 A b^2+2 b^2 C\right )}{3 b^3 d}+\frac {\left (-2 a^3 C+2 a^2 b B-a b^2 (2 A+C)+b^3 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}+\frac {(b B-a C) \tan (c+d x) \sec (c+d x)}{2 b^2 d}+\frac {C \tan (c+d x) \sec ^2(c+d x)}{3 b d} \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 4187
Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx &=\frac {C \sec ^2(c+d x) \tan (c+d x)}{3 b d}+\frac {\int \frac {\sec ^2(c+d x) \left (2 a C+b (3 A+2 C) \sec (c+d x)+3 (b B-a C) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{3 b}\\ &=\frac {(b B-a C) \sec (c+d x) \tan (c+d x)}{2 b^2 d}+\frac {C \sec ^2(c+d x) \tan (c+d x)}{3 b d}+\frac {\int \frac {\sec (c+d x) \left (3 a (b B-a C)+b (3 b B+a C) \sec (c+d x)+2 \left (3 A b^2-3 a b B+3 a^2 C+2 b^2 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{6 b^2}\\ &=\frac {\left (3 A b^2-3 a b B+3 a^2 C+2 b^2 C\right ) \tan (c+d x)}{3 b^3 d}+\frac {(b B-a C) \sec (c+d x) \tan (c+d x)}{2 b^2 d}+\frac {C \sec ^2(c+d x) \tan (c+d x)}{3 b d}+\frac {\int \frac {\sec (c+d x) \left (3 a b (b B-a C)+3 \left (2 a^2 b B+b^3 B-2 a^3 C-a b^2 (2 A+C)\right ) \sec (c+d x)\right )}{a+b \sec (c+d x)} \, dx}{6 b^3}\\ &=\frac {\left (3 A b^2-3 a b B+3 a^2 C+2 b^2 C\right ) \tan (c+d x)}{3 b^3 d}+\frac {(b B-a C) \sec (c+d x) \tan (c+d x)}{2 b^2 d}+\frac {C \sec ^2(c+d x) \tan (c+d x)}{3 b d}+\frac {\left (2 a^2 b B+b^3 B-2 a^3 C-a b^2 (2 A+C)\right ) \int \sec (c+d x) \, dx}{2 b^4}+\frac {\left (a^2 \left (A b^2-a (b B-a C)\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{b^4}\\ &=\frac {\left (2 a^2 b B+b^3 B-2 a^3 C-a b^2 (2 A+C)\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}+\frac {\left (3 A b^2-3 a b B+3 a^2 C+2 b^2 C\right ) \tan (c+d x)}{3 b^3 d}+\frac {(b B-a C) \sec (c+d x) \tan (c+d x)}{2 b^2 d}+\frac {C \sec ^2(c+d x) \tan (c+d x)}{3 b d}+\frac {\left (a^2 \left (A b^2-a (b B-a C)\right )\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{b^5}\\ &=\frac {\left (2 a^2 b B+b^3 B-2 a^3 C-a b^2 (2 A+C)\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}+\frac {\left (3 A b^2-3 a b B+3 a^2 C+2 b^2 C\right ) \tan (c+d x)}{3 b^3 d}+\frac {(b B-a C) \sec (c+d x) \tan (c+d x)}{2 b^2 d}+\frac {C \sec ^2(c+d x) \tan (c+d x)}{3 b d}+\frac {\left (2 a^2 \left (A b^2-a (b B-a 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^5 d}\\ &=\frac {\left (2 a^2 b B+b^3 B-2 a^3 C-a b^2 (2 A+C)\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}+\frac {2 a^2 \left (A b^2-a (b B-a C)\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^4 \sqrt {a+b} d}+\frac {\left (3 A b^2-3 a b B+3 a^2 C+2 b^2 C\right ) \tan (c+d x)}{3 b^3 d}+\frac {(b B-a C) \sec (c+d x) \tan (c+d x)}{2 b^2 d}+\frac {C \sec ^2(c+d x) \tan (c+d x)}{3 b d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 3.79, size = 512, normalized size = 2.38 \begin {gather*} \frac {(b+a \cos (c+d x)) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (12 \left (-2 a^2 b B-b^3 B+2 a^3 C+a b^2 (2 A+C)\right ) \cos ^3(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-12 \left (-2 a^2 b B-b^3 B+2 a^3 C+a b^2 (2 A+C)\right ) \cos ^3(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-\frac {48 i a^2 \left (A b^2+a (-b B+a C)\right ) \text {ArcTan}\left (\frac {(i \cos (c)+\sin (c)) \left (a \sin (c)+(-b+a \cos (c)) \tan \left (\frac {d x}{2}\right )\right )}{\sqrt {a^2-b^2} \sqrt {(\cos (c)-i \sin (c))^2}}\right ) \cos ^3(c+d x) (\cos (c)-i \sin (c))}{\sqrt {a^2-b^2} \sqrt {(\cos (c)-i \sin (c))^2}}+b \sec (c) \left (12 \left (A b^2-a b B+a^2 C+b^2 C\right ) \sin (d x)-6 \left (A b^2-a b B+a^2 C\right ) \sin (2 c+d x)+3 b^2 B \sin (c+2 d x)-3 a b C \sin (c+2 d x)+3 b^2 B \sin (3 c+2 d x)-3 a b C \sin (3 c+2 d x)+6 A b^2 \sin (2 c+3 d x)-6 a b B \sin (2 c+3 d x)+6 a^2 C \sin (2 c+3 d x)+4 b^2 C \sin (2 c+3 d x)\right )\right )}{12 b^4 d (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x))) (a+b \sec (c+d x))} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.36, size = 369, normalized size = 1.72
method | result | size |
derivativedivides | \(\frac {-\frac {C}{3 b \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-b B +a C +C b}{2 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (2 a A \,b^{2}-2 a^{2} b B -b^{3} B +2 a^{3} C +C \,b^{2} a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{4}}-\frac {2 A \,b^{2}-2 a b B -b^{2} B +2 a^{2} C +a b C +2 b^{2} C}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {C}{3 b \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {b B -a C -C b}{2 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (-2 a A \,b^{2}+2 a^{2} b B +b^{3} B -2 a^{3} C -C \,b^{2} a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{4}}-\frac {2 A \,b^{2}-2 a b B -b^{2} B +2 a^{2} C +a b C +2 b^{2} C}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {2 a^{2} \left (A \,b^{2}-a b B +a^{2} C \right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{4} \sqrt {\left (a +b \right ) \left (a -b \right )}}}{d}\) | \(369\) |
default | \(\frac {-\frac {C}{3 b \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-b B +a C +C b}{2 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (2 a A \,b^{2}-2 a^{2} b B -b^{3} B +2 a^{3} C +C \,b^{2} a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{4}}-\frac {2 A \,b^{2}-2 a b B -b^{2} B +2 a^{2} C +a b C +2 b^{2} C}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {C}{3 b \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {b B -a C -C b}{2 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (-2 a A \,b^{2}+2 a^{2} b B +b^{3} B -2 a^{3} C -C \,b^{2} a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{4}}-\frac {2 A \,b^{2}-2 a b B -b^{2} B +2 a^{2} C +a b C +2 b^{2} C}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {2 a^{2} \left (A \,b^{2}-a b B +a^{2} C \right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{4} \sqrt {\left (a +b \right ) \left (a -b \right )}}}{d}\) | \(369\) |
risch | \(\frac {i \left (-3 B \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+3 C a b \,{\mathrm e}^{5 i \left (d x +c \right )}+6 A \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-6 B a b \,{\mathrm e}^{4 i \left (d x +c \right )}+6 C \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+12 A \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-12 B a b \,{\mathrm e}^{2 i \left (d x +c \right )}+12 C \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+12 C \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+3 B \,b^{2} {\mathrm e}^{i \left (d x +c \right )}-3 C b a \,{\mathrm e}^{i \left (d x +c \right )}+6 A \,b^{2}-6 a b B +6 a^{2} C +4 b^{2} C \right )}{3 d \,b^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{b^{2} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2} B}{b^{3} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{2 b d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{b^{4} d}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 b^{2} d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{b^{2} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2} B}{b^{3} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{2 b d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{b^{4} d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 b^{2} d}+\frac {a^{2} \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 \,b^{2}}-\frac {a^{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 ) B}{\sqrt {a^{2}-b^{2}}\, d \,b^{3}}+\frac {a^{4} \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 \,b^{4}}-\frac {a^{2} \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 \,b^{2}}+\frac {a^{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 ) B}{\sqrt {a^{2}-b^{2}}\, d \,b^{3}}-\frac {a^{4} \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 \,b^{4}}\) | \(889\) |
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 = 38.97, size = 817, normalized size = 3.80 \begin {gather*} \left [\frac {6 \, {\left (C a^{4} - B a^{3} b + A a^{2} b^{2}\right )} \sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )^{3} \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 ) - 3 \, {\left (2 \, C a^{5} - 2 \, B a^{4} b + {\left (2 \, A - C\right )} a^{3} b^{2} + B a^{2} b^{3} - {\left (2 \, A + C\right )} a b^{4} + B b^{5}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (2 \, C a^{5} - 2 \, B a^{4} b + {\left (2 \, A - C\right )} a^{3} b^{2} + B a^{2} b^{3} - {\left (2 \, A + C\right )} a b^{4} + B b^{5}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, C a^{2} b^{3} - 2 \, C b^{5} + 2 \, {\left (3 \, C a^{4} b - 3 \, B a^{3} b^{2} + {\left (3 \, A - C\right )} a^{2} b^{3} + 3 \, B a b^{4} - {\left (3 \, A + 2 \, C\right )} b^{5}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (C a^{3} b^{2} - B a^{2} b^{3} - C a b^{4} + B b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, {\left (a^{2} b^{4} - b^{6}\right )} d \cos \left (d x + c\right )^{3}}, \frac {12 \, {\left (C a^{4} - B a^{3} b + A a^{2} b^{2}\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 ) \cos \left (d x + c\right )^{3} - 3 \, {\left (2 \, C a^{5} - 2 \, B a^{4} b + {\left (2 \, A - C\right )} a^{3} b^{2} + B a^{2} b^{3} - {\left (2 \, A + C\right )} a b^{4} + B b^{5}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (2 \, C a^{5} - 2 \, B a^{4} b + {\left (2 \, A - C\right )} a^{3} b^{2} + B a^{2} b^{3} - {\left (2 \, A + C\right )} a b^{4} + B b^{5}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, C a^{2} b^{3} - 2 \, C b^{5} + 2 \, {\left (3 \, C a^{4} b - 3 \, B a^{3} b^{2} + {\left (3 \, A - C\right )} a^{2} b^{3} + 3 \, B a b^{4} - {\left (3 \, A + 2 \, C\right )} b^{5}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (C a^{3} b^{2} - B a^{2} b^{3} - C a b^{4} + B b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, {\left (a^{2} b^{4} - b^{6}\right )} d \cos \left (d x + c\right )^{3}}\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 + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{3}{\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. 483 vs.
\(2 (197) = 394\).
time = 0.55, size = 483, normalized size = 2.25 \begin {gather*} -\frac {\frac {3 \, {\left (2 \, C a^{3} - 2 \, B a^{2} b + 2 \, A a b^{2} + C a b^{2} - B b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{4}} - \frac {3 \, {\left (2 \, C a^{3} - 2 \, B a^{2} b + 2 \, A a b^{2} + C a b^{2} - B b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{4}} - \frac {12 \, {\left (C a^{4} - B a^{3} b + A a^{2} b^{2}\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}} b^{4}} + \frac {2 \, {\left (6 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C b^{2} \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 )}^{3} b^{3}}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 14.38, size = 2500, normalized size = 11.63 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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