Optimal. Leaf size=293 \[ \frac {1}{8} \left (8 A b^4+16 a^3 b B+32 a b^3 B+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) x+\frac {b^3 (b B+4 a C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a \left (12 A b^3+8 a^3 B+36 a b^2 B+a^2 b (23 A+36 C)\right ) \sin (c+d x)}{12 d}+\frac {\left (4 A b^2+8 a b B+a^2 (3 A+4 C)\right ) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{8 d}+\frac {(A b+a B) \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{4 d}-\frac {b^2 \left (32 a b B+2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \tan (c+d x)}{24 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.68, antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {4179, 4161,
4132, 8, 4130, 3855} \begin {gather*} -\frac {b^2 \tan (c+d x) \left (3 a^2 (3 A+4 C)+32 a b B+2 b^2 (13 A-12 C)\right )}{24 d}+\frac {\sin (c+d x) \cos (c+d x) \left (a^2 (3 A+4 C)+8 a b B+4 A b^2\right ) (a+b \sec (c+d x))^2}{8 d}+\frac {a \sin (c+d x) \left (8 a^3 B+a^2 b (23 A+36 C)+36 a b^2 B+12 A b^3\right )}{12 d}+\frac {1}{8} x \left (a^4 (3 A+4 C)+16 a^3 b B+24 a^2 b^2 (A+2 C)+32 a b^3 B+8 A b^4\right )+\frac {(a B+A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^4}{4 d}+\frac {b^3 (4 a C+b B) \tanh ^{-1}(\sin (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3855
Rule 4130
Rule 4132
Rule 4161
Rule 4179
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{4 d}+\frac {1}{4} \int \cos ^3(c+d x) (a+b \sec (c+d x))^3 \left (4 (A b+a B)+(3 a A+4 b B+4 a C) \sec (c+d x)-b (A-4 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {(A b+a B) \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{4 d}+\frac {1}{12} \int \cos ^2(c+d x) (a+b \sec (c+d x))^2 \left (3 \left (4 A b^2+8 a b B+a^2 (3 A+4 C)\right )+2 \left (4 a^2 B+6 b^2 B+a b (7 A+12 C)\right ) \sec (c+d x)-b (7 A b+4 a B-12 b C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {\left (4 A b^2+8 a b B+a^2 (3 A+4 C)\right ) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{8 d}+\frac {(A b+a B) \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{4 d}+\frac {1}{24} \int \cos (c+d x) (a+b \sec (c+d x)) \left (2 \left (12 A b^3+8 a^3 B+36 a b^2 B+\frac {1}{2} a^2 (46 A b+72 b C)\right )+\left (32 a^2 b B+24 b^3 B+3 a^3 (3 A+4 C)+2 a b^2 (13 A+36 C)\right ) \sec (c+d x)-b \left (32 a b B+2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {\left (4 A b^2+8 a b B+a^2 (3 A+4 C)\right ) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{8 d}+\frac {(A b+a B) \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{4 d}-\frac {b^2 \left (32 a b B+2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \tan (c+d x)}{24 d}+\frac {1}{24} \int \cos (c+d x) \left (2 a \left (12 A b^3+8 a^3 B+36 a b^2 B+a^2 b (23 A+36 C)\right )+3 \left (8 A b^4+16 a^3 b B+32 a b^3 B+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) \sec (c+d x)+24 b^3 (b B+4 a C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {\left (4 A b^2+8 a b B+a^2 (3 A+4 C)\right ) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{8 d}+\frac {(A b+a B) \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{4 d}-\frac {b^2 \left (32 a b B+2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \tan (c+d x)}{24 d}+\frac {1}{24} \int \cos (c+d x) \left (2 a \left (12 A b^3+8 a^3 B+36 a b^2 B+a^2 b (23 A+36 C)\right )+24 b^3 (b B+4 a C) \sec ^2(c+d x)\right ) \, dx+\frac {1}{8} \left (8 A b^4+16 a^3 b B+32 a b^3 B+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) \int 1 \, dx\\ &=\frac {1}{8} \left (8 A b^4+16 a^3 b B+32 a b^3 B+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) x+\frac {a \left (12 A b^3+8 a^3 B+36 a b^2 B+a^2 b (23 A+36 C)\right ) \sin (c+d x)}{12 d}+\frac {\left (4 A b^2+8 a b B+a^2 (3 A+4 C)\right ) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{8 d}+\frac {(A b+a B) \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{4 d}-\frac {b^2 \left (32 a b B+2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \tan (c+d x)}{24 d}+\left (b^3 (b B+4 a C)\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{8} \left (8 A b^4+16 a^3 b B+32 a b^3 B+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) x+\frac {b^3 (b B+4 a C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a \left (12 A b^3+8 a^3 B+36 a b^2 B+a^2 b (23 A+36 C)\right ) \sin (c+d x)}{12 d}+\frac {\left (4 A b^2+8 a b B+a^2 (3 A+4 C)\right ) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{8 d}+\frac {(A b+a B) \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{4 d}-\frac {b^2 \left (32 a b B+2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \tan (c+d x)}{24 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 3.88, size = 382, normalized size = 1.30 \begin {gather*} \frac {32 a \left (24 A b^3+5 a^3 B+36 a b^2 B+4 a^2 b (5 A+6 C)\right ) \sin (c+d x)+a^2 \sec (c+d x) \left (3 \left (48 A b^2+32 a b B+a^2 (9 A+8 C)\right ) \sin (3 (c+d x))+a (8 (4 A b+a B) \sin (4 (c+d x))+3 a A \sin (5 (c+d x)))\right )+24 \left (3 a^4 A c+24 a^2 A b^2 c+8 A b^4 c+16 a^3 b B c+32 a b^3 B c+4 a^4 c C+48 a^2 b^2 c C+3 a^4 A d x+24 a^2 A b^2 d x+8 A b^4 d x+16 a^3 b B d x+32 a b^3 B d x+4 a^4 C d x+48 a^2 b^2 C d x-8 b^3 (b B+4 a C) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+8 b^4 B \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+32 a b^3 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\left (6 a^2 A b^2+4 a^3 b B+8 b^4 C+a^4 (A+C)\right ) \tan (c+d x)\right )}{192 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.09, size = 304, normalized size = 1.04
method | result | size |
derivativedivides | \(\frac {A \,b^{4} \left (d x +c \right )+b^{4} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C \,b^{4} \tan \left (d x +c \right )+4 a A \,b^{3} \sin \left (d x +c \right )+4 a \,b^{3} B \left (d x +c \right )+4 C \,b^{3} a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 a^{2} A \,b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 a^{2} b^{2} B \sin \left (d x +c \right )+6 C \,a^{2} b^{2} \left (d x +c \right )+\frac {4 A \,a^{3} b \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 a^{3} b B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{3} b C \sin \left (d x +c \right )+A \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{4} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{4} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(304\) |
default | \(\frac {A \,b^{4} \left (d x +c \right )+b^{4} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C \,b^{4} \tan \left (d x +c \right )+4 a A \,b^{3} \sin \left (d x +c \right )+4 a \,b^{3} B \left (d x +c \right )+4 C \,b^{3} a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 a^{2} A \,b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 a^{2} b^{2} B \sin \left (d x +c \right )+6 C \,a^{2} b^{2} \left (d x +c \right )+\frac {4 A \,a^{3} b \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 a^{3} b B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{3} b C \sin \left (d x +c \right )+A \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{4} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{4} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(304\) |
risch | \(\frac {3 a^{4} A x}{8}+\frac {2 i C \,b^{4}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} a^{4} C}{8 d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} a^{4} C}{8 d}+\frac {A \,a^{3} b \sin \left (3 d x +3 c \right )}{3 d}+3 A \,a^{2} b^{2} x +\frac {\sin \left (3 d x +3 c \right ) a^{4} B}{12 d}+4 x a \,b^{3} B -\frac {3 i A \,a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {3 i A \,a^{3} b \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}+6 x C \,a^{2} b^{2}+\frac {A \,a^{4} \sin \left (4 d x +4 c \right )}{32 d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} a^{3} b B}{2 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} a^{2} b^{2} B}{d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} a^{2} b^{2} B}{d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} a^{3} b B}{2 d}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} a A \,b^{3}}{d}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} a A \,b^{3}}{d}-\frac {3 i {\mathrm e}^{2 i \left (d x +c \right )} a^{2} A \,b^{2}}{4 d}+\frac {3 i {\mathrm e}^{-2 i \left (d x +c \right )} a^{2} A \,b^{2}}{4 d}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} a^{3} b C}{d}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} a^{3} b C}{d}+\frac {C \,a^{4} x}{2}+\frac {4 b^{3} a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}-\frac {4 b^{3} a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}+x A \,b^{4}-\frac {i A \,a^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {i A \,a^{4} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} a^{4} B}{8 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} a^{4} B}{8 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b^{4} B}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b^{4} B}{d}+2 B \,a^{3} b x\) | \(587\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 305, normalized size = 1.04 \begin {gather*} \frac {3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 128 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} b + 96 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} b + 144 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} b^{2} + 576 \, {\left (d x + c\right )} C a^{2} b^{2} + 384 \, {\left (d x + c\right )} B a b^{3} + 96 \, {\left (d x + c\right )} A b^{4} + 192 \, C a b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, B b^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 384 \, C a^{3} b \sin \left (d x + c\right ) + 576 \, B a^{2} b^{2} \sin \left (d x + c\right ) + 384 \, A a b^{3} \sin \left (d x + c\right ) + 96 \, C b^{4} \tan \left (d x + c\right )}{96 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.52, size = 263, normalized size = 0.90 \begin {gather*} \frac {3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{4} + 16 \, B a^{3} b + 24 \, {\left (A + 2 \, C\right )} a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} d x \cos \left (d x + c\right ) + 12 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 12 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (6 \, A a^{4} \cos \left (d x + c\right )^{4} + 24 \, C b^{4} + 8 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 16 \, {\left (B a^{4} + 2 \, {\left (2 \, A + 3 \, C\right )} a^{3} b + 9 \, B a^{2} b^{2} + 6 \, A a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d \cos \left (d x + c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 802 vs.
\(2 (281) = 562\).
time = 0.53, size = 802, normalized size = 2.74 \begin {gather*} -\frac {\frac {48 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - 3 \, {\left (3 \, A a^{4} + 4 \, C a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2} + 48 \, C a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} {\left (d x + c\right )} - 24 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 24 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (15 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 48 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 144 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 9 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 160 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 48 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 288 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 432 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 288 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 160 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 288 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 432 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 288 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 48 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 144 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, A 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}}}{24 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 8.62, size = 2500, normalized size = 8.53 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________