Optimal. Leaf size=219 \[ \frac {1}{2} a^2 \left (12 A b^2+a^2 (A+2 C)\right ) x+\frac {2 a b \left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {2 A b (a+b \sec (c+d x))^3 \sin (c+d x)}{d}+\frac {A \cos (c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{2 d}-\frac {b^2 \left (a^2 (39 A-34 C)-2 b^2 (3 A+2 C)\right ) \tan (c+d x)}{6 d}-\frac {a b^3 (9 A-4 C) \sec (c+d x) \tan (c+d x)}{3 d}-\frac {b^2 (15 A-2 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{6 d} \]
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Rubi [A]
time = 0.41, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4180, 4179,
4141, 4133, 3855, 3852, 8} \begin {gather*} -\frac {b^2 \left (a^2 (39 A-34 C)-2 b^2 (3 A+2 C)\right ) \tan (c+d x)}{6 d}+\frac {2 a b \left (C \left (2 a^2+b^2\right )+2 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {1}{2} a^2 x \left (a^2 (A+2 C)+12 A b^2\right )-\frac {a b^3 (9 A-4 C) \tan (c+d x) \sec (c+d x)}{3 d}-\frac {b^2 (15 A-2 C) \tan (c+d x) (a+b \sec (c+d x))^2}{6 d}+\frac {2 A b \sin (c+d x) (a+b \sec (c+d x))^3}{d}+\frac {A \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^4}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3852
Rule 3855
Rule 4133
Rule 4141
Rule 4179
Rule 4180
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos (c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{2 d}+\frac {1}{2} \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (4 A b+a (A+2 C) \sec (c+d x)-b (3 A-2 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {2 A b (a+b \sec (c+d x))^3 \sin (c+d x)}{d}+\frac {A \cos (c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{2 d}+\frac {1}{2} \int (a+b \sec (c+d x))^2 \left (12 A b^2+a^2 (A+2 C)-2 a b (A-2 C) \sec (c+d x)-b^2 (15 A-2 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {2 A b (a+b \sec (c+d x))^3 \sin (c+d x)}{d}+\frac {A \cos (c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{2 d}-\frac {b^2 (15 A-2 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac {1}{6} \int (a+b \sec (c+d x)) \left (3 a \left (12 A b^2+a^2 (A+2 C)\right )-b \left (3 a^2 (A-6 C)-2 b^2 (3 A+2 C)\right ) \sec (c+d x)-4 a b^2 (9 A-4 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {2 A b (a+b \sec (c+d x))^3 \sin (c+d x)}{d}+\frac {A \cos (c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{2 d}-\frac {a b^3 (9 A-4 C) \sec (c+d x) \tan (c+d x)}{3 d}-\frac {b^2 (15 A-2 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac {1}{12} \int \left (6 a^2 \left (12 A b^2+a^2 (A+2 C)\right )+24 a b \left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) \sec (c+d x)-2 b^2 \left (a^2 (39 A-34 C)-2 b^2 (3 A+2 C)\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {1}{2} a^2 \left (12 A b^2+a^2 (A+2 C)\right ) x+\frac {2 A b (a+b \sec (c+d x))^3 \sin (c+d x)}{d}+\frac {A \cos (c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{2 d}-\frac {a b^3 (9 A-4 C) \sec (c+d x) \tan (c+d x)}{3 d}-\frac {b^2 (15 A-2 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{6 d}+\left (2 a b \left (2 A b^2+\left (2 a^2+b^2\right ) C\right )\right ) \int \sec (c+d x) \, dx-\frac {1}{6} \left (b^2 \left (a^2 (39 A-34 C)-2 b^2 (3 A+2 C)\right )\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac {1}{2} a^2 \left (12 A b^2+a^2 (A+2 C)\right ) x+\frac {2 a b \left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {2 A b (a+b \sec (c+d x))^3 \sin (c+d x)}{d}+\frac {A \cos (c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{2 d}-\frac {a b^3 (9 A-4 C) \sec (c+d x) \tan (c+d x)}{3 d}-\frac {b^2 (15 A-2 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac {\left (b^2 \left (a^2 (39 A-34 C)-2 b^2 (3 A+2 C)\right )\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{6 d}\\ &=\frac {1}{2} a^2 \left (12 A b^2+a^2 (A+2 C)\right ) x+\frac {2 a b \left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {2 A b (a+b \sec (c+d x))^3 \sin (c+d x)}{d}+\frac {A \cos (c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{2 d}-\frac {b^2 \left (a^2 (39 A-34 C)-2 b^2 (3 A+2 C)\right ) \tan (c+d x)}{6 d}-\frac {a b^3 (9 A-4 C) \sec (c+d x) \tan (c+d x)}{3 d}-\frac {b^2 (15 A-2 C) (a+b \sec (c+d x))^2 \tan (c+d x)}{6 d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(864\) vs. \(2(219)=438\).
time = 6.32, size = 864, normalized size = 3.95 \begin {gather*} \frac {a^2 \left (a^2 A+12 A b^2+2 a^2 C\right ) (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{2 d (b+a \cos (c+d x))^4}-\frac {2 \left (2 a A b^3+2 a^3 b C+a b^3 C\right ) \cos ^4(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \sec (c+d x))^4}{d (b+a \cos (c+d x))^4}+\frac {2 \left (2 a A b^3+2 a^3 b C+a b^3 C\right ) \cos ^4(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \sec (c+d x))^4}{d (b+a \cos (c+d x))^4}+\frac {\left (12 a b^3 C+b^4 C\right ) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{12 d (b+a \cos (c+d x))^4 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {b^4 C \cos ^4(c+d x) (a+b \sec (c+d x))^4 \sin \left (\frac {1}{2} (c+d x)\right )}{6 d (b+a \cos (c+d x))^4 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {b^4 C \cos ^4(c+d x) (a+b \sec (c+d x))^4 \sin \left (\frac {1}{2} (c+d x)\right )}{6 d (b+a \cos (c+d x))^4 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {\left (-12 a b^3 C-b^4 C\right ) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{12 d (b+a \cos (c+d x))^4 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {\cos ^4(c+d x) (a+b \sec (c+d x))^4 \left (3 A b^4 \sin \left (\frac {1}{2} (c+d x)\right )+18 a^2 b^2 C \sin \left (\frac {1}{2} (c+d x)\right )+2 b^4 C \sin \left (\frac {1}{2} (c+d x)\right )\right )}{3 d (b+a \cos (c+d x))^4 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {\cos ^4(c+d x) (a+b \sec (c+d x))^4 \left (3 A b^4 \sin \left (\frac {1}{2} (c+d x)\right )+18 a^2 b^2 C \sin \left (\frac {1}{2} (c+d x)\right )+2 b^4 C \sin \left (\frac {1}{2} (c+d x)\right )\right )}{3 d (b+a \cos (c+d x))^4 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {4 a^3 A b \cos ^4(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{d (b+a \cos (c+d x))^4}+\frac {a^4 A \cos ^4(c+d x) (a+b \sec (c+d x))^4 \sin (2 (c+d x))}{4 d (b+a \cos (c+d x))^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 200, normalized size = 0.91 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 221, normalized size = 1.01 \begin {gather*} \frac {3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 12 \, {\left (d x + c\right )} C a^{4} + 72 \, {\left (d x + c\right )} A a^{2} b^{2} + 4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C b^{4} - 12 \, C a b^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, C a^{3} b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, A 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 \, A a^{3} b \sin \left (d x + c\right ) + 72 \, C a^{2} b^{2} \tan \left (d x + c\right ) + 12 \, A b^{4} \tan \left (d x + c\right )}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.42, size = 208, normalized size = 0.95 \begin {gather*} \frac {3 \, {\left ({\left (A + 2 \, C\right )} a^{4} + 12 \, A a^{2} b^{2}\right )} d x \cos \left (d x + c\right )^{3} + 6 \, {\left (2 \, C a^{3} b + {\left (2 \, A + C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 6 \, {\left (2 \, C a^{3} b + {\left (2 \, A + C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (3 \, A a^{4} \cos \left (d x + c\right )^{4} + 24 \, A a^{3} b \cos \left (d x + c\right )^{3} + 12 \, C a b^{3} \cos \left (d x + c\right ) + 2 \, C b^{4} + 2 \, {\left (18 \, C a^{2} b^{2} + {\left (3 \, A + 2 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.51, size = 397, normalized size = 1.81 \begin {gather*} \frac {3 \, {\left (A a^{4} + 2 \, C a^{4} + 12 \, A a^{2} b^{2}\right )} {\left (d x + c\right )} + 12 \, {\left (2 \, C a^{3} b + 2 \, A a b^{3} + C a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 12 \, {\left (2 \, C a^{3} b + 2 \, A a b^{3} + C a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {6 \, {\left (A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, A a^{3} b \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 )}^{2}} - \frac {4 \, {\left (18 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 36 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C b^{4} \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}}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.90, size = 2664, normalized size = 12.16 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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