Optimal. Leaf size=242 \[ \frac {x}{a^4}-\frac {b \left (8 a^6-8 a^4 b^2+7 a^2 b^4-2 b^6\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 (a-b)^{7/2} (a+b)^{7/2} d}+\frac {b^2 \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b^2 \left (8 a^2-3 b^2\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (26 a^4-17 a^2 b^2+6 b^4\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \]
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Rubi [A]
time = 0.40, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3870, 4145,
4004, 3916, 2738, 214} \begin {gather*} \frac {x}{a^4}+\frac {b^2 \left (8 a^2-3 b^2\right ) \tan (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {b \left (8 a^6-8 a^4 b^2+7 a^2 b^4-2 b^6\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}+\frac {b^2 \left (26 a^4-17 a^2 b^2+6 b^4\right ) \tan (c+d x)}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 2738
Rule 3870
Rule 3916
Rule 4004
Rule 4145
Rubi steps
\begin {align*} \int \frac {1}{(a+b \sec (c+d x))^4} \, dx &=\frac {b^2 \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\int \frac {-3 \left (a^2-b^2\right )+3 a b \sec (c+d x)-2 b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx}{3 a \left (a^2-b^2\right )}\\ &=\frac {b^2 \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b^2 \left (8 a^2-3 b^2\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\int \frac {6 \left (a^2-b^2\right )^2-2 a b \left (6 a^2-b^2\right ) \sec (c+d x)+b^2 \left (8 a^2-3 b^2\right ) \sec ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx}{6 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {b^2 \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b^2 \left (8 a^2-3 b^2\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (26 a^4-17 a^2 b^2+6 b^4\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\int \frac {-6 \left (a^2-b^2\right )^3+3 a b \left (6 a^4-2 a^2 b^2+b^4\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )^3}\\ &=\frac {x}{a^4}+\frac {b^2 \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b^2 \left (8 a^2-3 b^2\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (26 a^4-17 a^2 b^2+6 b^4\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\left (b \left (8 a^6-8 a^4 b^2+7 a^2 b^4-2 b^6\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^4 \left (a^2-b^2\right )^3}\\ &=\frac {x}{a^4}+\frac {b^2 \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b^2 \left (8 a^2-3 b^2\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (26 a^4-17 a^2 b^2+6 b^4\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\left (8 a^6-8 a^4 b^2+7 a^2 b^4-2 b^6\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 a^4 \left (a^2-b^2\right )^3}\\ &=\frac {x}{a^4}+\frac {b^2 \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b^2 \left (8 a^2-3 b^2\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (26 a^4-17 a^2 b^2+6 b^4\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\left (8 a^6-8 a^4 b^2+7 a^2 b^4-2 b^6\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^4 \left (a^2-b^2\right )^3 d}\\ &=\frac {x}{a^4}-\frac {b \left (8 a^6-8 a^4 b^2+7 a^2 b^4-2 b^6\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 (a-b)^{7/2} (a+b)^{7/2} d}+\frac {b^2 \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b^2 \left (8 a^2-3 b^2\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (26 a^4-17 a^2 b^2+6 b^4\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 1.50, size = 268, normalized size = 1.11 \begin {gather*} \frac {(b+a \cos (c+d x)) \sec ^4(c+d x) \left (6 (c+d x) (b+a \cos (c+d x))^3-\frac {6 b \left (-8 a^6+8 a^4 b^2-7 a^2 b^4+2 b^6\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))^3}{\left (a^2-b^2\right )^{7/2}}+\frac {2 a b^4 \sin (c+d x)}{(a-b) (a+b)}-\frac {a b^3 \left (12 a^2-7 b^2\right ) (b+a \cos (c+d x)) \sin (c+d x)}{(a-b)^2 (a+b)^2}+\frac {a b^2 \left (36 a^4-32 a^2 b^2+11 b^4\right ) (b+a \cos (c+d x))^2 \sin (c+d x)}{(a-b)^3 (a+b)^3}\right )}{6 a^4 d (a+b \sec (c+d x))^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 365, normalized size = 1.51 Too large to display
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. 699 vs.
\(2 (227) = 454\).
time = 3.10, size = 1456, normalized size = 6.02 \begin {gather*} \left [\frac {12 \, {\left (a^{11} - 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} - 4 \, a^{5} b^{6} + a^{3} b^{8}\right )} d x \cos \left (d x + c\right )^{3} + 36 \, {\left (a^{10} b - 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} - 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} d x \cos \left (d x + c\right )^{2} + 36 \, {\left (a^{9} b^{2} - 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} - 4 \, a^{3} b^{8} + a b^{10}\right )} d x \cos \left (d x + c\right ) + 12 \, {\left (a^{8} b^{3} - 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} - 4 \, a^{2} b^{9} + b^{11}\right )} d x + 3 \, {\left (8 \, a^{6} b^{4} - 8 \, a^{4} b^{6} + 7 \, a^{2} b^{8} - 2 \, b^{10} + {\left (8 \, a^{9} b - 8 \, a^{7} b^{3} + 7 \, a^{5} b^{5} - 2 \, a^{3} b^{7}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, a^{8} b^{2} - 8 \, a^{6} b^{4} + 7 \, a^{4} b^{6} - 2 \, a^{2} b^{8}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (8 \, a^{7} b^{3} - 8 \, a^{5} b^{5} + 7 \, a^{3} b^{7} - 2 \, a b^{9}\right )} \cos \left (d x + c\right )\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 ) + 2 \, {\left (26 \, a^{7} b^{4} - 43 \, a^{5} b^{6} + 23 \, a^{3} b^{8} - 6 \, a b^{10} + {\left (36 \, a^{9} b^{2} - 68 \, a^{7} b^{4} + 43 \, a^{5} b^{6} - 11 \, a^{3} b^{8}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (4 \, a^{8} b^{3} - 7 \, a^{6} b^{5} + 4 \, a^{4} b^{7} - a^{2} b^{9}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, {\left ({\left (a^{15} - 4 \, a^{13} b^{2} + 6 \, a^{11} b^{4} - 4 \, a^{9} b^{6} + a^{7} b^{8}\right )} d \cos \left (d x + c\right )^{3} + 3 \, {\left (a^{14} b - 4 \, a^{12} b^{3} + 6 \, a^{10} b^{5} - 4 \, a^{8} b^{7} + a^{6} b^{9}\right )} d \cos \left (d x + c\right )^{2} + 3 \, {\left (a^{13} b^{2} - 4 \, a^{11} b^{4} + 6 \, a^{9} b^{6} - 4 \, a^{7} b^{8} + a^{5} b^{10}\right )} d \cos \left (d x + c\right ) + {\left (a^{12} b^{3} - 4 \, a^{10} b^{5} + 6 \, a^{8} b^{7} - 4 \, a^{6} b^{9} + a^{4} b^{11}\right )} d\right )}}, \frac {6 \, {\left (a^{11} - 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} - 4 \, a^{5} b^{6} + a^{3} b^{8}\right )} d x \cos \left (d x + c\right )^{3} + 18 \, {\left (a^{10} b - 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} - 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} d x \cos \left (d x + c\right )^{2} + 18 \, {\left (a^{9} b^{2} - 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} - 4 \, a^{3} b^{8} + a b^{10}\right )} d x \cos \left (d x + c\right ) + 6 \, {\left (a^{8} b^{3} - 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} - 4 \, a^{2} b^{9} + b^{11}\right )} d x - 3 \, {\left (8 \, a^{6} b^{4} - 8 \, a^{4} b^{6} + 7 \, a^{2} b^{8} - 2 \, b^{10} + {\left (8 \, a^{9} b - 8 \, a^{7} b^{3} + 7 \, a^{5} b^{5} - 2 \, a^{3} b^{7}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, a^{8} b^{2} - 8 \, a^{6} b^{4} + 7 \, a^{4} b^{6} - 2 \, a^{2} b^{8}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (8 \, a^{7} b^{3} - 8 \, a^{5} b^{5} + 7 \, a^{3} b^{7} - 2 \, a b^{9}\right )} \cos \left (d x + c\right )\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 (26 \, a^{7} b^{4} - 43 \, a^{5} b^{6} + 23 \, a^{3} b^{8} - 6 \, a b^{10} + {\left (36 \, a^{9} b^{2} - 68 \, a^{7} b^{4} + 43 \, a^{5} b^{6} - 11 \, a^{3} b^{8}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (4 \, a^{8} b^{3} - 7 \, a^{6} b^{5} + 4 \, a^{4} b^{7} - a^{2} b^{9}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left ({\left (a^{15} - 4 \, a^{13} b^{2} + 6 \, a^{11} b^{4} - 4 \, a^{9} b^{6} + a^{7} b^{8}\right )} d \cos \left (d x + c\right )^{3} + 3 \, {\left (a^{14} b - 4 \, a^{12} b^{3} + 6 \, a^{10} b^{5} - 4 \, a^{8} b^{7} + a^{6} b^{9}\right )} d \cos \left (d x + c\right )^{2} + 3 \, {\left (a^{13} b^{2} - 4 \, a^{11} b^{4} + 6 \, a^{9} b^{6} - 4 \, a^{7} b^{8} + a^{5} b^{10}\right )} d \cos \left (d x + c\right ) + {\left (a^{12} b^{3} - 4 \, a^{10} b^{5} + 6 \, a^{8} b^{7} - 4 \, a^{6} b^{9} + a^{4} b^{11}\right )} d\right )}}\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 {1}{\left (a + b \sec {\left (c + d x \right )}\right )^{4}}\, 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. 532 vs.
\(2 (227) = 454\).
time = 0.47, size = 532, normalized size = 2.20 \begin {gather*} \frac {\frac {3 \, {\left (8 \, a^{6} b - 8 \, a^{4} b^{3} + 7 \, a^{2} b^{5} - 2 \, b^{7}\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^{10} - 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} - a^{4} b^{6}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {36 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 60 \, a^{5} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 45 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 72 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 116 \, a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 56 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 60 \, a^{5} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 45 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}^{3}} + \frac {3 \, {\left (d x + c\right )}}{a^{4}}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 12.79, size = 2500, normalized size = 10.33 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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