Optimal. Leaf size=161 \[ -\frac {2 b (a-b)^{3/2} (a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d}+\frac {\sin ^3(c+d x) (4 b-3 a \cos (c+d x))}{12 a^2 d}+\frac {\sin (c+d x) \left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cos (c+d x)\right )}{8 a^4 d}+\frac {x \left (3 a^4-12 a^2 b^2+8 b^4\right )}{8 a^5} \]
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Rubi [A] time = 0.38, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3872, 2865, 2735, 2659, 208} \[ \frac {\sin (c+d x) \left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cos (c+d x)\right )}{8 a^4 d}+\frac {x \left (-12 a^2 b^2+3 a^4+8 b^4\right )}{8 a^5}+\frac {\sin ^3(c+d x) (4 b-3 a \cos (c+d x))}{12 a^2 d}-\frac {2 b (a-b)^{3/2} (a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d} \]
Antiderivative was successfully verified.
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Rule 208
Rule 2659
Rule 2735
Rule 2865
Rule 3872
Rubi steps
\begin {align*} \int \frac {\sin ^4(c+d x)}{a+b \sec (c+d x)} \, dx &=-\int \frac {\cos (c+d x) \sin ^4(c+d x)}{-b-a \cos (c+d x)} \, dx\\ &=\frac {(4 b-3 a \cos (c+d x)) \sin ^3(c+d x)}{12 a^2 d}-\frac {\int \frac {\left (-a b+\left (3 a^2-4 b^2\right ) \cos (c+d x)\right ) \sin ^2(c+d x)}{-b-a \cos (c+d x)} \, dx}{4 a^2}\\ &=\frac {\left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cos (c+d x)\right ) \sin (c+d x)}{8 a^4 d}+\frac {(4 b-3 a \cos (c+d x)) \sin ^3(c+d x)}{12 a^2 d}-\frac {\int \frac {-a b \left (5 a^2-4 b^2\right )+\left (3 a^4-12 a^2 b^2+8 b^4\right ) \cos (c+d x)}{-b-a \cos (c+d x)} \, dx}{8 a^4}\\ &=\frac {\left (3 a^4-12 a^2 b^2+8 b^4\right ) x}{8 a^5}+\frac {\left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cos (c+d x)\right ) \sin (c+d x)}{8 a^4 d}+\frac {(4 b-3 a \cos (c+d x)) \sin ^3(c+d x)}{12 a^2 d}+\frac {\left (b \left (a^2-b^2\right )^2\right ) \int \frac {1}{-b-a \cos (c+d x)} \, dx}{a^5}\\ &=\frac {\left (3 a^4-12 a^2 b^2+8 b^4\right ) x}{8 a^5}+\frac {\left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cos (c+d x)\right ) \sin (c+d x)}{8 a^4 d}+\frac {(4 b-3 a \cos (c+d x)) \sin ^3(c+d x)}{12 a^2 d}+\frac {\left (2 b \left (a^2-b^2\right )^2\right ) \operatorname {Subst}\left (\int \frac {1}{-a-b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 d}\\ &=\frac {\left (3 a^4-12 a^2 b^2+8 b^4\right ) x}{8 a^5}-\frac {2 (a-b)^{3/2} b (a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d}+\frac {\left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cos (c+d x)\right ) \sin (c+d x)}{8 a^4 d}+\frac {(4 b-3 a \cos (c+d x)) \sin ^3(c+d x)}{12 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.82, size = 172, normalized size = 1.07 \[ \frac {3 a^4 \sin (4 (c+d x))+36 a^4 c+36 a^4 d x-8 a^3 b \sin (3 (c+d x))+24 a b \left (5 a^2-4 b^2\right ) \sin (c+d x)+192 b \left (a^2-b^2\right )^{3/2} \tanh ^{-1}\left (\frac {(b-a) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )-144 a^2 b^2 c-144 a^2 b^2 d x-24 \left (a^4-a^2 b^2\right ) \sin (2 (c+d x))+96 b^4 c+96 b^4 d x}{96 a^5 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 393, normalized size = 2.44 \[ \left [\frac {3 \, {\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\right )} d x - 12 \, {\left (a^{2} b - b^{3}\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) + {\left (6 \, a^{4} \cos \left (d x + c\right )^{3} - 8 \, a^{3} b \cos \left (d x + c\right )^{2} + 32 \, a^{3} b - 24 \, a b^{3} - 3 \, {\left (5 \, a^{4} - 4 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, a^{5} d}, \frac {3 \, {\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\right )} d x - 24 \, {\left (a^{2} b - b^{3}\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) + {\left (6 \, a^{4} \cos \left (d x + c\right )^{3} - 8 \, a^{3} b \cos \left (d x + c\right )^{2} + 32 \, a^{3} b - 24 \, a b^{3} - 3 \, {\left (5 \, a^{4} - 4 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, a^{5} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 407, normalized size = 2.53 \[ \frac {\frac {3 \, {\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\right )} {\left (d x + c\right )}}{a^{5}} - \frac {48 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{\sqrt {-a^{2} + b^{2}} a^{5}} + \frac {2 \, {\left (9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 12 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 33 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 104 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 72 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 33 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 104 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a^{4}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.45, size = 769, normalized size = 4.78 \[ \text {result too large to display} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.29, size = 317, normalized size = 1.97 \[ \frac {\frac {5\,b\,\sin \left (c+d\,x\right )}{4}-\frac {b\,\sin \left (3\,c+3\,d\,x\right )}{12}}{a^2\,d}-\frac {3\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )-\frac {b^2\,\sin \left (2\,c+2\,d\,x\right )}{4}}{a^3\,d}+\frac {\frac {3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4}-\frac {\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {\sin \left (4\,c+4\,d\,x\right )}{32}}{a\,d}-\frac {b^3\,\sin \left (c+d\,x\right )}{a^4\,d}+\frac {2\,b^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{a^5\,d}-\frac {2\,b\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3+\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b-\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b^2-\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^3}\right )\,\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}}{a^5\,d} \]
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 \[ \int \frac {\sin ^{4}{\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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