Optimal. Leaf size=191 \[ \frac {8 (A+2 C) \sin ^3(c+d x)}{3 a^2 d}-\frac {8 (A+2 C) \sin (c+d x)}{a^2 d}-\frac {2 (A+2 C) \sin (c+d x) \cos ^4(c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac {(28 A+55 C) \sin (c+d x) \cos ^3(c+d x)}{12 a^2 d}+\frac {(28 A+55 C) \sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac {x (28 A+55 C)}{8 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.34, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3042, 2977, 2748, 2633, 2635, 8} \[ \frac {8 (A+2 C) \sin ^3(c+d x)}{3 a^2 d}-\frac {8 (A+2 C) \sin (c+d x)}{a^2 d}-\frac {2 (A+2 C) \sin (c+d x) \cos ^4(c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac {(28 A+55 C) \sin (c+d x) \cos ^3(c+d x)}{12 a^2 d}+\frac {(28 A+55 C) \sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac {x (28 A+55 C)}{8 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2748
Rule 2977
Rule 3042
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx &=-\frac {(A+C) \cos ^5(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {\int \frac {\cos ^4(c+d x) (-a (2 A+5 C)+a (4 A+7 C) \cos (c+d x))}{a+a \cos (c+d x)} \, dx}{3 a^2}\\ &=-\frac {2 (A+2 C) \cos ^4(c+d x) \sin (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac {(A+C) \cos ^5(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {\int \cos ^3(c+d x) \left (-24 a^2 (A+2 C)+a^2 (28 A+55 C) \cos (c+d x)\right ) \, dx}{3 a^4}\\ &=-\frac {2 (A+2 C) \cos ^4(c+d x) \sin (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac {(A+C) \cos ^5(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac {(8 (A+2 C)) \int \cos ^3(c+d x) \, dx}{a^2}+\frac {(28 A+55 C) \int \cos ^4(c+d x) \, dx}{3 a^2}\\ &=\frac {(28 A+55 C) \cos ^3(c+d x) \sin (c+d x)}{12 a^2 d}-\frac {2 (A+2 C) \cos ^4(c+d x) \sin (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac {(A+C) \cos ^5(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {(28 A+55 C) \int \cos ^2(c+d x) \, dx}{4 a^2}+\frac {(8 (A+2 C)) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{a^2 d}\\ &=-\frac {8 (A+2 C) \sin (c+d x)}{a^2 d}+\frac {(28 A+55 C) \cos (c+d x) \sin (c+d x)}{8 a^2 d}+\frac {(28 A+55 C) \cos ^3(c+d x) \sin (c+d x)}{12 a^2 d}-\frac {2 (A+2 C) \cos ^4(c+d x) \sin (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac {(A+C) \cos ^5(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {8 (A+2 C) \sin ^3(c+d x)}{3 a^2 d}+\frac {(28 A+55 C) \int 1 \, dx}{8 a^2}\\ &=\frac {(28 A+55 C) x}{8 a^2}-\frac {8 (A+2 C) \sin (c+d x)}{a^2 d}+\frac {(28 A+55 C) \cos (c+d x) \sin (c+d x)}{8 a^2 d}+\frac {(28 A+55 C) \cos ^3(c+d x) \sin (c+d x)}{12 a^2 d}-\frac {2 (A+2 C) \cos ^4(c+d x) \sin (c+d x)}{a^2 d (1+\cos (c+d x))}-\frac {(A+C) \cos ^5(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {8 (A+2 C) \sin ^3(c+d x)}{3 a^2 d}\\ \end {align*}
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Mathematica [B] time = 0.88, size = 399, normalized size = 2.09 \[ \frac {\sec \left (\frac {c}{2}\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \left (72 d x (28 A+55 C) \cos \left (c+\frac {d x}{2}\right )+1176 A \sin \left (c+\frac {d x}{2}\right )-1912 A \sin \left (c+\frac {3 d x}{2}\right )-504 A \sin \left (2 c+\frac {3 d x}{2}\right )-120 A \sin \left (2 c+\frac {5 d x}{2}\right )-120 A \sin \left (3 c+\frac {5 d x}{2}\right )+24 A \sin \left (3 c+\frac {7 d x}{2}\right )+24 A \sin \left (4 c+\frac {7 d x}{2}\right )+672 A d x \cos \left (c+\frac {3 d x}{2}\right )+672 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+72 d x (28 A+55 C) \cos \left (\frac {d x}{2}\right )-3048 A \sin \left (\frac {d x}{2}\right )+1344 C \sin \left (c+\frac {d x}{2}\right )-3488 C \sin \left (c+\frac {3 d x}{2}\right )-1312 C \sin \left (2 c+\frac {3 d x}{2}\right )-285 C \sin \left (2 c+\frac {5 d x}{2}\right )-285 C \sin \left (3 c+\frac {5 d x}{2}\right )+57 C \sin \left (3 c+\frac {7 d x}{2}\right )+57 C \sin \left (4 c+\frac {7 d x}{2}\right )-7 C \sin \left (4 c+\frac {9 d x}{2}\right )-7 C \sin \left (5 c+\frac {9 d x}{2}\right )+3 C \sin \left (5 c+\frac {11 d x}{2}\right )+3 C \sin \left (6 c+\frac {11 d x}{2}\right )+1320 C d x \cos \left (c+\frac {3 d x}{2}\right )+1320 C d x \cos \left (2 c+\frac {3 d x}{2}\right )-5184 C \sin \left (\frac {d x}{2}\right )\right )}{384 a^2 d (\cos (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 167, normalized size = 0.87 \[ \frac {3 \, {\left (28 \, A + 55 \, C\right )} d x \cos \left (d x + c\right )^{2} + 6 \, {\left (28 \, A + 55 \, C\right )} d x \cos \left (d x + c\right ) + 3 \, {\left (28 \, A + 55 \, C\right )} d x + {\left (6 \, C \cos \left (d x + c\right )^{5} - 4 \, C \cos \left (d x + c\right )^{4} + {\left (12 \, A + 19 \, C\right )} \cos \left (d x + c\right )^{3} - 6 \, {\left (4 \, A + 9 \, C\right )} \cos \left (d x + c\right )^{2} - {\left (172 \, A + 347 \, C\right )} \cos \left (d x + c\right ) - 128 \, A - 256 \, C\right )} \sin \left (d x + c\right )}{24 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 220, normalized size = 1.15 \[ \frac {\frac {3 \, {\left (d x + c\right )} {\left (28 \, A + 55 \, C\right )}}{a^{2}} + \frac {4 \, {\left (A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 21 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 33 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{6}} - \frac {2 \, {\left (60 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 195 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 156 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 395 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 132 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 341 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 93 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a^{2}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 392, normalized size = 2.05 \[ \frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{6 d \,a^{2}}+\frac {C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d \,a^{2}}-\frac {7 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{2}}-\frac {11 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{2}}-\frac {5 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {65 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{4 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {13 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {395 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{12 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {341 C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {3 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {31 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {7 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{d \,a^{2}}+\frac {55 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{4 d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 415, normalized size = 2.17 \[ -\frac {C {\left (\frac {\frac {93 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {341 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {395 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {195 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{2} + \frac {4 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} + \frac {2 \, {\left (\frac {33 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{2}} - \frac {165 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )} + 2 \, A {\left (\frac {6 \, {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {5 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{2} + \frac {2 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {42 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.04, size = 220, normalized size = 1.15 \[ \frac {x\,\left (28\,A+55\,C\right )}{8\,a^2}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,\left (A+C\right )}{2\,a^2}+\frac {2\,A+6\,C}{2\,a^2}\right )}{d}-\frac {\left (5\,A+\frac {65\,C}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (13\,A+\frac {395\,C}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (11\,A+\frac {341\,C}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (3\,A+\frac {31\,C}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+4\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^2\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (A+C\right )}{6\,a^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 17.38, size = 2161, normalized size = 11.31 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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