Optimal. Leaf size=182 \[ \frac {a \left (2 a^2+3 b^2\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{d \left (a^2-b^2\right )^{7/2}}+\frac {b \left (11 a^2+4 b^2\right ) \cos (c+d x)}{6 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}+\frac {5 a b \cos (c+d x)}{6 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}+\frac {b \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3} \]
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Rubi [A] time = 0.22, antiderivative size = 182, 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 = {2664, 2754, 12, 2660, 618, 204} \[ \frac {a \left (2 a^2+3 b^2\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{d \left (a^2-b^2\right )^{7/2}}+\frac {b \left (11 a^2+4 b^2\right ) \cos (c+d x)}{6 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}+\frac {5 a b \cos (c+d x)}{6 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}+\frac {b \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 618
Rule 2660
Rule 2664
Rule 2754
Rubi steps
\begin {align*} \int \frac {1}{(a+b \sin (c+d x))^4} \, dx &=\frac {b \cos (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^3}-\frac {\int \frac {-3 a+2 b \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx}{3 \left (a^2-b^2\right )}\\ &=\frac {b \cos (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^3}+\frac {5 a b \cos (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}+\frac {\int \frac {2 \left (3 a^2+2 b^2\right )-5 a b \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx}{6 \left (a^2-b^2\right )^2}\\ &=\frac {b \cos (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^3}+\frac {5 a b \cos (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}+\frac {b \left (11 a^2+4 b^2\right ) \cos (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac {\int -\frac {3 a \left (2 a^2+3 b^2\right )}{a+b \sin (c+d x)} \, dx}{6 \left (a^2-b^2\right )^3}\\ &=\frac {b \cos (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^3}+\frac {5 a b \cos (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}+\frac {b \left (11 a^2+4 b^2\right ) \cos (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}+\frac {\left (a \left (2 a^2+3 b^2\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{2 \left (a^2-b^2\right )^3}\\ &=\frac {b \cos (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^3}+\frac {5 a b \cos (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}+\frac {b \left (11 a^2+4 b^2\right ) \cos (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}+\frac {\left (a \left (2 a^2+3 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{\left (a^2-b^2\right )^3 d}\\ &=\frac {b \cos (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^3}+\frac {5 a b \cos (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}+\frac {b \left (11 a^2+4 b^2\right ) \cos (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac {\left (2 a \left (2 a^2+3 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{\left (a^2-b^2\right )^3 d}\\ &=\frac {a \left (2 a^2+3 b^2\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2} d}+\frac {b \cos (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^3}+\frac {5 a b \cos (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}+\frac {b \left (11 a^2+4 b^2\right ) \cos (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 1.02, size = 157, normalized size = 0.86 \[ \frac {\frac {6 a \left (2 a^2+3 b^2\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}+\frac {b \cos (c+d x) \left (18 a^4+b^2 \left (11 a^2+4 b^2\right ) \sin ^2(c+d x)+3 a b \left (9 a^2+b^2\right ) \sin (c+d x)-5 a^2 b^2+2 b^4\right )}{(a-b)^3 (a+b)^3 (a+b \sin (c+d x))^3}}{6 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 965, normalized size = 5.30 \[ \left [\frac {2 \, {\left (11 \, a^{4} b^{3} - 7 \, a^{2} b^{5} - 4 \, b^{7}\right )} \cos \left (d x + c\right )^{3} - 6 \, {\left (9 \, a^{5} b^{2} - 8 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 3 \, {\left (2 \, a^{6} + 9 \, a^{4} b^{2} + 9 \, a^{2} b^{4} - 3 \, {\left (2 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (6 \, a^{5} b + 11 \, a^{3} b^{3} + 3 \, a b^{5} - {\left (2 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} - 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - 12 \, {\left (3 \, a^{6} b - 2 \, a^{4} b^{3} - b^{7}\right )} \cos \left (d x + c\right )}{12 \, {\left (3 \, {\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 \cos \left (d x + c\right )^{2} - {\left (a^{11} - a^{9} b^{2} - 6 \, a^{7} b^{4} + 14 \, a^{5} b^{6} - 11 \, a^{3} b^{8} + 3 \, a b^{10}\right )} d + {\left ({\left (a^{8} b^{3} - 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} - 4 \, a^{2} b^{9} + b^{11}\right )} d \cos \left (d x + c\right )^{2} - {\left (3 \, a^{10} b - 11 \, a^{8} b^{3} + 14 \, a^{6} b^{5} - 6 \, a^{4} b^{7} - a^{2} b^{9} + b^{11}\right )} d\right )} \sin \left (d x + c\right )\right )}}, \frac {{\left (11 \, a^{4} b^{3} - 7 \, a^{2} b^{5} - 4 \, b^{7}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (9 \, a^{5} b^{2} - 8 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 3 \, {\left (2 \, a^{6} + 9 \, a^{4} b^{2} + 9 \, a^{2} b^{4} - 3 \, {\left (2 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (6 \, a^{5} b + 11 \, a^{3} b^{3} + 3 \, a b^{5} - {\left (2 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) - 6 \, {\left (3 \, a^{6} b - 2 \, a^{4} b^{3} - b^{7}\right )} \cos \left (d x + c\right )}{6 \, {\left (3 \, {\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 \cos \left (d x + c\right )^{2} - {\left (a^{11} - a^{9} b^{2} - 6 \, a^{7} b^{4} + 14 \, a^{5} b^{6} - 11 \, a^{3} b^{8} + 3 \, a b^{10}\right )} d + {\left ({\left (a^{8} b^{3} - 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} - 4 \, a^{2} b^{9} + b^{11}\right )} d \cos \left (d x + c\right )^{2} - {\left (3 \, a^{10} b - 11 \, a^{8} b^{3} + 14 \, a^{6} b^{5} - 6 \, a^{4} b^{7} - a^{2} b^{9} + b^{11}\right )} d\right )} \sin \left (d x + c\right )\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 510, normalized size = 2.80 \[ \frac {\frac {3 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sqrt {a^{2} - b^{2}}} + \frac {27 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 18 \, a^{4} b^{4} \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} + 18 \, a^{7} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 81 \, a^{5} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 36 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 12 \, a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 108 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 42 \, a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, a^{7} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, a^{5} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 18 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 81 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, a^{4} b^{4} \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 ) + 18 \, a^{7} b - 5 \, a^{5} b^{3} + 2 \, a^{3} b^{5}}{{\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} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.21, size = 1733, normalized size = 9.52 \[ \text {result too large to display} \]
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 = 10.38, size = 708, normalized size = 3.89 \[ \frac {\frac {18\,a^4\,b-5\,a^2\,b^3+2\,b^5}{3\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (6\,a^6\,b+20\,a^4\,b^3-3\,a^2\,b^5+2\,b^7\right )}{a^2\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (6\,a^6\,b+27\,a^4\,b^3-12\,a^2\,b^5+4\,b^7\right )}{a^2\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (27\,a^4\,b-4\,a^2\,b^3+2\,b^5\right )}{a\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (9\,a^4\,b-6\,a^2\,b^3+2\,b^5\right )}{a\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (3\,a^2+2\,b^2\right )\,\left (18\,a^4\,b-5\,a^2\,b^3+2\,b^5\right )}{3\,a^3\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (3\,a^3+12\,a\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (3\,a^3+12\,a\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (12\,a^2\,b+8\,b^3\right )+a^3+6\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+6\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )}+\frac {a\,\mathrm {atan}\left (\frac {\left (\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^2+3\,b^2\right )}{{\left (a+b\right )}^{7/2}\,{\left (a-b\right )}^{7/2}}+\frac {a\,\left (2\,a^2+3\,b^2\right )\,\left (2\,a^6\,b-6\,a^4\,b^3+6\,a^2\,b^5-2\,b^7\right )}{2\,{\left (a+b\right )}^{7/2}\,{\left (a-b\right )}^{7/2}\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}\right )\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}{2\,a^3+3\,a\,b^2}\right )\,\left (2\,a^2+3\,b^2\right )}{d\,{\left (a+b\right )}^{7/2}\,{\left (a-b\right )}^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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