Optimal. Leaf size=238 \[ \frac {2 \left (a^4-5 a^2 b^2+4 b^4\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \sqrt {a^2-b^2} d}-\frac {b \left (3 a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac {\left (7 a^2-12 b^2\right ) \cot (c+d x)}{3 a^4 d}-\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^3 b d}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))} \]
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Rubi [A]
time = 0.47, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2803, 3134,
3080, 3855, 2739, 632, 210} \begin {gather*} \frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {b \left (3 a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac {\left (7 a^2-12 b^2\right ) \cot (c+d x)}{3 a^4 d}-\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^3 b d}+\frac {2 \left (a^4-5 a^2 b^2+4 b^4\right ) \text {ArcTan}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^5 d \sqrt {a^2-b^2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 2803
Rule 3080
Rule 3134
Rule 3855
Rubi steps
\begin {align*} \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^3(c+d x) \left (6 \left (a^2-2 b^2\right )-a b \sin (c+d x)-\left (3 a^2-8 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{3 a^2 b}\\ &=-\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^3 b d}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^2(c+d x) \left (-2 b \left (7 a^2-12 b^2\right )+4 a b^2 \sin (c+d x)+6 b \left (a^2-2 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{6 a^3 b}\\ &=\frac {\left (7 a^2-12 b^2\right ) \cot (c+d x)}{3 a^4 d}-\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^3 b d}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x) \left (6 b^2 \left (3 a^2-4 b^2\right )+6 a b \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{6 a^4 b}\\ &=\frac {\left (7 a^2-12 b^2\right ) \cot (c+d x)}{3 a^4 d}-\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^3 b d}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}+\frac {\left (b \left (3 a^2-4 b^2\right )\right ) \int \csc (c+d x) \, dx}{a^5}+\frac {\left (a^4-5 a^2 b^2+4 b^4\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^5}\\ &=-\frac {b \left (3 a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac {\left (7 a^2-12 b^2\right ) \cot (c+d x)}{3 a^4 d}-\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^3 b d}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}+\frac {\left (2 \left (a^4-5 a^2 b^2+4 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 d}\\ &=-\frac {b \left (3 a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac {\left (7 a^2-12 b^2\right ) \cot (c+d x)}{3 a^4 d}-\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^3 b d}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac {\left (4 \left (a^4-5 a^2 b^2+4 b^4\right )\right ) \text {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 )}{a^5 d}\\ &=\frac {2 \left (a^4-5 a^2 b^2+4 b^4\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \sqrt {a^2-b^2} d}-\frac {b \left (3 a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac {\left (7 a^2-12 b^2\right ) \cot (c+d x)}{3 a^4 d}-\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^3 b d}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 6.17, size = 403, normalized size = 1.69 \begin {gather*} \frac {2 \left (a^4-5 a^2 b^2+4 b^4\right ) \tan ^{-1}\left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \sqrt {a^2-b^2} d}+\frac {\left (4 a^2 \cos \left (\frac {1}{2} (c+d x)\right )-9 b^2 \cos \left (\frac {1}{2} (c+d x)\right )\right ) \csc \left (\frac {1}{2} (c+d x)\right )}{6 a^4 d}+\frac {b \csc ^2\left (\frac {1}{2} (c+d x)\right )}{4 a^3 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{24 a^2 d}+\frac {\left (-3 a^2 b+4 b^3\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 d}+\frac {\left (3 a^2 b-4 b^3\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 d}-\frac {b \sec ^2\left (\frac {1}{2} (c+d x)\right )}{4 a^3 d}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (-4 a^2 \sin \left (\frac {1}{2} (c+d x)\right )+9 b^2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{6 a^4 d}+\frac {a^2 b \cos (c+d x)-b^3 \cos (c+d x)}{a^4 d (a+b \sin (c+d x))}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{24 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.49, size = 287, normalized size = 1.21
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{3}-2 a b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+12 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{4}}+\frac {\frac {2 \left (b^{2} \left (a^{2}-b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a b \left (a^{2}-b^{2}\right )\right )}{a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {2 \left (a^{4}-5 a^{2} b^{2}+4 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{a^{5}}-\frac {1}{24 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-5 a^{2}+12 b^{2}}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{4 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {b \left (3 a^{2}-4 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{5}}}{d}\) | \(287\) |
default | \(\frac {\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{3}-2 a b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+12 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{4}}+\frac {\frac {2 \left (b^{2} \left (a^{2}-b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a b \left (a^{2}-b^{2}\right )\right )}{a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {2 \left (a^{4}-5 a^{2} b^{2}+4 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{a^{5}}-\frac {1}{24 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-5 a^{2}+12 b^{2}}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{4 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {b \left (3 a^{2}-4 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{5}}}{d}\) | \(287\) |
risch | \(\frac {12 b^{2} a \,{\mathrm e}^{i \left (d x +c \right )}+\frac {50 i a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}}{3}-\frac {22 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{3}-28 a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+24 i b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-\frac {14 i a^{2} b}{3}-4 a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+20 a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-24 i b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+2 a^{3} {\mathrm e}^{7 i \left (d x +c \right )}-14 i a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+8 i b^{3}+2 i a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-8 i b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-14 a^{3} {\mathrm e}^{5 i \left (d x +c \right )}+14 a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right ) d \,a^{4}}+\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a^{3} d}-\frac {4 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{5}}-\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{3} d}+\frac {4 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{5}}+\frac {\sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,a^{3}}-\frac {4 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right ) b^{2}}{d \,a^{5}}-\frac {\sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {-i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,a^{3}}+\frac {4 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {-i a +\sqrt {-a^{2}+b^{2}}}{b}\right ) b^{2}}{d \,a^{5}}\) | \(571\) |
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. 533 vs.
\(2 (227) = 454\).
time = 0.46, size = 1149, normalized size = 4.83 \begin {gather*} \left [-\frac {4 \, {\left (2 \, a^{4} - 3 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left ({\left (a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{4} + a^{2} b - 4 \, b^{3} - 2 \, {\left (a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{3} - 4 \, a b^{2} - {\left (a^{3} - 4 \, a b^{2}\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 ) - 6 \, {\left (a^{4} - 2 \, a^{2} b^{2}\right )} \cos \left (d x + c\right ) + 3 \, {\left ({\left (3 \, a^{2} b^{2} - 4 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 3 \, a^{2} b^{2} - 4 \, b^{4} - 2 \, {\left (3 \, a^{2} b^{2} - 4 \, b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (3 \, a^{3} b - 4 \, a b^{3} - {\left (3 \, a^{3} b - 4 \, a b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left ({\left (3 \, a^{2} b^{2} - 4 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 3 \, a^{2} b^{2} - 4 \, b^{4} - 2 \, {\left (3 \, a^{2} b^{2} - 4 \, b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (3 \, a^{3} b - 4 \, a b^{3} - {\left (3 \, a^{3} b - 4 \, a b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left ({\left (7 \, a^{3} b - 12 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (3 \, a^{3} b - 4 \, a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left (a^{5} b d \cos \left (d x + c\right )^{4} - 2 \, a^{5} b d \cos \left (d x + c\right )^{2} + a^{5} b d - {\left (a^{6} d \cos \left (d x + c\right )^{2} - a^{6} d\right )} \sin \left (d x + c\right )\right )}}, -\frac {4 \, {\left (2 \, a^{4} - 3 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left ({\left (a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{4} + a^{2} b - 4 \, b^{3} - 2 \, {\left (a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{3} - 4 \, a b^{2} - {\left (a^{3} - 4 \, a b^{2}\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 (a^{4} - 2 \, a^{2} b^{2}\right )} \cos \left (d x + c\right ) + 3 \, {\left ({\left (3 \, a^{2} b^{2} - 4 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 3 \, a^{2} b^{2} - 4 \, b^{4} - 2 \, {\left (3 \, a^{2} b^{2} - 4 \, b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (3 \, a^{3} b - 4 \, a b^{3} - {\left (3 \, a^{3} b - 4 \, a b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left ({\left (3 \, a^{2} b^{2} - 4 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 3 \, a^{2} b^{2} - 4 \, b^{4} - 2 \, {\left (3 \, a^{2} b^{2} - 4 \, b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (3 \, a^{3} b - 4 \, a b^{3} - {\left (3 \, a^{3} b - 4 \, a b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left ({\left (7 \, a^{3} b - 12 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (3 \, a^{3} b - 4 \, a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left (a^{5} b d \cos \left (d x + c\right )^{4} - 2 \, a^{5} b d \cos \left (d x + c\right )^{2} + a^{5} b d - {\left (a^{6} d \cos \left (d x + c\right )^{2} - a^{6} d\right )} \sin \left (d x + c\right )\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 {\cot ^{4}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 17.41, size = 356, normalized size = 1.50 \begin {gather*} \frac {\frac {24 \, {\left (3 \, a^{2} b - 4 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{5}} + \frac {48 \, {\left (a^{4} - 5 \, a^{2} b^{2} + 4 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{5}} + \frac {a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}} + \frac {48 \, {\left (a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3} b - a b^{3}\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 )} a^{5}} - \frac {132 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 176 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}}{a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.83, size = 973, normalized size = 4.09 \begin {gather*} \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (28\,a^2\,b-40\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (8\,a\,b^2-\frac {14\,a^3}{3}\right )-\frac {a^3}{3}+\frac {4\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (5\,a^4+4\,a^2\,b^2-16\,b^4\right )}{a}}{d\,\left (8\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+16\,b\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a^2\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {16\,a^2+32\,b^2}{64\,a^4}+\frac {3}{8\,a^2}-\frac {2\,b^2}{a^4}\right )}{d}-\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,a^3\,d}+\frac {b\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (3\,a^2-4\,b^2\right )}{a^5\,d}+\frac {\mathrm {atan}\left (\frac {\frac {\sqrt {b^2-a^2}\,\left (a^2-4\,b^2\right )\,\left (\frac {2\,\left (a^9-8\,a^7\,b^2+8\,a^5\,b^4\right )}{a^8}+\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (5\,a^7\,b-20\,a^5\,b^3+16\,a^3\,b^5\right )}{a^7}+\frac {\left (2\,a^2\,b-\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,a^{10}-4\,a^8\,b^2\right )}{a^7}\right )\,\sqrt {b^2-a^2}\,\left (a^2-4\,b^2\right )}{a^5}\right )\,1{}\mathrm {i}}{a^5}+\frac {\sqrt {b^2-a^2}\,\left (a^2-4\,b^2\right )\,\left (\frac {2\,\left (a^9-8\,a^7\,b^2+8\,a^5\,b^4\right )}{a^8}+\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (5\,a^7\,b-20\,a^5\,b^3+16\,a^3\,b^5\right )}{a^7}-\frac {\left (2\,a^2\,b-\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,a^{10}-4\,a^8\,b^2\right )}{a^7}\right )\,\sqrt {b^2-a^2}\,\left (a^2-4\,b^2\right )}{a^5}\right )\,1{}\mathrm {i}}{a^5}}{\frac {4\,\left (3\,a^6\,b-19\,a^4\,b^3+32\,a^2\,b^5-16\,b^7\right )}{a^8}+\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^6-14\,a^4\,b^2+28\,a^2\,b^4-16\,b^6\right )}{a^7}-\frac {\sqrt {b^2-a^2}\,\left (a^2-4\,b^2\right )\,\left (\frac {2\,\left (a^9-8\,a^7\,b^2+8\,a^5\,b^4\right )}{a^8}+\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (5\,a^7\,b-20\,a^5\,b^3+16\,a^3\,b^5\right )}{a^7}+\frac {\left (2\,a^2\,b-\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,a^{10}-4\,a^8\,b^2\right )}{a^7}\right )\,\sqrt {b^2-a^2}\,\left (a^2-4\,b^2\right )}{a^5}\right )}{a^5}+\frac {\sqrt {b^2-a^2}\,\left (a^2-4\,b^2\right )\,\left (\frac {2\,\left (a^9-8\,a^7\,b^2+8\,a^5\,b^4\right )}{a^8}+\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (5\,a^7\,b-20\,a^5\,b^3+16\,a^3\,b^5\right )}{a^7}-\frac {\left (2\,a^2\,b-\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,a^{10}-4\,a^8\,b^2\right )}{a^7}\right )\,\sqrt {b^2-a^2}\,\left (a^2-4\,b^2\right )}{a^5}\right )}{a^5}}\right )\,\sqrt {b^2-a^2}\,\left (a^2-4\,b^2\right )\,2{}\mathrm {i}}{a^5\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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