Optimal. Leaf size=195 \[ -\frac {x}{b}+\frac {2 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 b d}-\frac {\left (15 a^4-20 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^5 d}+\frac {b \left (-2 a^2+b^2\right ) \cot (c+d x)}{a^4 d}+\frac {b \cot ^3(c+d x)}{3 a^2 d}+\frac {\left (7 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^3 d}-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 a d} \]
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Rubi [A]
time = 0.20, antiderivative size = 275, normalized size of antiderivative = 1.41, number of steps
used = 15, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2976, 3855,
3852, 8, 3853, 2739, 632, 210} \begin {gather*} \frac {b \cot ^3(c+d x)}{3 a^2 d}+\frac {b \cot (c+d x)}{a^2 d}+\frac {2 \left (a^2-b^2\right )^{5/2} \text {ArcTan}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^5 b d}-\frac {b \left (3 a^2-b^2\right ) \cot (c+d x)}{a^4 d}+\frac {\left (3 a^2-b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}+\frac {\left (3 a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac {\left (3 a^4-3 a^2 b^2+b^4\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}-\frac {3 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac {x}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 210
Rule 632
Rule 2739
Rule 2976
Rule 3852
Rule 3853
Rule 3855
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+b \sin (c+d x)} \, dx &=\int \left (-\frac {1}{b}+\frac {\left (3 a^4-3 a^2 b^2+b^4\right ) \csc (c+d x)}{a^5}+\frac {\left (3 a^2 b-b^3\right ) \csc ^2(c+d x)}{a^4}+\frac {\left (-3 a^2+b^2\right ) \csc ^3(c+d x)}{a^3}-\frac {b \csc ^4(c+d x)}{a^2}+\frac {\csc ^5(c+d x)}{a}+\frac {\left (a^2-b^2\right )^3}{a^5 b (a+b \sin (c+d x))}\right ) \, dx\\ &=-\frac {x}{b}+\frac {\int \csc ^5(c+d x) \, dx}{a}-\frac {b \int \csc ^4(c+d x) \, dx}{a^2}+\frac {\left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^5 b}-\frac {\left (3 a^2-b^2\right ) \int \csc ^3(c+d x) \, dx}{a^3}+\frac {\left (b \left (3 a^2-b^2\right )\right ) \int \csc ^2(c+d x) \, dx}{a^4}+\frac {\left (3 a^4-3 a^2 b^2+b^4\right ) \int \csc (c+d x) \, dx}{a^5}\\ &=-\frac {x}{b}-\frac {\left (3 a^4-3 a^2 b^2+b^4\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac {\left (3 a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac {3 \int \csc ^3(c+d x) \, dx}{4 a}-\frac {\left (3 a^2-b^2\right ) \int \csc (c+d x) \, dx}{2 a^3}+\frac {b \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^2 d}+\frac {\left (2 \left (a^2-b^2\right )^3\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 b d}-\frac {\left (b \left (3 a^2-b^2\right )\right ) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^4 d}\\ &=-\frac {x}{b}+\frac {\left (3 a^2-b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac {\left (3 a^4-3 a^2 b^2+b^4\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac {b \cot (c+d x)}{a^2 d}-\frac {b \left (3 a^2-b^2\right ) \cot (c+d x)}{a^4 d}+\frac {b \cot ^3(c+d x)}{3 a^2 d}-\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}+\frac {\left (3 a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac {3 \int \csc (c+d x) \, dx}{8 a}-\frac {\left (4 \left (a^2-b^2\right )^3\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 b d}\\ &=-\frac {x}{b}+\frac {2 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 b d}-\frac {3 \tanh ^{-1}(\cos (c+d x))}{8 a d}+\frac {\left (3 a^2-b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac {\left (3 a^4-3 a^2 b^2+b^4\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac {b \cot (c+d x)}{a^2 d}-\frac {b \left (3 a^2-b^2\right ) \cot (c+d x)}{a^4 d}+\frac {b \cot ^3(c+d x)}{3 a^2 d}-\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}+\frac {\left (3 a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(448\) vs. \(2(195)=390\).
time = 6.16, size = 448, normalized size = 2.30 \begin {gather*} -\frac {c+d x}{b d}+\frac {2 \left (a^2-b^2\right )^{5/2} \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 b d}+\frac {\left (-7 a^2 b \cos \left (\frac {1}{2} (c+d x)\right )+3 b^3 \cos \left (\frac {1}{2} (c+d x)\right )\right ) \csc \left (\frac {1}{2} (c+d x)\right )}{6 a^4 d}+\frac {\left (9 a^2-4 b^2\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 a^3 d}+\frac {b \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{24 a^2 d}-\frac {\csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 a d}+\frac {\left (-15 a^4+20 a^2 b^2-8 b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 a^5 d}+\frac {\left (15 a^4-20 a^2 b^2+8 b^4\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 a^5 d}+\frac {\left (-9 a^2+4 b^2\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 a^3 d}+\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 a d}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (7 a^2 b \sin \left (\frac {1}{2} (c+d x)\right )-3 b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{6 a^4 d}-\frac {b \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 = 0.51, size = 320, normalized size = 1.64
method | result | size |
derivativedivides | \(\frac {\frac {\frac {a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {2 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{3}-4 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \,b^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+18 a^{2} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-8 b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a^{4}}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b}-\frac {1}{64 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {-8 a^{2}+4 b^{2}}{32 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (30 a^{4}-40 a^{2} b^{2}+16 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{5}}+\frac {b}{24 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b \left (9 a^{2}-4 b^{2}\right )}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {\left (32 a^{6}-96 a^{4} b^{2}+96 a^{2} b^{4}-32 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{16 a^{5} b \sqrt {a^{2}-b^{2}}}}{d}\) | \(320\) |
default | \(\frac {\frac {\frac {a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {2 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{3}-4 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \,b^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+18 a^{2} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-8 b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a^{4}}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b}-\frac {1}{64 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {-8 a^{2}+4 b^{2}}{32 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (30 a^{4}-40 a^{2} b^{2}+16 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{5}}+\frac {b}{24 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b \left (9 a^{2}-4 b^{2}\right )}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {\left (32 a^{6}-96 a^{4} b^{2}+96 a^{2} b^{4}-32 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{16 a^{5} b \sqrt {a^{2}-b^{2}}}}{d}\) | \(320\) |
risch | \(-\frac {x}{b}+\frac {i \left (-12 i a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-3 i a^{3} {\mathrm e}^{5 i \left (d x +c \right )}+27 i a^{3} {\mathrm e}^{i \left (d x +c \right )}+12 i a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-72 b \,{\mathrm e}^{6 i \left (d x +c \right )} a^{2}+24 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+12 i a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-12 i b^{2} a \,{\mathrm e}^{i \left (d x +c \right )}+168 b \,{\mathrm e}^{4 i \left (d x +c \right )} a^{2}-72 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+27 i a^{3} {\mathrm e}^{7 i \left (d x +c \right )}-3 i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-152 b \,{\mathrm e}^{2 i \left (d x +c \right )} a^{2}+72 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+56 a^{2} b -24 b^{3}\right )}{12 d \,a^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 a d}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{2 a^{3} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{4}}{a^{5} d}-\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 a d}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{2 a^{3} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{4}}{a^{5} d}+\frac {i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d b a}-\frac {2 i \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,a^{3}}+\frac {i \sqrt {a^{2}-b^{2}}\, b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,a^{5}}-\frac {i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d b a}+\frac {2 i \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,a^{3}}-\frac {i \sqrt {a^{2}-b^{2}}\, b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,a^{5}}\) | \(704\) |
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. 475 vs.
\(2 (185) = 370\).
time = 0.68, size = 1034, normalized size = 5.30 \begin {gather*} \left [-\frac {48 \, a^{5} d x \cos \left (d x + c\right )^{4} - 96 \, a^{5} d x \cos \left (d x + c\right )^{2} + 48 \, a^{5} d x + 6 \, {\left (9 \, a^{4} b - 4 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )^{3} - 24 \, {\left ({\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + a^{4} - 2 \, a^{2} b^{2} + b^{4} - 2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}\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 (7 \, a^{4} b - 4 \, a^{2} b^{3}\right )} \cos \left (d x + c\right ) + 3 \, {\left (15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5} + {\left (15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left (15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5} + {\left (15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 16 \, {\left ({\left (7 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (2 \, a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, {\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\right )}}, -\frac {48 \, a^{5} d x \cos \left (d x + c\right )^{4} - 96 \, a^{5} d x \cos \left (d x + c\right )^{2} + 48 \, a^{5} d x + 6 \, {\left (9 \, a^{4} b - 4 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )^{3} + 48 \, {\left ({\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + a^{4} - 2 \, a^{2} b^{2} + b^{4} - 2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}\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 (7 \, a^{4} b - 4 \, a^{2} b^{3}\right )} \cos \left (d x + c\right ) + 3 \, {\left (15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5} + {\left (15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left (15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5} + {\left (15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 16 \, {\left ({\left (7 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (2 \, a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, {\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\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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. 396 vs.
\(2 (185) = 370\).
time = 0.49, size = 396, normalized size = 2.03 \begin {gather*} -\frac {\frac {192 \, {\left (d x + c\right )}}{b} - \frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 216 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{4}} - \frac {24 \, {\left (15 \, a^{4} - 20 \, a^{2} b^{2} + 8 \, b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{5}} - \frac {384 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\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} b} + \frac {750 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1000 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 400 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 216 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 96 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{4}}{a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 13.40, size = 2500, normalized size = 12.82 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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