Optimal. Leaf size=198 \[ -\frac {2 b \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 d}-\frac {\left (3 a^4-12 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^5 d}-\frac {b \left (4 a^2-3 b^2\right ) \cot (c+d x)}{3 a^4 d}+\frac {\left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d} \]
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Rubi [A]
time = 0.48, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2972, 3134,
3080, 3855, 2739, 632, 210} \begin {gather*} \frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {2 b \left (a^2-b^2\right )^{3/2} \text {ArcTan}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^5 d}-\frac {b \left (4 a^2-3 b^2\right ) \cot (c+d x)}{3 a^4 d}+\frac {\left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^3 d}-\frac {\left (3 a^4-12 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^5 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 2972
Rule 3080
Rule 3134
Rule 3855
Rubi steps
\begin {align*} \int \frac {\cot ^4(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {\int \frac {\csc ^3(c+d x) \left (3 \left (5 a^2-4 b^2\right )-a b \sin (c+d x)-4 \left (3 a^2-2 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{12 a^2}\\ &=\frac {\left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {\int \frac {\csc ^2(c+d x) \left (-8 b \left (4 a^2-3 b^2\right )-a \left (9 a^2-4 b^2\right ) \sin (c+d x)+3 b \left (5 a^2-4 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{24 a^3}\\ &=-\frac {b \left (4 a^2-3 b^2\right ) \cot (c+d x)}{3 a^4 d}+\frac {\left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {\int \frac {\csc (c+d x) \left (-3 \left (3 a^4-12 a^2 b^2+8 b^4\right )+3 a b \left (5 a^2-4 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{24 a^4}\\ &=-\frac {b \left (4 a^2-3 b^2\right ) \cot (c+d x)}{3 a^4 d}+\frac {\left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {\left (b \left (a^2-b^2\right )^2\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^5}+\frac {\left (3 a^4-12 a^2 b^2+8 b^4\right ) \int \csc (c+d x) \, dx}{8 a^5}\\ &=-\frac {\left (3 a^4-12 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^5 d}-\frac {b \left (4 a^2-3 b^2\right ) \cot (c+d x)}{3 a^4 d}+\frac {\left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {\left (2 b \left (a^2-b^2\right )^2\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 {\left (3 a^4-12 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^5 d}-\frac {b \left (4 a^2-3 b^2\right ) \cot (c+d x)}{3 a^4 d}+\frac {\left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac {\left (4 b \left (a^2-b^2\right )^2\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 b \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 d}-\frac {\left (3 a^4-12 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^5 d}-\frac {b \left (4 a^2-3 b^2\right ) \cot (c+d x)}{3 a^4 d}+\frac {\left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 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. \(433\) vs. \(2(198)=396\).
time = 6.18, size = 433, normalized size = 2.19 \begin {gather*} -\frac {2 b \left (a^2-b^2\right )^{3/2} \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 d}+\frac {\left (-4 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 (5 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 (-3 a^4+12 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 (3 a^4-12 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 (-5 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 (4 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 = 0.46, size = 291, normalized size = 1.47
method | result | size |
derivativedivides | \(\frac {\frac {\frac {a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {2 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{3}-2 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \,b^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 a^{2} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-8 b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a^{4}}-\frac {1}{64 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {-4 a^{2}+4 b^{2}}{32 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (6 a^{4}-24 a^{2} b^{2}+16 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{5}}+\frac {b}{24 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b \left (5 a^{2}-4 b^{2}\right )}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{5} \sqrt {a^{2}-b^{2}}}}{d}\) | \(291\) |
default | \(\frac {\frac {\frac {a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {2 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{3}-2 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \,b^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 a^{2} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-8 b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a^{4}}-\frac {1}{64 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {-4 a^{2}+4 b^{2}}{32 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (6 a^{4}-24 a^{2} b^{2}+16 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{5}}+\frac {b}{24 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b \left (5 a^{2}-4 b^{2}\right )}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{5} \sqrt {a^{2}-b^{2}}}}{d}\) | \(291\) |
risch | \(\frac {i \left (-12 i a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+9 i a^{3} {\mathrm e}^{5 i \left (d x +c \right )}+15 i a^{3} {\mathrm e}^{i \left (d x +c \right )}+12 i a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-48 b \,{\mathrm e}^{6 i \left (d x +c \right )} a^{2}+24 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+12 i a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-12 i b^{2} a \,{\mathrm e}^{i \left (d x +c \right )}+96 b \,{\mathrm e}^{4 i \left (d x +c \right )} a^{2}-72 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+15 i a^{3} {\mathrm e}^{7 i \left (d x +c \right )}+9 i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-80 b \,{\mathrm e}^{2 i \left (d x +c \right )} a^{2}+72 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+32 a^{2} b -24 b^{3}\right )}{12 d \,a^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 a d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{2 a^{3} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{4}}{a^{5} d}+\frac {i \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,a^{3}}-\frac {i \sqrt {a^{2}-b^{2}}\, b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,a^{5}}-\frac {i \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,a^{3}}+\frac {i \sqrt {a^{2}-b^{2}}\, b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,a^{5}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 a d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{2 a^{3} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{4}}{a^{5} d}\) | \(590\) |
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. 410 vs.
\(2 (183) = 366\).
time = 0.56, size = 904, normalized size = 4.57 \begin {gather*} \left [-\frac {6 \, {\left (5 \, a^{4} - 4 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3} + 24 \, {\left ({\left (a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{4} + a^{2} b - b^{3} - 2 \, {\left (a^{2} b - b^{3}\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 (3 \, a^{4} - 4 \, a^{2} b^{2}\right )} \cos \left (d x + c\right ) + 3 \, {\left ({\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4} - 2 \, {\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\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 ({\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4} - 2 \, {\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\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 (4 \, a^{3} b - 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, {\left (a^{5} d \cos \left (d x + c\right )^{4} - 2 \, a^{5} d \cos \left (d x + c\right )^{2} + a^{5} d\right )}}, -\frac {6 \, {\left (5 \, a^{4} - 4 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3} - 48 \, {\left ({\left (a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{4} + a^{2} b - b^{3} - 2 \, {\left (a^{2} b - b^{3}\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 (3 \, a^{4} - 4 \, a^{2} b^{2}\right )} \cos \left (d x + c\right ) + 3 \, {\left ({\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4} - 2 \, {\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\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 ({\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4} - 2 \, {\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\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 (4 \, a^{3} b - 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, {\left (a^{5} d \cos \left (d x + c\right )^{4} - 2 \, a^{5} d \cos \left (d x + c\right )^{2} + a^{5} 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. 375 vs.
\(2 (183) = 366\).
time = 0.48, size = 375, normalized size = 1.89 \begin {gather*} \frac {\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} - 24 \, 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} + 120 \, 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 (3 \, a^{4} - 12 \, 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^{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 (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 {150 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 600 \, 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} + 120 \, 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} - 24 \, 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 = 12.16, size = 953, normalized size = 4.81 \begin {gather*} \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {1}{8\,a}-\frac {b^2}{8\,a^3}\right )}{d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {b}{8\,a^2}+\frac {2\,b\,\left (\frac {1}{4\,a}-\frac {b^2}{4\,a^3}\right )}{a}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (2\,a\,b^2-2\,a^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (10\,a^2\,b-8\,b^3\right )+\frac {a^3}{4}-\frac {2\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}}{16\,a^4\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,a^4}{8}-\frac {3\,a^2\,b^2}{2}+b^4\right )}{a^5\,d}-\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a^2\,d}-\frac {b\,\mathrm {atan}\left (\frac {\frac {b\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,a^9-34\,a^7\,b^2+64\,a^5\,b^4-32\,a^3\,b^6\right )}{4\,a^7}-\frac {11\,a^9\,b-28\,a^7\,b^3+16\,a^5\,b^5}{4\,a^8}+\frac {b\,\left (2\,a^2\,b-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (24\,a^{10}-32\,a^8\,b^2\right )}{4\,a^7}\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}}{a^5}\right )\,1{}\mathrm {i}}{a^5}-\frac {b\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,\left (\frac {11\,a^9\,b-28\,a^7\,b^3+16\,a^5\,b^5}{4\,a^8}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,a^9-34\,a^7\,b^2+64\,a^5\,b^4-32\,a^3\,b^6\right )}{4\,a^7}+\frac {b\,\left (2\,a^2\,b-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (24\,a^{10}-32\,a^8\,b^2\right )}{4\,a^7}\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}}{a^5}\right )\,1{}\mathrm {i}}{a^5}}{\frac {3\,a^8\,b-18\,a^6\,b^3+35\,a^4\,b^5-28\,a^2\,b^7+8\,b^9}{2\,a^8}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (-10\,a^6\,b^2+28\,a^4\,b^4-26\,a^2\,b^6+8\,b^8\right )}{2\,a^7}+\frac {b\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,a^9-34\,a^7\,b^2+64\,a^5\,b^4-32\,a^3\,b^6\right )}{4\,a^7}-\frac {11\,a^9\,b-28\,a^7\,b^3+16\,a^5\,b^5}{4\,a^8}+\frac {b\,\left (2\,a^2\,b-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (24\,a^{10}-32\,a^8\,b^2\right )}{4\,a^7}\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}}{a^5}\right )}{a^5}+\frac {b\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,\left (\frac {11\,a^9\,b-28\,a^7\,b^3+16\,a^5\,b^5}{4\,a^8}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,a^9-34\,a^7\,b^2+64\,a^5\,b^4-32\,a^3\,b^6\right )}{4\,a^7}+\frac {b\,\left (2\,a^2\,b-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (24\,a^{10}-32\,a^8\,b^2\right )}{4\,a^7}\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}}{a^5}\right )}{a^5}}\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,2{}\mathrm {i}}{a^5\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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