Optimal. Leaf size=332 \[ -\frac {2 b^7 \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{5/2} d}-\frac {5 \tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac {b^2 \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac {b \cot (c+d x)}{a^2 d}+\frac {5 \sec (c+d x)}{2 a d}+\frac {b^2 \sec (c+d x)}{a^3 d}+\frac {5 \sec ^3(c+d x)}{6 a d}+\frac {b^2 \sec ^3(c+d x)}{3 a^3 d}-\frac {\csc ^2(c+d x) \sec ^3(c+d x)}{2 a d}+\frac {b^3 \sec ^3(c+d x) (b-a \sin (c+d x))}{3 a^3 \left (a^2-b^2\right ) d}-\frac {b^3 \sec (c+d x) \left (3 b^3+a \left (2 a^2-5 b^2\right ) \sin (c+d x)\right )}{3 a^3 \left (a^2-b^2\right )^2 d}-\frac {2 b \tan (c+d x)}{a^2 d}-\frac {b \tan ^3(c+d x)}{3 a^2 d} \]
[Out]
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Rubi [A]
time = 0.35, antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps
used = 20, number of rules used = 13, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {2977, 2702,
308, 213, 2700, 276, 294, 2775, 2945, 12, 2739, 632, 210} \begin {gather*} \frac {b^2 \sec ^3(c+d x)}{3 a^3 d}+\frac {b^2 \sec (c+d x)}{a^3 d}-\frac {b^2 \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac {b \tan ^3(c+d x)}{3 a^2 d}-\frac {2 b \tan (c+d x)}{a^2 d}+\frac {b \cot (c+d x)}{a^2 d}-\frac {2 b^7 \text {ArcTan}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^3 d \left (a^2-b^2\right )^{5/2}}+\frac {b^3 \sec ^3(c+d x) (b-a \sin (c+d x))}{3 a^3 d \left (a^2-b^2\right )}-\frac {b^3 \sec (c+d x) \left (a \left (2 a^2-5 b^2\right ) \sin (c+d x)+3 b^3\right )}{3 a^3 d \left (a^2-b^2\right )^2}+\frac {5 \sec ^3(c+d x)}{6 a d}+\frac {5 \sec (c+d x)}{2 a d}-\frac {5 \tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac {\csc ^2(c+d x) \sec ^3(c+d x)}{2 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 210
Rule 213
Rule 276
Rule 294
Rule 308
Rule 632
Rule 2700
Rule 2702
Rule 2739
Rule 2775
Rule 2945
Rule 2977
Rubi steps
\begin {align*} \int \frac {\csc ^3(c+d x) \sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx &=\int \left (\frac {b^2 \csc (c+d x) \sec ^4(c+d x)}{a^3}-\frac {b \csc ^2(c+d x) \sec ^4(c+d x)}{a^2}+\frac {\csc ^3(c+d x) \sec ^4(c+d x)}{a}-\frac {b^3 \sec ^4(c+d x)}{a^3 (a+b \sin (c+d x))}\right ) \, dx\\ &=\frac {\int \csc ^3(c+d x) \sec ^4(c+d x) \, dx}{a}-\frac {b \int \csc ^2(c+d x) \sec ^4(c+d x) \, dx}{a^2}+\frac {b^2 \int \csc (c+d x) \sec ^4(c+d x) \, dx}{a^3}-\frac {b^3 \int \frac {\sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx}{a^3}\\ &=\frac {b^3 \sec ^3(c+d x) (b-a \sin (c+d x))}{3 a^3 \left (a^2-b^2\right ) d}+\frac {b^3 \int \frac {\sec ^2(c+d x) \left (-2 a^2+3 b^2-2 a b \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{3 a^3 \left (a^2-b^2\right )}+\frac {\text {Subst}\left (\int \frac {x^6}{\left (-1+x^2\right )^2} \, dx,x,\sec (c+d x)\right )}{a d}-\frac {b \text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^2} \, dx,x,\tan (c+d x)\right )}{a^2 d}+\frac {b^2 \text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^3 d}\\ &=-\frac {\csc ^2(c+d x) \sec ^3(c+d x)}{2 a d}+\frac {b^3 \sec ^3(c+d x) (b-a \sin (c+d x))}{3 a^3 \left (a^2-b^2\right ) d}-\frac {b^3 \sec (c+d x) \left (3 b^3+a \left (2 a^2-5 b^2\right ) \sin (c+d x)\right )}{3 a^3 \left (a^2-b^2\right )^2 d}-\frac {b^3 \int \frac {3 b^4}{a+b \sin (c+d x)} \, dx}{3 a^3 \left (a^2-b^2\right )^2}+\frac {5 \text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 a d}-\frac {b \text {Subst}\left (\int \left (2+\frac {1}{x^2}+x^2\right ) \, dx,x,\tan (c+d x)\right )}{a^2 d}+\frac {b^2 \text {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\sec (c+d x)\right )}{a^3 d}\\ &=\frac {b \cot (c+d x)}{a^2 d}+\frac {b^2 \sec (c+d x)}{a^3 d}+\frac {b^2 \sec ^3(c+d x)}{3 a^3 d}-\frac {\csc ^2(c+d x) \sec ^3(c+d x)}{2 a d}+\frac {b^3 \sec ^3(c+d x) (b-a \sin (c+d x))}{3 a^3 \left (a^2-b^2\right ) d}-\frac {b^3 \sec (c+d x) \left (3 b^3+a \left (2 a^2-5 b^2\right ) \sin (c+d x)\right )}{3 a^3 \left (a^2-b^2\right )^2 d}-\frac {2 b \tan (c+d x)}{a^2 d}-\frac {b \tan ^3(c+d x)}{3 a^2 d}-\frac {b^7 \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^3 \left (a^2-b^2\right )^2}+\frac {5 \text {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\sec (c+d x)\right )}{2 a d}+\frac {b^2 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^3 d}\\ &=-\frac {b^2 \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac {b \cot (c+d x)}{a^2 d}+\frac {5 \sec (c+d x)}{2 a d}+\frac {b^2 \sec (c+d x)}{a^3 d}+\frac {5 \sec ^3(c+d x)}{6 a d}+\frac {b^2 \sec ^3(c+d x)}{3 a^3 d}-\frac {\csc ^2(c+d x) \sec ^3(c+d x)}{2 a d}+\frac {b^3 \sec ^3(c+d x) (b-a \sin (c+d x))}{3 a^3 \left (a^2-b^2\right ) d}-\frac {b^3 \sec (c+d x) \left (3 b^3+a \left (2 a^2-5 b^2\right ) \sin (c+d x)\right )}{3 a^3 \left (a^2-b^2\right )^2 d}-\frac {2 b \tan (c+d x)}{a^2 d}-\frac {b \tan ^3(c+d x)}{3 a^2 d}+\frac {5 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 a d}-\frac {\left (2 b^7\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^3 \left (a^2-b^2\right )^2 d}\\ &=-\frac {5 \tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac {b^2 \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac {b \cot (c+d x)}{a^2 d}+\frac {5 \sec (c+d x)}{2 a d}+\frac {b^2 \sec (c+d x)}{a^3 d}+\frac {5 \sec ^3(c+d x)}{6 a d}+\frac {b^2 \sec ^3(c+d x)}{3 a^3 d}-\frac {\csc ^2(c+d x) \sec ^3(c+d x)}{2 a d}+\frac {b^3 \sec ^3(c+d x) (b-a \sin (c+d x))}{3 a^3 \left (a^2-b^2\right ) d}-\frac {b^3 \sec (c+d x) \left (3 b^3+a \left (2 a^2-5 b^2\right ) \sin (c+d x)\right )}{3 a^3 \left (a^2-b^2\right )^2 d}-\frac {2 b \tan (c+d x)}{a^2 d}-\frac {b \tan ^3(c+d x)}{3 a^2 d}+\frac {\left (4 b^7\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^3 \left (a^2-b^2\right )^2 d}\\ &=-\frac {2 b^7 \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{5/2} d}-\frac {5 \tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac {b^2 \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac {b \cot (c+d x)}{a^2 d}+\frac {5 \sec (c+d x)}{2 a d}+\frac {b^2 \sec (c+d x)}{a^3 d}+\frac {5 \sec ^3(c+d x)}{6 a d}+\frac {b^2 \sec ^3(c+d x)}{3 a^3 d}-\frac {\csc ^2(c+d x) \sec ^3(c+d x)}{2 a d}+\frac {b^3 \sec ^3(c+d x) (b-a \sin (c+d x))}{3 a^3 \left (a^2-b^2\right ) d}-\frac {b^3 \sec (c+d x) \left (3 b^3+a \left (2 a^2-5 b^2\right ) \sin (c+d x)\right )}{3 a^3 \left (a^2-b^2\right )^2 d}-\frac {2 b \tan (c+d x)}{a^2 d}-\frac {b \tan ^3(c+d x)}{3 a^2 d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(947\) vs. \(2(332)=664\).
time = 6.16, size = 947, normalized size = 2.85 \begin {gather*} 16 \left (\frac {a \left (13 a^2-19 b^2\right ) \csc (c+d x) (a+b \sin (c+d x))}{96 \left (a^2-b^2\right )^2 d (b+a \csc (c+d x))}-\frac {b^7 \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 ) \csc (c+d x) (a+b \sin (c+d x))}{8 a^3 \left (a^2-b^2\right )^{5/2} d (b+a \csc (c+d x))}+\frac {b \cot \left (\frac {1}{2} (c+d x)\right ) \csc (c+d x) (a+b \sin (c+d x))}{32 a^2 d (b+a \csc (c+d x))}-\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right ) \csc (c+d x) (a+b \sin (c+d x))}{128 a d (b+a \csc (c+d x))}+\frac {\left (-5 a^2-2 b^2\right ) \csc (c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \sin (c+d x))}{32 a^3 d (b+a \csc (c+d x))}+\frac {\left (5 a^2+2 b^2\right ) \csc (c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \sin (c+d x))}{32 a^3 d (b+a \csc (c+d x))}+\frac {\csc (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+b \sin (c+d x))}{128 a d (b+a \csc (c+d x))}+\frac {\csc (c+d x) (a+b \sin (c+d x))}{192 (a+b) d (b+a \csc (c+d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {\csc (c+d x) \sin \left (\frac {1}{2} (c+d x)\right ) (a+b \sin (c+d x))}{96 (a+b) d (b+a \csc (c+d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}-\frac {\csc (c+d x) \sin \left (\frac {1}{2} (c+d x)\right ) (a+b \sin (c+d x))}{96 (a-b) d (b+a \csc (c+d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {\csc (c+d x) (a+b \sin (c+d x))}{192 (a-b) d (b+a \csc (c+d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {\csc (c+d x) \left (-13 a \sin \left (\frac {1}{2} (c+d x)\right )+16 b \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \sin (c+d x))}{96 (a-b)^2 d (b+a \csc (c+d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {\csc (c+d x) \left (13 a \sin \left (\frac {1}{2} (c+d x)\right )+16 b \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \sin (c+d x))}{96 (a+b)^2 d (b+a \csc (c+d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {b \csc (c+d x) (a+b \sin (c+d x)) \tan \left (\frac {1}{2} (c+d x)\right )}{32 a^2 d (b+a \csc (c+d x))}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.82, size = 298, normalized size = 0.90
method | result | size |
derivativedivides | \(\frac {\frac {\frac {a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}-\frac {-10 a +12 b}{4 \left (a -b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {1}{3 \left (a -b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{2 \left (a -b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {1}{8 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (10 a^{2}+4 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b^{7} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{3} \left (a +b \right )^{2} \left (a -b \right )^{2} \sqrt {a^{2}-b^{2}}}-\frac {10 a +12 b}{4 \left (a +b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{3 \left (a +b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (a +b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}}{d}\) | \(298\) |
default | \(\frac {\frac {\frac {a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}-\frac {-10 a +12 b}{4 \left (a -b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {1}{3 \left (a -b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{2 \left (a -b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {1}{8 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (10 a^{2}+4 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b^{7} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{3} \left (a +b \right )^{2} \left (a -b \right )^{2} \sqrt {a^{2}-b^{2}}}-\frac {10 a +12 b}{4 \left (a +b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{3 \left (a +b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (a +b \right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}}{d}\) | \(298\) |
risch | \(\frac {i \left (12 b^{5} {\mathrm e}^{6 i \left (d x +c \right )}+6 b^{5} {\mathrm e}^{8 i \left (d x +c \right )}-12 b^{5} {\mathrm e}^{2 i \left (d x +c \right )}-6 b^{5}-16 a^{4} b +28 a^{2} b^{3}+32 a^{4} b \,{\mathrm e}^{4 i \left (d x +c \right )}-16 a^{4} b \,{\mathrm e}^{2 i \left (d x +c \right )}-15 i a^{5} {\mathrm e}^{i \left (d x +c \right )}-15 i a^{5} {\mathrm e}^{9 i \left (d x +c \right )}+22 i a^{5} {\mathrm e}^{5 i \left (d x +c \right )}-20 i a^{5} {\mathrm e}^{3 i \left (d x +c \right )}-20 i a^{5} {\mathrm e}^{7 i \left (d x +c \right )}-12 a^{2} b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-44 a^{2} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+28 a^{2} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+24 i a^{3} b^{2} {\mathrm e}^{i \left (d x +c \right )}-3 i a \,b^{4} {\mathrm e}^{i \left (d x +c \right )}+24 i a^{3} b^{2} {\mathrm e}^{9 i \left (d x +c \right )}-3 i a \,b^{4} {\mathrm e}^{9 i \left (d x +c \right )}+32 i a^{3} b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-12 i a \,b^{4} {\mathrm e}^{7 i \left (d x +c \right )}-16 i a^{3} b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-18 i a \,b^{4} {\mathrm e}^{5 i \left (d x +c \right )}+32 i a^{3} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-12 i a \,b^{4} {\mathrm e}^{3 i \left (d x +c \right )}\right )}{3 \left (-a^{2}+b^{2}\right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} a^{2} d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 a d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{a^{3} d}-\frac {i b^{7} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,a^{3}}+\frac {i b^{7} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,a^{3}}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 a d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{a^{3} d}\) | \(691\) |
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 [A]
time = 1.22, size = 1182, normalized size = 3.56 \begin {gather*} \left [-\frac {4 \, a^{8} - 8 \, a^{6} b^{2} + 4 \, a^{4} b^{4} - 6 \, {\left (5 \, a^{8} - 13 \, a^{6} b^{2} + 9 \, a^{4} b^{4} - a^{2} b^{6}\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (5 \, a^{8} - 13 \, a^{6} b^{2} + 8 \, a^{4} b^{4}\right )} \cos \left (d x + c\right )^{2} + 6 \, {\left (b^{7} \cos \left (d x + c\right )^{5} - b^{7} \cos \left (d x + c\right )^{3}\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 ) + 3 \, {\left ({\left (5 \, a^{8} - 13 \, a^{6} b^{2} + 9 \, a^{4} b^{4} + a^{2} b^{6} - 2 \, b^{8}\right )} \cos \left (d x + c\right )^{5} - {\left (5 \, a^{8} - 13 \, a^{6} b^{2} + 9 \, a^{4} b^{4} + a^{2} b^{6} - 2 \, b^{8}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left ({\left (5 \, a^{8} - 13 \, a^{6} b^{2} + 9 \, a^{4} b^{4} + a^{2} b^{6} - 2 \, b^{8}\right )} \cos \left (d x + c\right )^{5} - {\left (5 \, a^{8} - 13 \, a^{6} b^{2} + 9 \, a^{4} b^{4} + a^{2} b^{6} - 2 \, b^{8}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 4 \, {\left (a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5} - {\left (8 \, a^{7} b - 22 \, a^{5} b^{3} + 17 \, a^{3} b^{5} - 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} + {\left (4 \, a^{7} b - 11 \, a^{5} b^{3} + 7 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{12 \, {\left ({\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} d \cos \left (d x + c\right )^{5} - {\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} d \cos \left (d x + c\right )^{3}\right )}}, -\frac {4 \, a^{8} - 8 \, a^{6} b^{2} + 4 \, a^{4} b^{4} - 6 \, {\left (5 \, a^{8} - 13 \, a^{6} b^{2} + 9 \, a^{4} b^{4} - a^{2} b^{6}\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (5 \, a^{8} - 13 \, a^{6} b^{2} + 8 \, a^{4} b^{4}\right )} \cos \left (d x + c\right )^{2} - 12 \, {\left (b^{7} \cos \left (d x + c\right )^{5} - b^{7} \cos \left (d x + c\right )^{3}\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 ) + 3 \, {\left ({\left (5 \, a^{8} - 13 \, a^{6} b^{2} + 9 \, a^{4} b^{4} + a^{2} b^{6} - 2 \, b^{8}\right )} \cos \left (d x + c\right )^{5} - {\left (5 \, a^{8} - 13 \, a^{6} b^{2} + 9 \, a^{4} b^{4} + a^{2} b^{6} - 2 \, b^{8}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left ({\left (5 \, a^{8} - 13 \, a^{6} b^{2} + 9 \, a^{4} b^{4} + a^{2} b^{6} - 2 \, b^{8}\right )} \cos \left (d x + c\right )^{5} - {\left (5 \, a^{8} - 13 \, a^{6} b^{2} + 9 \, a^{4} b^{4} + a^{2} b^{6} - 2 \, b^{8}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 4 \, {\left (a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5} - {\left (8 \, a^{7} b - 22 \, a^{5} b^{3} + 17 \, a^{3} b^{5} - 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} + {\left (4 \, a^{7} b - 11 \, a^{5} b^{3} + 7 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{12 \, {\left ({\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} d \cos \left (d x + c\right )^{5} - {\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} d \cos \left (d x + c\right )^{3}\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 [A]
time = 0.50, size = 417, normalized size = 1.26 \begin {gather*} -\frac {\frac {48 \, {\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 )} b^{7}}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \sqrt {a^{2} - b^{2}}} - \frac {3 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{2}} - \frac {12 \, {\left (5 \, a^{2} + 2 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {16 \, {\left (6 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 12 \, a b^{2} \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} + 14 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 18 \, 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 ) - 9 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 7 \, a^{3} + 10 \, a b^{2}\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}} + \frac {3 \, {\left (30 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{2}\right )}}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{24 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 18.01, size = 2500, normalized size = 7.53 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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