Optimal. Leaf size=600 \[ -\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {a^{3/2} \left (3 a^4 A b+6 a^2 A b^3+35 A b^5-15 a^5 B-46 a^3 b^2 B-63 a b^4 B\right ) \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 b^{7/2} \left (a^2+b^2\right )^3 d}-\frac {3 a^3 A b+11 a A b^3-15 a^4 B-31 a^2 b^2 B-8 b^4 B}{4 b^3 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)}}+\frac {a (A b-a B)}{2 b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))^2}+\frac {a \left (a^2 A b+9 A b^3-5 a^3 B-13 a b^2 B\right )}{4 b^2 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d} \]
[Out]
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Rubi [A]
time = 1.22, antiderivative size = 600, normalized size of antiderivative = 1.00, number
of steps used = 18, number of rules used = 14, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.424, Rules
used = {3662, 3690, 3730, 3734, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211}
\begin {gather*} \frac {a (A b-a B)}{2 b d \left (a^2+b^2\right ) \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}-\frac {\left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {\left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \text {ArcTan}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {a \left (-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right )}{4 b^2 d \left (a^2+b^2\right )^2 \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)}-\frac {\left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {\left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {-15 a^4 B+3 a^3 A b-31 a^2 b^2 B+11 a A b^3-8 b^4 B}{4 b^3 d \left (a^2+b^2\right )^2 \sqrt {\cot (c+d x)}}-\frac {a^{3/2} \left (-15 a^5 B+3 a^4 A b-46 a^3 b^2 B+6 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 b^{7/2} d \left (a^2+b^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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[Out]
Rule 65
Rule 210
Rule 211
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3615
Rule 3662
Rule 3690
Rule 3715
Rule 3730
Rule 3734
Rubi steps
\begin {align*} \int \frac {A+B \tan (c+d x)}{\cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx &=\int \frac {B+A \cot (c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^3} \, dx\\ &=\frac {a (A b-a B)}{2 b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))^2}-\frac {\int \frac {\frac {1}{2} \left (a A b-5 a^2 B-4 b^2 B\right )-2 b (A b-a B) \cot (c+d x)+\frac {5}{2} a (A b-a B) \cot ^2(c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^2} \, dx}{2 b \left (a^2+b^2\right )}\\ &=\frac {a (A b-a B)}{2 b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))^2}+\frac {a \left (a^2 A b+9 A b^3-5 a^3 B-13 a b^2 B\right )}{4 b^2 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}+\frac {\int \frac {\frac {1}{4} \left (-3 a^3 A b-11 a A b^3+15 a^4 B+31 a^2 b^2 B+8 b^4 B\right )-2 b^2 \left (a^2 A-A b^2+2 a b B\right ) \cot (c+d x)-\frac {3}{4} a \left (a^2 A b+9 A b^3-5 a^3 B-13 a b^2 B\right ) \cot ^2(c+d x)}{\cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))} \, dx}{2 b^2 \left (a^2+b^2\right )^2}\\ &=-\frac {3 a^3 A b+11 a A b^3-15 a^4 B-31 a^2 b^2 B-8 b^4 B}{4 b^3 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)}}+\frac {a (A b-a B)}{2 b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))^2}+\frac {a \left (a^2 A b+9 A b^3-5 a^3 B-13 a b^2 B\right )}{4 b^2 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}+\frac {\int \frac {\frac {1}{8} \left (3 a^4 A b+3 a^2 A b^3+8 A b^5-15 a^5 B-31 a^3 b^2 B-24 a b^4 B\right )-b^3 \left (2 a A b-a^2 B+b^2 B\right ) \cot (c+d x)+\frac {1}{8} a \left (3 a^3 A b+11 a A b^3-15 a^4 B-31 a^2 b^2 B-8 b^4 B\right ) \cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{b^3 \left (a^2+b^2\right )^2}\\ &=-\frac {3 a^3 A b+11 a A b^3-15 a^4 B-31 a^2 b^2 B-8 b^4 B}{4 b^3 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)}}+\frac {a (A b-a B)}{2 b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))^2}+\frac {a \left (a^2 A b+9 A b^3-5 a^3 B-13 a b^2 B\right )}{4 b^2 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}+\frac {\int \frac {-b^3 \left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right )+b^3 \left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{b^3 \left (a^2+b^2\right )^3}+\frac {\left (a^2 \left (3 a^4 A b+6 a^2 A b^3+35 A b^5-15 a^5 B-46 a^3 b^2 B-63 a b^4 B\right )\right ) \int \frac {1+\cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{8 b^3 \left (a^2+b^2\right )^3}\\ &=-\frac {3 a^3 A b+11 a A b^3-15 a^4 B-31 a^2 b^2 B-8 b^4 B}{4 b^3 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)}}+\frac {a (A b-a B)}{2 b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))^2}+\frac {a \left (a^2 A b+9 A b^3-5 a^3 B-13 a b^2 B\right )}{4 b^2 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}+\frac {2 \text {Subst}\left (\int \frac {b^3 \left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right )-b^3 \left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right ) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{b^3 \left (a^2+b^2\right )^3 d}+\frac {\left (a^2 \left (3 a^4 A b+6 a^2 A b^3+35 A b^5-15 a^5 B-46 a^3 b^2 B-63 a b^4 B\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-x} (b-a x)} \, dx,x,-\cot (c+d x)\right )}{8 b^3 \left (a^2+b^2\right )^3 d}\\ &=-\frac {3 a^3 A b+11 a A b^3-15 a^4 B-31 a^2 b^2 B-8 b^4 B}{4 b^3 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)}}+\frac {a (A b-a B)}{2 b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))^2}+\frac {a \left (a^2 A b+9 A b^3-5 a^3 B-13 a b^2 B\right )}{4 b^2 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {\left (a^2 \left (3 a^4 A b+6 a^2 A b^3+35 A b^5-15 a^5 B-46 a^3 b^2 B-63 a b^4 B\right )\right ) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{4 b^3 \left (a^2+b^2\right )^3 d}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{\left (a^2+b^2\right )^3 d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{\left (a^2+b^2\right )^3 d}\\ &=-\frac {a^{3/2} \left (3 a^4 A b+6 a^2 A b^3+35 A b^5-15 a^5 B-46 a^3 b^2 B-63 a b^4 B\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 b^{7/2} \left (a^2+b^2\right )^3 d}-\frac {3 a^3 A b+11 a A b^3-15 a^4 B-31 a^2 b^2 B-8 b^4 B}{4 b^3 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)}}+\frac {a (A b-a B)}{2 b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))^2}+\frac {a \left (a^2 A b+9 A b^3-5 a^3 B-13 a b^2 B\right )}{4 b^2 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \left (a^2+b^2\right )^3 d}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \left (a^2+b^2\right )^3 d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}\\ &=-\frac {a^{3/2} \left (3 a^4 A b+6 a^2 A b^3+35 A b^5-15 a^5 B-46 a^3 b^2 B-63 a b^4 B\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 b^{7/2} \left (a^2+b^2\right )^3 d}-\frac {3 a^3 A b+11 a A b^3-15 a^4 B-31 a^2 b^2 B-8 b^4 B}{4 b^3 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)}}+\frac {a (A b-a B)}{2 b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))^2}+\frac {a \left (a^2 A b+9 A b^3-5 a^3 B-13 a b^2 B\right )}{4 b^2 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}\\ &=-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {a^{3/2} \left (3 a^4 A b+6 a^2 A b^3+35 A b^5-15 a^5 B-46 a^3 b^2 B-63 a b^4 B\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 b^{7/2} \left (a^2+b^2\right )^3 d}-\frac {3 a^3 A b+11 a A b^3-15 a^4 B-31 a^2 b^2 B-8 b^4 B}{4 b^3 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)}}+\frac {a (A b-a B)}{2 b \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))^2}+\frac {a \left (a^2 A b+9 A b^3-5 a^3 B-13 a b^2 B\right )}{4 b^2 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}\\ \end {align*}
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Mathematica [A]
time = 5.97, size = 599, normalized size = 1.00 \begin {gather*} \frac {\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (-\frac {2 \sqrt {2} \left (b^3 (A-B)+3 a^2 b (-A+B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \left (\text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )\right )}{\left (a^2+b^2\right )^3}+\frac {4 a^{3/2} \left (-2 a^2 A b-4 A b^3+3 a^3 B+5 a b^2 B\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{7/2} \left (a^2+b^2\right )^2}+\frac {8 a^{3/2} \left (a^4 A b+3 a^2 A b^3+6 A b^5-3 a^5 B-9 a^3 b^2 B-10 a b^4 B\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{7/2} \left (a^2+b^2\right )^3}-\frac {\sqrt {2} \left (a^3 (A-B)+3 a b^2 (-A+B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \left (\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )}{\left (a^2+b^2\right )^3}+\frac {8 B \sqrt {\tan (c+d x)}}{b^3}+\frac {2 a^3 (A b-a B) \sqrt {\tan (c+d x)}}{b^3 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {4 a^2 \left (-2 a^2 A b-4 A b^3+3 a^3 B+5 a b^2 B\right ) \sqrt {\tan (c+d x)}}{b^3 \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac {3 (A b-a B) \left (\frac {a^{3/2} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {a^2 \sqrt {\tan (c+d x)}}{a+b \tan (c+d x)}\right )}{b^3 \left (a^2+b^2\right )}\right )}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 51.42, size = 104911, normalized size = 174.85
method | result | size |
default | \(\text {Expression too large to display}\) | \(104911\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 612, normalized size = 1.02 \begin {gather*} \frac {\frac {{\left (15 \, B a^{7} - 3 \, A a^{6} b + 46 \, B a^{5} b^{2} - 6 \, A a^{4} b^{3} + 63 \, B a^{3} b^{4} - 35 \, A a^{2} b^{5}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}\right )} \sqrt {a b}} - \frac {2 \, \sqrt {2} {\left ({\left (A + B\right )} a^{3} - 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} + {\left (A - B\right )} b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left ({\left (A + B\right )} a^{3} - 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} + {\left (A - B\right )} b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} {\left ({\left (A - B\right )} a^{3} + 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} - {\left (A + B\right )} b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} {\left ({\left (A - B\right )} a^{3} + 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} - {\left (A + B\right )} b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {8 \, B a^{4} b^{2} + 16 \, B a^{2} b^{4} + 8 \, B b^{6} + \frac {25 \, B a^{5} b - 5 \, A a^{4} b^{2} + 49 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4} + 16 \, B a b^{5}}{\tan \left (d x + c\right )} + \frac {15 \, B a^{6} - 3 \, A a^{5} b + 31 \, B a^{4} b^{2} - 11 \, A a^{3} b^{3} + 8 \, B a^{2} b^{4}}{\tan \left (d x + c\right )^{2}}}{\frac {a^{4} b^{5} + 2 \, a^{2} b^{7} + b^{9}}{\sqrt {\tan \left (d x + c\right )}} + \frac {2 \, {\left (a^{5} b^{4} + 2 \, a^{3} b^{6} + a b^{8}\right )}}{\tan \left (d x + c\right )^{\frac {3}{2}}} + \frac {a^{6} b^{3} + 2 \, a^{4} b^{5} + a^{2} b^{7}}{\tan \left (d x + c\right )^{\frac {5}{2}}}}}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,\mathrm {tan}\left (c+d\,x\right )}{{\mathrm {cot}\left (c+d\,x\right )}^{7/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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