Optimal. Leaf size=249 \[ \frac {\left (a^2-2 a b-b^2\right ) \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2-2 a b-b^2\right ) \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {2 \left (a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {4 a b \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d} \]
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Rubi [A]
time = 0.19, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3754, 3624,
3609, 3615, 1182, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {\left (a^2-2 a b-b^2\right ) \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2-2 a b-b^2\right ) \text {ArcTan}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d}+\frac {2 \left (a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {4 a b \cot ^{\frac {3}{2}}(c+d x)}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3609
Rule 3615
Rule 3624
Rule 3754
Rubi steps
\begin {align*} \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx &=\int \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^2 \, dx\\ &=-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\int \cot ^{\frac {3}{2}}(c+d x) \left (-a^2+b^2+2 a b \cot (c+d x)\right ) \, dx\\ &=-\frac {4 a b \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\int \sqrt {\cot (c+d x)} \left (-2 a b-\left (a^2-b^2\right ) \cot (c+d x)\right ) \, dx\\ &=\frac {2 \left (a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {4 a b \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\int \frac {a^2-b^2-2 a b \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx\\ &=\frac {2 \left (a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {4 a b \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 \text {Subst}\left (\int \frac {-a^2+b^2+2 a b x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d}\\ &=\frac {2 \left (a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {4 a b \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d}-\frac {\left (a^2+2 a b-b^2\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d}\\ &=\frac {2 \left (a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {4 a b \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 d}-\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 d}+\frac {\left (a^2+2 a b-b^2\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} d}+\frac {\left (a^2+2 a b-b^2\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} d}\\ &=\frac {2 \left (a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {4 a b \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}-\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}\\ &=\frac {\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {2 \left (a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {4 a b \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 1.45, size = 202, normalized size = 0.81 \begin {gather*} -\frac {\frac {2}{5} a^2 \cot ^{\frac {5}{2}}(c+d x)-\frac {4}{3} a b \cot ^{\frac {3}{2}}(c+d x) \left (-1+\, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )\right )-\frac {1}{4} \left (a^2-b^2\right ) \left (2 \sqrt {2} \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \sqrt {2} \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+8 \sqrt {\cot (c+d x)}+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{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 = 18.64, size = 6328, normalized size = 25.41
method | result | size |
default | \(\text {Expression too large to display}\) | \(6328\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 209, normalized size = 0.84 \begin {gather*} -\frac {30 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 30 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 15 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - 15 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \frac {80 \, a b}{\tan \left (d x + c\right )^{\frac {3}{2}}} - \frac {120 \, {\left (a^{2} - b^{2}\right )}}{\sqrt {\tan \left (d x + c\right )}} + \frac {24 \, a^{2}}{\tan \left (d x + c\right )^{\frac {5}{2}}}}{60 \, 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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {\mathrm {cot}\left (c+d\,x\right )}^{7/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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