Integrand size = 23, antiderivative size = 223 \[ \int \frac {(a+b \tan (c+d x))^2}{\tan ^{\frac {5}{2}}(c+d x)} \, dx=\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a^2-2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {\left (a^2-2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {2 a^2}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\tan (c+d x)}} \]
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Time = 0.21 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3623, 3610, 3615, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {(a+b \tan (c+d x))^2}{\tan ^{\frac {5}{2}}(c+d x)} \, dx=\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}+\frac {\left (a^2-2 a b-b^2\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {\left (a^2-2 a b-b^2\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {2 a^2}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\tan (c+d x)}} \]
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Rule 210
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3610
Rule 3615
Rule 3623
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\int \frac {2 a b-\left (a^2-b^2\right ) \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x)} \, dx \\ & = -\frac {2 a^2}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\tan (c+d x)}}+\int \frac {-a^2+b^2-2 a b \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx \\ & = -\frac {2 a^2}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\tan (c+d x)}}+\frac {2 \text {Subst}\left (\int \frac {-a^2+b^2-2 a b x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = -\frac {2 a^2}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\tan (c+d x)}}-\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (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 {\tan (c+d x)}\right )}{d} \\ & = -\frac {2 a^2}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\tan (c+d x)}}+\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 {\tan (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 {\tan (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-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (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 {\tan (c+d x)}\right )}{2 d} \\ & = \frac {\left (a^2-2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {\left (a^2-2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {2 a^2}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\tan (c+d x)}}-\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 {\tan (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 {\tan (c+d x)}\right )}{\sqrt {2} d} \\ & = \frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a^2-2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {\left (a^2-2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {2 a^2}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\tan (c+d x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.35 \[ \int \frac {(a+b \tan (c+d x))^2}{\tan ^{\frac {5}{2}}(c+d x)} \, dx=-\frac {2 \left (\left (a^2-b^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},-\tan ^2(c+d x)\right )+b \left (b+6 a \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\tan ^2(c+d x)\right ) \tan (c+d x)\right )\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)} \]
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Time = 0.06 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {\frac {\left (-a^{2}+b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}-\frac {a b \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{2}-\frac {2 a^{2}}{3 \tan \left (d x +c \right )^{\frac {3}{2}}}-\frac {4 a b}{\sqrt {\tan \left (d x +c \right )}}}{d}\) | \(212\) |
default | \(\frac {\frac {\left (-a^{2}+b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}-\frac {a b \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{2}-\frac {2 a^{2}}{3 \tan \left (d x +c \right )^{\frac {3}{2}}}-\frac {4 a b}{\sqrt {\tan \left (d x +c \right )}}}{d}\) | \(212\) |
parts | \(\frac {a^{2} \left (-\frac {2}{3 \tan \left (d x +c \right )^{\frac {3}{2}}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}+\frac {b^{2} \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4 d}+\frac {2 a b \left (-\frac {2}{\sqrt {\tan \left (d x +c \right )}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}\) | \(299\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1050 vs. \(2 (189) = 378\).
Time = 0.25 (sec) , antiderivative size = 1050, normalized size of antiderivative = 4.71 \[ \int \frac {(a+b \tan (c+d x))^2}{\tan ^{\frac {5}{2}}(c+d x)} \, dx=-\frac {3 \, d \sqrt {-\frac {4 \, a^{3} b - 4 \, a b^{3} + d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} \log \left ({\left (2 \, a b d^{3} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}} + {\left (a^{6} - 7 \, a^{4} b^{2} + 7 \, a^{2} b^{4} - b^{6}\right )} d\right )} \sqrt {-\frac {4 \, a^{3} b - 4 \, a b^{3} + d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} + {\left (a^{8} - 4 \, a^{6} b^{2} - 10 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \sqrt {\tan \left (d x + c\right )}\right ) \tan \left (d x + c\right )^{2} - 3 \, d \sqrt {-\frac {4 \, a^{3} b - 4 \, a b^{3} + d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} \log \left (-{\left (2 \, a b d^{3} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}} + {\left (a^{6} - 7 \, a^{4} b^{2} + 7 \, a^{2} b^{4} - b^{6}\right )} d\right )} \sqrt {-\frac {4 \, a^{3} b - 4 \, a b^{3} + d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} + {\left (a^{8} - 4 \, a^{6} b^{2} - 10 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \sqrt {\tan \left (d x + c\right )}\right ) \tan \left (d x + c\right )^{2} - 3 \, d \sqrt {-\frac {4 \, a^{3} b - 4 \, a b^{3} - d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} \log \left ({\left (2 \, a b d^{3} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}} - {\left (a^{6} - 7 \, a^{4} b^{2} + 7 \, a^{2} b^{4} - b^{6}\right )} d\right )} \sqrt {-\frac {4 \, a^{3} b - 4 \, a b^{3} - d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} + {\left (a^{8} - 4 \, a^{6} b^{2} - 10 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \sqrt {\tan \left (d x + c\right )}\right ) \tan \left (d x + c\right )^{2} + 3 \, d \sqrt {-\frac {4 \, a^{3} b - 4 \, a b^{3} - d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} \log \left (-{\left (2 \, a b d^{3} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}} - {\left (a^{6} - 7 \, a^{4} b^{2} + 7 \, a^{2} b^{4} - b^{6}\right )} d\right )} \sqrt {-\frac {4 \, a^{3} b - 4 \, a b^{3} - d^{2} \sqrt {-\frac {a^{8} - 12 \, a^{6} b^{2} + 38 \, a^{4} b^{4} - 12 \, a^{2} b^{6} + b^{8}}{d^{4}}}}{d^{2}}} + {\left (a^{8} - 4 \, a^{6} b^{2} - 10 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \sqrt {\tan \left (d x + c\right )}\right ) \tan \left (d x + c\right )^{2} + 4 \, {\left (6 \, a b \tan \left (d x + c\right ) + a^{2}\right )} \sqrt {\tan \left (d x + c\right )}}{6 \, d \tan \left (d x + c\right )^{2}} \]
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\[ \int \frac {(a+b \tan (c+d x))^2}{\tan ^{\frac {5}{2}}(c+d x)} \, dx=\int \frac {\left (a + b \tan {\left (c + d x \right )}\right )^{2}}{\tan ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx \]
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Time = 0.41 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b \tan (c+d x))^2}{\tan ^{\frac {5}{2}}(c+d x)} \, dx=-\frac {6 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 6 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 3 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - 3 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \frac {8 \, {\left (6 \, a b \tan \left (d x + c\right ) + a^{2}\right )}}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{12 \, d} \]
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Timed out. \[ \int \frac {(a+b \tan (c+d x))^2}{\tan ^{\frac {5}{2}}(c+d x)} \, dx=\text {Timed out} \]
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Time = 6.16 (sec) , antiderivative size = 968, normalized size of antiderivative = 4.34 \[ \int \frac {(a+b \tan (c+d x))^2}{\tan ^{\frac {5}{2}}(c+d x)} \, dx=2\,\mathrm {atanh}\left (\frac {32\,a^4\,d^3\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {a\,b^3}{d^2}-\frac {b^4\,1{}\mathrm {i}}{4\,d^2}-\frac {a^4\,1{}\mathrm {i}}{4\,d^2}-\frac {a^3\,b}{d^2}+\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2}}}{a^6\,d^2\,16{}\mathrm {i}+32\,a^5\,b\,d^2-a^4\,b^2\,d^2\,112{}\mathrm {i}-192\,a^3\,b^3\,d^2+a^2\,b^4\,d^2\,112{}\mathrm {i}+32\,a\,b^5\,d^2-b^6\,d^2\,16{}\mathrm {i}}+\frac {32\,b^4\,d^3\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {a\,b^3}{d^2}-\frac {b^4\,1{}\mathrm {i}}{4\,d^2}-\frac {a^4\,1{}\mathrm {i}}{4\,d^2}-\frac {a^3\,b}{d^2}+\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2}}}{a^6\,d^2\,16{}\mathrm {i}+32\,a^5\,b\,d^2-a^4\,b^2\,d^2\,112{}\mathrm {i}-192\,a^3\,b^3\,d^2+a^2\,b^4\,d^2\,112{}\mathrm {i}+32\,a\,b^5\,d^2-b^6\,d^2\,16{}\mathrm {i}}-\frac {192\,a^2\,b^2\,d^3\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {a\,b^3}{d^2}-\frac {b^4\,1{}\mathrm {i}}{4\,d^2}-\frac {a^4\,1{}\mathrm {i}}{4\,d^2}-\frac {a^3\,b}{d^2}+\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2}}}{a^6\,d^2\,16{}\mathrm {i}+32\,a^5\,b\,d^2-a^4\,b^2\,d^2\,112{}\mathrm {i}-192\,a^3\,b^3\,d^2+a^2\,b^4\,d^2\,112{}\mathrm {i}+32\,a\,b^5\,d^2-b^6\,d^2\,16{}\mathrm {i}}\right )\,\sqrt {-\frac {a^4\,1{}\mathrm {i}+4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}-4\,a\,b^3+b^4\,1{}\mathrm {i}}{4\,d^2}}+2\,\mathrm {atanh}\left (\frac {32\,a^4\,d^3\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {a^4\,1{}\mathrm {i}}{4\,d^2}+\frac {b^4\,1{}\mathrm {i}}{4\,d^2}+\frac {a\,b^3}{d^2}-\frac {a^3\,b}{d^2}-\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2}}}{-a^6\,d^2\,16{}\mathrm {i}+32\,a^5\,b\,d^2+a^4\,b^2\,d^2\,112{}\mathrm {i}-192\,a^3\,b^3\,d^2-a^2\,b^4\,d^2\,112{}\mathrm {i}+32\,a\,b^5\,d^2+b^6\,d^2\,16{}\mathrm {i}}+\frac {32\,b^4\,d^3\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {a^4\,1{}\mathrm {i}}{4\,d^2}+\frac {b^4\,1{}\mathrm {i}}{4\,d^2}+\frac {a\,b^3}{d^2}-\frac {a^3\,b}{d^2}-\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2}}}{-a^6\,d^2\,16{}\mathrm {i}+32\,a^5\,b\,d^2+a^4\,b^2\,d^2\,112{}\mathrm {i}-192\,a^3\,b^3\,d^2-a^2\,b^4\,d^2\,112{}\mathrm {i}+32\,a\,b^5\,d^2+b^6\,d^2\,16{}\mathrm {i}}-\frac {192\,a^2\,b^2\,d^3\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {a^4\,1{}\mathrm {i}}{4\,d^2}+\frac {b^4\,1{}\mathrm {i}}{4\,d^2}+\frac {a\,b^3}{d^2}-\frac {a^3\,b}{d^2}-\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2}}}{-a^6\,d^2\,16{}\mathrm {i}+32\,a^5\,b\,d^2+a^4\,b^2\,d^2\,112{}\mathrm {i}-192\,a^3\,b^3\,d^2-a^2\,b^4\,d^2\,112{}\mathrm {i}+32\,a\,b^5\,d^2+b^6\,d^2\,16{}\mathrm {i}}\right )\,\sqrt {\frac {a^4\,1{}\mathrm {i}-4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}+4\,a\,b^3+b^4\,1{}\mathrm {i}}{4\,d^2}}-\frac {\frac {2\,a^2}{3}+4\,b\,\mathrm {tan}\left (c+d\,x\right )\,a}{d\,{\mathrm {tan}\left (c+d\,x\right )}^{3/2}} \]
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