Integrand size = 23, antiderivative size = 258 \[ \int (a+a \sec (c+d x))^{3/2} \tan ^6(c+d x) \, dx=-\frac {2 a^{3/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}+\frac {2 a^2 \tan (c+d x)}{d \sqrt {a+a \sec (c+d x)}}-\frac {2 a^3 \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac {2 a^4 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac {30 a^5 \tan ^7(c+d x)}{7 d (a+a \sec (c+d x))^{7/2}}+\frac {34 a^6 \tan ^9(c+d x)}{9 d (a+a \sec (c+d x))^{9/2}}+\frac {14 a^7 \tan ^{11}(c+d x)}{11 d (a+a \sec (c+d x))^{11/2}}+\frac {2 a^8 \tan ^{13}(c+d x)}{13 d (a+a \sec (c+d x))^{13/2}} \]
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Time = 0.16 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3972, 472, 209} \[ \int (a+a \sec (c+d x))^{3/2} \tan ^6(c+d x) \, dx=-\frac {2 a^{3/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}+\frac {2 a^8 \tan ^{13}(c+d x)}{13 d (a \sec (c+d x)+a)^{13/2}}+\frac {14 a^7 \tan ^{11}(c+d x)}{11 d (a \sec (c+d x)+a)^{11/2}}+\frac {34 a^6 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac {30 a^5 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac {2 a^4 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac {2 a^3 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}+\frac {2 a^2 \tan (c+d x)}{d \sqrt {a \sec (c+d x)+a}} \]
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Rule 209
Rule 472
Rule 3972
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (2 a^5\right ) \text {Subst}\left (\int \frac {x^6 \left (2+a x^2\right )^4}{1+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d} \\ & = -\frac {\left (2 a^5\right ) \text {Subst}\left (\int \left (\frac {1}{a^3}-\frac {x^2}{a^2}+\frac {x^4}{a}+15 x^6+17 a x^8+7 a^2 x^{10}+a^3 x^{12}-\frac {1}{a^3 \left (1+a x^2\right )}\right ) \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d} \\ & = \frac {2 a^2 \tan (c+d x)}{d \sqrt {a+a \sec (c+d x)}}-\frac {2 a^3 \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac {2 a^4 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac {30 a^5 \tan ^7(c+d x)}{7 d (a+a \sec (c+d x))^{7/2}}+\frac {34 a^6 \tan ^9(c+d x)}{9 d (a+a \sec (c+d x))^{9/2}}+\frac {14 a^7 \tan ^{11}(c+d x)}{11 d (a+a \sec (c+d x))^{11/2}}+\frac {2 a^8 \tan ^{13}(c+d x)}{13 d (a+a \sec (c+d x))^{13/2}}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d} \\ & = -\frac {2 a^{3/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}+\frac {2 a^2 \tan (c+d x)}{d \sqrt {a+a \sec (c+d x)}}-\frac {2 a^3 \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac {2 a^4 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac {30 a^5 \tan ^7(c+d x)}{7 d (a+a \sec (c+d x))^{7/2}}+\frac {34 a^6 \tan ^9(c+d x)}{9 d (a+a \sec (c+d x))^{9/2}}+\frac {14 a^7 \tan ^{11}(c+d x)}{11 d (a+a \sec (c+d x))^{11/2}}+\frac {2 a^8 \tan ^{13}(c+d x)}{13 d (a+a \sec (c+d x))^{13/2}} \\ \end{align*}
Time = 9.38 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.57 \[ \int (a+a \sec (c+d x))^{3/2} \tan ^6(c+d x) \, dx=\frac {a \sec \left (\frac {1}{2} (c+d x)\right ) \sec ^6(c+d x) \sqrt {a (1+\sec (c+d x))} \left (-1441440 \sqrt {2} \arcsin \left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^{\frac {13}{2}}(c+d x)+164736 \sin \left (\frac {1}{2} (c+d x)\right )+81081 \sin \left (\frac {3}{2} (c+d x)\right )+134849 \sin \left (\frac {5}{2} (c+d x)\right )+98176 \sin \left (\frac {9}{2} (c+d x)\right )+45045 \sin \left (\frac {11}{2} (c+d x)\right )+32429 \sin \left (\frac {13}{2} (c+d x)\right )\right )}{1441440 d} \]
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Time = 4.03 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.97
method | result | size |
default | \(-\frac {2 a \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (45045 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )+45045 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )-32429 \sin \left (d x +c \right )-38737 \tan \left (d x +c \right )+4731 \sec \left (d x +c \right ) \tan \left (d x +c \right )+26465 \tan \left (d x +c \right ) \sec \left (d x +c \right )^{2}+6265 \tan \left (d x +c \right ) \sec \left (d x +c \right )^{3}-7875 \sec \left (d x +c \right )^{4} \tan \left (d x +c \right )-3465 \tan \left (d x +c \right ) \sec \left (d x +c \right )^{5}\right )}{45045 d \left (\cos \left (d x +c \right )+1\right )}\) | \(250\) |
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Time = 0.36 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.61 \[ \int (a+a \sec (c+d x))^{3/2} \tan ^6(c+d x) \, dx=\left [\frac {45045 \, {\left (a \cos \left (d x + c\right )^{7} + a \cos \left (d x + c\right )^{6}\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} + 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \, {\left (32429 \, a \cos \left (d x + c\right )^{6} + 38737 \, a \cos \left (d x + c\right )^{5} - 4731 \, a \cos \left (d x + c\right )^{4} - 26465 \, a \cos \left (d x + c\right )^{3} - 6265 \, a \cos \left (d x + c\right )^{2} + 7875 \, a \cos \left (d x + c\right ) + 3465 \, a\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{45045 \, {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\right )}}, \frac {2 \, {\left (45045 \, {\left (a \cos \left (d x + c\right )^{7} + a \cos \left (d x + c\right )^{6}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) + {\left (32429 \, a \cos \left (d x + c\right )^{6} + 38737 \, a \cos \left (d x + c\right )^{5} - 4731 \, a \cos \left (d x + c\right )^{4} - 26465 \, a \cos \left (d x + c\right )^{3} - 6265 \, a \cos \left (d x + c\right )^{2} + 7875 \, a \cos \left (d x + c\right ) + 3465 \, a\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )\right )}}{45045 \, {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\right )}}\right ] \]
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\[ \int (a+a \sec (c+d x))^{3/2} \tan ^6(c+d x) \, dx=\int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}} \tan ^{6}{\left (c + d x \right )}\, dx \]
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Timed out. \[ \int (a+a \sec (c+d x))^{3/2} \tan ^6(c+d x) \, dx=\text {Timed out} \]
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\[ \int (a+a \sec (c+d x))^{3/2} \tan ^6(c+d x) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \tan \left (d x + c\right )^{6} \,d x } \]
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Timed out. \[ \int (a+a \sec (c+d x))^{3/2} \tan ^6(c+d x) \, dx=\int {\mathrm {tan}\left (c+d\,x\right )}^6\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
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