Optimal. Leaf size=290 \[ \frac{2 a^{10} \tan ^{15}(c+d x)}{15 d (a \sec (c+d x)+a)^{15/2}}+\frac{18 a^9 \tan ^{13}(c+d x)}{13 d (a \sec (c+d x)+a)^{13/2}}+\frac{62 a^8 \tan ^{11}(c+d x)}{11 d (a \sec (c+d x)+a)^{11/2}}+\frac{98 a^7 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac{62 a^6 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{2 a^5 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 a^4 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}+\frac{2 a^3 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.127909, antiderivative size = 290, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3887, 461, 203} \[ \frac{2 a^{10} \tan ^{15}(c+d x)}{15 d (a \sec (c+d x)+a)^{15/2}}+\frac{18 a^9 \tan ^{13}(c+d x)}{13 d (a \sec (c+d x)+a)^{13/2}}+\frac{62 a^8 \tan ^{11}(c+d x)}{11 d (a \sec (c+d x)+a)^{11/2}}+\frac{98 a^7 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac{62 a^6 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{2 a^5 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 a^4 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}+\frac{2 a^3 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 3887
Rule 461
Rule 203
Rubi steps
\begin{align*} \int (a+a \sec (c+d x))^{5/2} \tan ^6(c+d x) \, dx &=-\frac{\left (2 a^6\right ) \operatorname{Subst}\left (\int \frac{x^6 \left (2+a x^2\right )^5}{1+a x^2} \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}\\ &=-\frac{\left (2 a^6\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^3}-\frac{x^2}{a^2}+\frac{x^4}{a}+31 x^6+49 a x^8+31 a^2 x^{10}+9 a^3 x^{12}+a^4 x^{14}-\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^3 \tan (c+d x)}{d \sqrt{a+a \sec (c+d x)}}-\frac{2 a^4 \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac{2 a^5 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac{62 a^6 \tan ^7(c+d x)}{7 d (a+a \sec (c+d x))^{7/2}}+\frac{98 a^7 \tan ^9(c+d x)}{9 d (a+a \sec (c+d x))^{9/2}}+\frac{62 a^8 \tan ^{11}(c+d x)}{11 d (a+a \sec (c+d x))^{11/2}}+\frac{18 a^9 \tan ^{13}(c+d x)}{13 d (a+a \sec (c+d x))^{13/2}}+\frac{2 a^{10} \tan ^{15}(c+d x)}{15 d (a+a \sec (c+d x))^{15/2}}+\frac{\left (2 a^3\right ) \operatorname{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^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}+\frac{2 a^3 \tan (c+d x)}{d \sqrt{a+a \sec (c+d x)}}-\frac{2 a^4 \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac{2 a^5 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac{62 a^6 \tan ^7(c+d x)}{7 d (a+a \sec (c+d x))^{7/2}}+\frac{98 a^7 \tan ^9(c+d x)}{9 d (a+a \sec (c+d x))^{9/2}}+\frac{62 a^8 \tan ^{11}(c+d x)}{11 d (a+a \sec (c+d x))^{11/2}}+\frac{18 a^9 \tan ^{13}(c+d x)}{13 d (a+a \sec (c+d x))^{13/2}}+\frac{2 a^{10} \tan ^{15}(c+d x)}{15 d (a+a \sec (c+d x))^{15/2}}\\ \end{align*}
Mathematica [A] time = 9.97692, size = 173, normalized size = 0.6 \[ \frac{a^2 \sec \left (\frac{1}{2} (c+d x)\right ) \sec ^7(c+d x) \sqrt{a (\sec (c+d x)+1)} \left (604890 \sin \left (\frac{1}{2} (c+d x)\right )-87230 \sin \left (\frac{3}{2} (c+d x)\right )+450450 \sin \left (\frac{5}{2} (c+d x)\right )-137670 \sin \left (\frac{7}{2} (c+d x)\right )+210210 \sin \left (\frac{9}{2} (c+d x)\right )+75450 \sin \left (\frac{11}{2} (c+d x)\right )+90090 \sin \left (\frac{13}{2} (c+d x)\right )+16066 \sin \left (\frac{15}{2} (c+d x)\right )-2882880 \sqrt{2} \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right ) \cos ^{\frac{15}{2}}(c+d x)\right )}{2882880 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.286, size = 747, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09329, size = 1283, normalized size = 4.42 \begin{align*} \left [\frac{45045 \,{\left (a^{2} \cos \left (d x + c\right )^{8} + a^{2} \cos \left (d x + c\right )^{7}\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 (16066 \, a^{2} \cos \left (d x + c\right )^{7} + 53078 \, a^{2} \cos \left (d x + c\right )^{6} + 17286 \, a^{2} \cos \left (d x + c\right )^{5} - 30640 \, a^{2} \cos \left (d x + c\right )^{4} - 26810 \, a^{2} \cos \left (d x + c\right )^{3} + 2898 \, a^{2} \cos \left (d x + c\right )^{2} + 10164 \, a^{2} \cos \left (d x + c\right ) + 3003 \, a^{2}\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 )^{8} + d \cos \left (d x + c\right )^{7}\right )}}, \frac{2 \,{\left (45045 \,{\left (a^{2} \cos \left (d x + c\right )^{8} + a^{2} \cos \left (d x + c\right )^{7}\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 (16066 \, a^{2} \cos \left (d x + c\right )^{7} + 53078 \, a^{2} \cos \left (d x + c\right )^{6} + 17286 \, a^{2} \cos \left (d x + c\right )^{5} - 30640 \, a^{2} \cos \left (d x + c\right )^{4} - 26810 \, a^{2} \cos \left (d x + c\right )^{3} + 2898 \, a^{2} \cos \left (d x + c\right )^{2} + 10164 \, a^{2} \cos \left (d x + c\right ) + 3003 \, a^{2}\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 )^{8} + d \cos \left (d x + c\right )^{7}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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