Optimal. Leaf size=279 \[ \frac{128 \tan ^3(c+d x)}{12597 a^8 d}+\frac{128 \tan (c+d x)}{4199 a^8 d}-\frac{32 \sec ^3(c+d x)}{4199 d \left (a^8 \sin (c+d x)+a^8\right )}-\frac{32 \sec ^3(c+d x)}{4199 d \left (a^4 \sin (c+d x)+a^4\right )^2}-\frac{112 \sec ^3(c+d x)}{12597 a^2 d \left (a^2 \sin (c+d x)+a^2\right )^3}-\frac{48 \sec ^3(c+d x)}{4199 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac{66 \sec ^3(c+d x)}{4199 a^3 d (a \sin (c+d x)+a)^5}-\frac{22 \sec ^3(c+d x)}{969 a^2 d (a \sin (c+d x)+a)^6}-\frac{11 \sec ^3(c+d x)}{323 a d (a \sin (c+d x)+a)^7}-\frac{\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.418573, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2672, 3767} \[ \frac{128 \tan ^3(c+d x)}{12597 a^8 d}+\frac{128 \tan (c+d x)}{4199 a^8 d}-\frac{32 \sec ^3(c+d x)}{4199 d \left (a^8 \sin (c+d x)+a^8\right )}-\frac{32 \sec ^3(c+d x)}{4199 d \left (a^4 \sin (c+d x)+a^4\right )^2}-\frac{112 \sec ^3(c+d x)}{12597 a^2 d \left (a^2 \sin (c+d x)+a^2\right )^3}-\frac{48 \sec ^3(c+d x)}{4199 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac{66 \sec ^3(c+d x)}{4199 a^3 d (a \sin (c+d x)+a)^5}-\frac{22 \sec ^3(c+d x)}{969 a^2 d (a \sin (c+d x)+a)^6}-\frac{11 \sec ^3(c+d x)}{323 a d (a \sin (c+d x)+a)^7}-\frac{\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2672
Rule 3767
Rubi steps
\begin{align*} \int \frac{\sec ^4(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=-\frac{\sec ^3(c+d x)}{19 d (a+a \sin (c+d x))^8}+\frac{11 \int \frac{\sec ^4(c+d x)}{(a+a \sin (c+d x))^7} \, dx}{19 a}\\ &=-\frac{\sec ^3(c+d x)}{19 d (a+a \sin (c+d x))^8}-\frac{11 \sec ^3(c+d x)}{323 a d (a+a \sin (c+d x))^7}+\frac{110 \int \frac{\sec ^4(c+d x)}{(a+a \sin (c+d x))^6} \, dx}{323 a^2}\\ &=-\frac{\sec ^3(c+d x)}{19 d (a+a \sin (c+d x))^8}-\frac{11 \sec ^3(c+d x)}{323 a d (a+a \sin (c+d x))^7}-\frac{22 \sec ^3(c+d x)}{969 a^2 d (a+a \sin (c+d x))^6}+\frac{66 \int \frac{\sec ^4(c+d x)}{(a+a \sin (c+d x))^5} \, dx}{323 a^3}\\ &=-\frac{\sec ^3(c+d x)}{19 d (a+a \sin (c+d x))^8}-\frac{11 \sec ^3(c+d x)}{323 a d (a+a \sin (c+d x))^7}-\frac{22 \sec ^3(c+d x)}{969 a^2 d (a+a \sin (c+d x))^6}-\frac{66 \sec ^3(c+d x)}{4199 a^3 d (a+a \sin (c+d x))^5}+\frac{528 \int \frac{\sec ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx}{4199 a^4}\\ &=-\frac{\sec ^3(c+d x)}{19 d (a+a \sin (c+d x))^8}-\frac{11 \sec ^3(c+d x)}{323 a d (a+a \sin (c+d x))^7}-\frac{22 \sec ^3(c+d x)}{969 a^2 d (a+a \sin (c+d x))^6}-\frac{66 \sec ^3(c+d x)}{4199 a^3 d (a+a \sin (c+d x))^5}-\frac{48 \sec ^3(c+d x)}{4199 d \left (a^2+a^2 \sin (c+d x)\right )^4}+\frac{336 \int \frac{\sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx}{4199 a^5}\\ &=-\frac{\sec ^3(c+d x)}{19 d (a+a \sin (c+d x))^8}-\frac{11 \sec ^3(c+d x)}{323 a d (a+a \sin (c+d x))^7}-\frac{22 \sec ^3(c+d x)}{969 a^2 d (a+a \sin (c+d x))^6}-\frac{66 \sec ^3(c+d x)}{4199 a^3 d (a+a \sin (c+d x))^5}-\frac{112 \sec ^3(c+d x)}{12597 a^5 d (a+a \sin (c+d x))^3}-\frac{48 \sec ^3(c+d x)}{4199 d \left (a^2+a^2 \sin (c+d x)\right )^4}+\frac{224 \int \frac{\sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx}{4199 a^6}\\ &=-\frac{\sec ^3(c+d x)}{19 d (a+a \sin (c+d x))^8}-\frac{11 \sec ^3(c+d x)}{323 a d (a+a \sin (c+d x))^7}-\frac{22 \sec ^3(c+d x)}{969 a^2 d (a+a \sin (c+d x))^6}-\frac{66 \sec ^3(c+d x)}{4199 a^3 d (a+a \sin (c+d x))^5}-\frac{112 \sec ^3(c+d x)}{12597 a^5 d (a+a \sin (c+d x))^3}-\frac{48 \sec ^3(c+d x)}{4199 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac{32 \sec ^3(c+d x)}{4199 d \left (a^4+a^4 \sin (c+d x)\right )^2}+\frac{160 \int \frac{\sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx}{4199 a^7}\\ &=-\frac{\sec ^3(c+d x)}{19 d (a+a \sin (c+d x))^8}-\frac{11 \sec ^3(c+d x)}{323 a d (a+a \sin (c+d x))^7}-\frac{22 \sec ^3(c+d x)}{969 a^2 d (a+a \sin (c+d x))^6}-\frac{66 \sec ^3(c+d x)}{4199 a^3 d (a+a \sin (c+d x))^5}-\frac{112 \sec ^3(c+d x)}{12597 a^5 d (a+a \sin (c+d x))^3}-\frac{48 \sec ^3(c+d x)}{4199 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac{32 \sec ^3(c+d x)}{4199 d \left (a^4+a^4 \sin (c+d x)\right )^2}-\frac{32 \sec ^3(c+d x)}{4199 d \left (a^8+a^8 \sin (c+d x)\right )}+\frac{128 \int \sec ^4(c+d x) \, dx}{4199 a^8}\\ &=-\frac{\sec ^3(c+d x)}{19 d (a+a \sin (c+d x))^8}-\frac{11 \sec ^3(c+d x)}{323 a d (a+a \sin (c+d x))^7}-\frac{22 \sec ^3(c+d x)}{969 a^2 d (a+a \sin (c+d x))^6}-\frac{66 \sec ^3(c+d x)}{4199 a^3 d (a+a \sin (c+d x))^5}-\frac{112 \sec ^3(c+d x)}{12597 a^5 d (a+a \sin (c+d x))^3}-\frac{48 \sec ^3(c+d x)}{4199 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac{32 \sec ^3(c+d x)}{4199 d \left (a^4+a^4 \sin (c+d x)\right )^2}-\frac{32 \sec ^3(c+d x)}{4199 d \left (a^8+a^8 \sin (c+d x)\right )}-\frac{128 \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{4199 a^8 d}\\ &=-\frac{\sec ^3(c+d x)}{19 d (a+a \sin (c+d x))^8}-\frac{11 \sec ^3(c+d x)}{323 a d (a+a \sin (c+d x))^7}-\frac{22 \sec ^3(c+d x)}{969 a^2 d (a+a \sin (c+d x))^6}-\frac{66 \sec ^3(c+d x)}{4199 a^3 d (a+a \sin (c+d x))^5}-\frac{112 \sec ^3(c+d x)}{12597 a^5 d (a+a \sin (c+d x))^3}-\frac{48 \sec ^3(c+d x)}{4199 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac{32 \sec ^3(c+d x)}{4199 d \left (a^4+a^4 \sin (c+d x)\right )^2}-\frac{32 \sec ^3(c+d x)}{4199 d \left (a^8+a^8 \sin (c+d x)\right )}+\frac{128 \tan (c+d x)}{4199 a^8 d}+\frac{128 \tan ^3(c+d x)}{12597 a^8 d}\\ \end{align*}
Mathematica [A] time = 0.419192, size = 125, normalized size = 0.45 \[ \frac{\sec ^3(c+d x) (8398 \sin (c+d x)-5814 \sin (3 (c+d x))-2907 \sin (5 (c+d x))+1463 \sin (7 (c+d x))-117 \sin (9 (c+d x))+\sin (11 (c+d x))-10336 \cos (2 (c+d x))+2736 \cos (6 (c+d x))-512 \cos (8 (c+d x))+16 \cos (10 (c+d x)))}{50388 a^8 d (\sin (c+d x)+1)^8} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.144, size = 340, normalized size = 1.2 \begin{align*} 2\,{\frac{1}{d{a}^{8}} \left ( -{\frac{1}{1536\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{3}}}-{\frac{1}{1024\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{2}}}-{\frac{3}{512\,\tan \left ( 1/2\,dx+c/2 \right ) -512}}-{\frac{128}{19\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{19}}}+64\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-18}-{\frac{5248}{17\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{17}}}+992\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-16}-{\frac{7096}{3\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{15}}}+4428\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-14}-{\frac{87508}{13\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{13}}}+{\frac{25468}{3\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{12}}}-{\frac{18011}{2\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{11}}}+{\frac{32417}{4\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{10}}}-6215\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-9}+{\frac{32525}{8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{8}}}-{\frac{72425}{32\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{7}}}+{\frac{204605}{192\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{6}}}-{\frac{26871}{64\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{5}}}+{\frac{2177}{16\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{4}}}-{\frac{54229}{1536\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{3}}}+{\frac{7181}{1024\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{2}}}-{\frac{509}{512\,\tan \left ( 1/2\,dx+c/2 \right ) +512}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.1709, size = 1169, normalized size = 4.19 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.01333, size = 695, normalized size = 2.49 \begin{align*} \frac{2048 \, \cos \left (d x + c\right )^{10} - 21504 \, \cos \left (d x + c\right )^{8} + 59136 \, \cos \left (d x + c\right )^{6} - 54912 \, \cos \left (d x + c\right )^{4} + 11440 \, \cos \left (d x + c\right )^{2} +{\left (256 \, \cos \left (d x + c\right )^{10} - 8064 \, \cos \left (d x + c\right )^{8} + 36960 \, \cos \left (d x + c\right )^{6} - 48048 \, \cos \left (d x + c\right )^{4} + 12870 \, \cos \left (d x + c\right )^{2} + 2431\right )} \sin \left (d x + c\right ) + 1768}{12597 \,{\left (a^{8} d \cos \left (d x + c\right )^{11} - 32 \, a^{8} d \cos \left (d x + c\right )^{9} + 160 \, a^{8} d \cos \left (d x + c\right )^{7} - 256 \, a^{8} d \cos \left (d x + c\right )^{5} + 128 \, a^{8} d \cos \left (d x + c\right )^{3} - 8 \,{\left (a^{8} d \cos \left (d x + c\right )^{9} - 10 \, a^{8} d \cos \left (d x + c\right )^{7} + 24 \, a^{8} d \cos \left (d x + c\right )^{5} - 16 \, a^{8} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.19403, size = 406, normalized size = 1.46 \begin{align*} -\frac{\frac{4199 \,{\left (18 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 33 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 17\right )}}{a^{8}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{3}} + \frac{12823746 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{18} + 140368371 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{17} + 879644311 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{16} + 3693272440 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{15} + 11467502592 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{14} + 27403194676 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 51919375300 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} + 79183835016 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 98304418212 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 99750226290 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 82860874122 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 56110430792 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 30766700912 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 13462452660 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 4616712644 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 1197851960 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 226248618 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 27911475 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2143959}{a^{8}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{19}}}{6449664 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]