Optimal. Leaf size=472 \[ \frac{10 a^2 b^3 \sec ^9(c+d x)}{9 d}-\frac{10 a^2 b^3 \sec ^7(c+d x)}{7 d}-\frac{25 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{64 d}+\frac{5 a^3 b^2 \tan (c+d x) \sec ^7(c+d x)}{4 d}-\frac{5 a^3 b^2 \tan (c+d x) \sec ^5(c+d x)}{24 d}-\frac{25 a^3 b^2 \tan (c+d x) \sec ^3(c+d x)}{96 d}-\frac{25 a^3 b^2 \tan (c+d x) \sec (c+d x)}{64 d}+\frac{5 a^4 b \sec ^7(c+d x)}{7 d}+\frac{5 a^5 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^5 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac{5 a^5 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{5 a^5 \tan (c+d x) \sec (c+d x)}{16 d}+\frac{15 a b^4 \tanh ^{-1}(\sin (c+d x))}{256 d}+\frac{a b^4 \tan ^3(c+d x) \sec ^7(c+d x)}{2 d}-\frac{3 a b^4 \tan (c+d x) \sec ^7(c+d x)}{16 d}+\frac{a b^4 \tan (c+d x) \sec ^5(c+d x)}{32 d}+\frac{5 a b^4 \tan (c+d x) \sec ^3(c+d x)}{128 d}+\frac{15 a b^4 \tan (c+d x) \sec (c+d x)}{256 d}+\frac{b^5 \sec ^{11}(c+d x)}{11 d}-\frac{2 b^5 \sec ^9(c+d x)}{9 d}+\frac{b^5 \sec ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.466743, antiderivative size = 472, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3090, 3768, 3770, 2606, 30, 2611, 14, 270} \[ \frac{10 a^2 b^3 \sec ^9(c+d x)}{9 d}-\frac{10 a^2 b^3 \sec ^7(c+d x)}{7 d}-\frac{25 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{64 d}+\frac{5 a^3 b^2 \tan (c+d x) \sec ^7(c+d x)}{4 d}-\frac{5 a^3 b^2 \tan (c+d x) \sec ^5(c+d x)}{24 d}-\frac{25 a^3 b^2 \tan (c+d x) \sec ^3(c+d x)}{96 d}-\frac{25 a^3 b^2 \tan (c+d x) \sec (c+d x)}{64 d}+\frac{5 a^4 b \sec ^7(c+d x)}{7 d}+\frac{5 a^5 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^5 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac{5 a^5 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{5 a^5 \tan (c+d x) \sec (c+d x)}{16 d}+\frac{15 a b^4 \tanh ^{-1}(\sin (c+d x))}{256 d}+\frac{a b^4 \tan ^3(c+d x) \sec ^7(c+d x)}{2 d}-\frac{3 a b^4 \tan (c+d x) \sec ^7(c+d x)}{16 d}+\frac{a b^4 \tan (c+d x) \sec ^5(c+d x)}{32 d}+\frac{5 a b^4 \tan (c+d x) \sec ^3(c+d x)}{128 d}+\frac{15 a b^4 \tan (c+d x) \sec (c+d x)}{256 d}+\frac{b^5 \sec ^{11}(c+d x)}{11 d}-\frac{2 b^5 \sec ^9(c+d x)}{9 d}+\frac{b^5 \sec ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 3090
Rule 3768
Rule 3770
Rule 2606
Rule 30
Rule 2611
Rule 14
Rule 270
Rubi steps
\begin{align*} \int \sec ^{12}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx &=\int \left (a^5 \sec ^7(c+d x)+5 a^4 b \sec ^7(c+d x) \tan (c+d x)+10 a^3 b^2 \sec ^7(c+d x) \tan ^2(c+d x)+10 a^2 b^3 \sec ^7(c+d x) \tan ^3(c+d x)+5 a b^4 \sec ^7(c+d x) \tan ^4(c+d x)+b^5 \sec ^7(c+d x) \tan ^5(c+d x)\right ) \, dx\\ &=a^5 \int \sec ^7(c+d x) \, dx+\left (5 a^4 b\right ) \int \sec ^7(c+d x) \tan (c+d x) \, dx+\left (10 a^3 b^2\right ) \int \sec ^7(c+d x) \tan ^2(c+d x) \, dx+\left (10 a^2 b^3\right ) \int \sec ^7(c+d x) \tan ^3(c+d x) \, dx+\left (5 a b^4\right ) \int \sec ^7(c+d x) \tan ^4(c+d x) \, dx+b^5 \int \sec ^7(c+d x) \tan ^5(c+d x) \, dx\\ &=\frac{a^5 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{5 a^3 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}+\frac{a b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{2 d}+\frac{1}{6} \left (5 a^5\right ) \int \sec ^5(c+d x) \, dx-\frac{1}{4} \left (5 a^3 b^2\right ) \int \sec ^7(c+d x) \, dx-\frac{1}{2} \left (3 a b^4\right ) \int \sec ^7(c+d x) \tan ^2(c+d x) \, dx+\frac{\left (5 a^4 b\right ) \operatorname{Subst}\left (\int x^6 \, dx,x,\sec (c+d x)\right )}{d}+\frac{\left (10 a^2 b^3\right ) \operatorname{Subst}\left (\int x^6 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{d}+\frac{b^5 \operatorname{Subst}\left (\int x^6 \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac{5 a^4 b \sec ^7(c+d x)}{7 d}+\frac{5 a^5 \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac{a^5 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac{5 a^3 b^2 \sec ^5(c+d x) \tan (c+d x)}{24 d}+\frac{5 a^3 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac{3 a b^4 \sec ^7(c+d x) \tan (c+d x)}{16 d}+\frac{a b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{2 d}+\frac{1}{8} \left (5 a^5\right ) \int \sec ^3(c+d x) \, dx-\frac{1}{24} \left (25 a^3 b^2\right ) \int \sec ^5(c+d x) \, dx+\frac{1}{16} \left (3 a b^4\right ) \int \sec ^7(c+d x) \, dx+\frac{\left (10 a^2 b^3\right ) \operatorname{Subst}\left (\int \left (-x^6+x^8\right ) \, dx,x,\sec (c+d x)\right )}{d}+\frac{b^5 \operatorname{Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac{5 a^4 b \sec ^7(c+d x)}{7 d}-\frac{10 a^2 b^3 \sec ^7(c+d x)}{7 d}+\frac{b^5 \sec ^7(c+d x)}{7 d}+\frac{10 a^2 b^3 \sec ^9(c+d x)}{9 d}-\frac{2 b^5 \sec ^9(c+d x)}{9 d}+\frac{b^5 \sec ^{11}(c+d x)}{11 d}+\frac{5 a^5 \sec (c+d x) \tan (c+d x)}{16 d}+\frac{5 a^5 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac{25 a^3 b^2 \sec ^3(c+d x) \tan (c+d x)}{96 d}+\frac{a^5 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac{5 a^3 b^2 \sec ^5(c+d x) \tan (c+d x)}{24 d}+\frac{a b^4 \sec ^5(c+d x) \tan (c+d x)}{32 d}+\frac{5 a^3 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac{3 a b^4 \sec ^7(c+d x) \tan (c+d x)}{16 d}+\frac{a b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{2 d}+\frac{1}{16} \left (5 a^5\right ) \int \sec (c+d x) \, dx-\frac{1}{32} \left (25 a^3 b^2\right ) \int \sec ^3(c+d x) \, dx+\frac{1}{32} \left (5 a b^4\right ) \int \sec ^5(c+d x) \, dx\\ &=\frac{5 a^5 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{5 a^4 b \sec ^7(c+d x)}{7 d}-\frac{10 a^2 b^3 \sec ^7(c+d x)}{7 d}+\frac{b^5 \sec ^7(c+d x)}{7 d}+\frac{10 a^2 b^3 \sec ^9(c+d x)}{9 d}-\frac{2 b^5 \sec ^9(c+d x)}{9 d}+\frac{b^5 \sec ^{11}(c+d x)}{11 d}+\frac{5 a^5 \sec (c+d x) \tan (c+d x)}{16 d}-\frac{25 a^3 b^2 \sec (c+d x) \tan (c+d x)}{64 d}+\frac{5 a^5 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac{25 a^3 b^2 \sec ^3(c+d x) \tan (c+d x)}{96 d}+\frac{5 a b^4 \sec ^3(c+d x) \tan (c+d x)}{128 d}+\frac{a^5 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac{5 a^3 b^2 \sec ^5(c+d x) \tan (c+d x)}{24 d}+\frac{a b^4 \sec ^5(c+d x) \tan (c+d x)}{32 d}+\frac{5 a^3 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac{3 a b^4 \sec ^7(c+d x) \tan (c+d x)}{16 d}+\frac{a b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{2 d}-\frac{1}{64} \left (25 a^3 b^2\right ) \int \sec (c+d x) \, dx+\frac{1}{128} \left (15 a b^4\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac{5 a^5 \tanh ^{-1}(\sin (c+d x))}{16 d}-\frac{25 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{64 d}+\frac{5 a^4 b \sec ^7(c+d x)}{7 d}-\frac{10 a^2 b^3 \sec ^7(c+d x)}{7 d}+\frac{b^5 \sec ^7(c+d x)}{7 d}+\frac{10 a^2 b^3 \sec ^9(c+d x)}{9 d}-\frac{2 b^5 \sec ^9(c+d x)}{9 d}+\frac{b^5 \sec ^{11}(c+d x)}{11 d}+\frac{5 a^5 \sec (c+d x) \tan (c+d x)}{16 d}-\frac{25 a^3 b^2 \sec (c+d x) \tan (c+d x)}{64 d}+\frac{15 a b^4 \sec (c+d x) \tan (c+d x)}{256 d}+\frac{5 a^5 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac{25 a^3 b^2 \sec ^3(c+d x) \tan (c+d x)}{96 d}+\frac{5 a b^4 \sec ^3(c+d x) \tan (c+d x)}{128 d}+\frac{a^5 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac{5 a^3 b^2 \sec ^5(c+d x) \tan (c+d x)}{24 d}+\frac{a b^4 \sec ^5(c+d x) \tan (c+d x)}{32 d}+\frac{5 a^3 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac{3 a b^4 \sec ^7(c+d x) \tan (c+d x)}{16 d}+\frac{a b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{2 d}+\frac{1}{256} \left (15 a b^4\right ) \int \sec (c+d x) \, dx\\ &=\frac{5 a^5 \tanh ^{-1}(\sin (c+d x))}{16 d}-\frac{25 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{64 d}+\frac{15 a b^4 \tanh ^{-1}(\sin (c+d x))}{256 d}+\frac{5 a^4 b \sec ^7(c+d x)}{7 d}-\frac{10 a^2 b^3 \sec ^7(c+d x)}{7 d}+\frac{b^5 \sec ^7(c+d x)}{7 d}+\frac{10 a^2 b^3 \sec ^9(c+d x)}{9 d}-\frac{2 b^5 \sec ^9(c+d x)}{9 d}+\frac{b^5 \sec ^{11}(c+d x)}{11 d}+\frac{5 a^5 \sec (c+d x) \tan (c+d x)}{16 d}-\frac{25 a^3 b^2 \sec (c+d x) \tan (c+d x)}{64 d}+\frac{15 a b^4 \sec (c+d x) \tan (c+d x)}{256 d}+\frac{5 a^5 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac{25 a^3 b^2 \sec ^3(c+d x) \tan (c+d x)}{96 d}+\frac{5 a b^4 \sec ^3(c+d x) \tan (c+d x)}{128 d}+\frac{a^5 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac{5 a^3 b^2 \sec ^5(c+d x) \tan (c+d x)}{24 d}+\frac{a b^4 \sec ^5(c+d x) \tan (c+d x)}{32 d}+\frac{5 a^3 b^2 \sec ^7(c+d x) \tan (c+d x)}{4 d}-\frac{3 a b^4 \sec ^7(c+d x) \tan (c+d x)}{16 d}+\frac{a b^4 \sec ^7(c+d x) \tan ^3(c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 1.85669, size = 374, normalized size = 0.79 \[ \frac{13860 a \left (2876 a^2 b^2+976 a^4+1207 b^4\right ) \tan (c+d x) \sec ^9(c+d x)-1774080 a \left (-20 a^2 b^2+16 a^4+3 b^4\right ) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+\sec ^{11}(c+d x) \left (5913600 a^3 b^2 \sin (4 (c+d x))-3571260 a^3 b^2 \sin (6 (c+d x))-739200 a^3 b^2 \sin (8 (c+d x))-69300 a^3 b^2 \sin (10 (c+d x))+3604480 \left (-4 a^2 b^3+9 a^4 b-b^5\right ) \cos (2 (c+d x))+1622016 \left (-10 a^2 b^3+5 a^4 b+b^5\right ) \cos (4 (c+d x))+1802240 a^2 b^3+24330240 a^4 b+6623232 a^5 \sin (4 (c+d x))+2857008 a^5 \sin (6 (c+d x))+591360 a^5 \sin (8 (c+d x))+55440 a^5 \sin (10 (c+d x))-6564096 a b^4 \sin (4 (c+d x))+535689 a b^4 \sin (6 (c+d x))+110880 a b^4 \sin (8 (c+d x))+10395 a b^4 \sin (10 (c+d x))+3031040 b^5\right )}{90832896 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.259, size = 814, normalized size = 1.7 \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 [A] time = 1.23221, size = 567, normalized size = 1.2 \begin{align*} -\frac{693 \, a b^{4}{\left (\frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{9} - 70 \, \sin \left (d x + c\right )^{7} + 128 \, \sin \left (d x + c\right )^{5} + 70 \, \sin \left (d x + c\right )^{3} - 15 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{10} - 5 \, \sin \left (d x + c\right )^{8} + 10 \, \sin \left (d x + c\right )^{6} - 10 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 4620 \, a^{3} b^{2}{\left (\frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{7} - 55 \, \sin \left (d x + c\right )^{5} + 73 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 3696 \, a^{5}{\left (\frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - \frac{253440 \, a^{4} b}{\cos \left (d x + c\right )^{7}} + \frac{56320 \,{\left (9 \, \cos \left (d x + c\right )^{2} - 7\right )} a^{2} b^{3}}{\cos \left (d x + c\right )^{9}} - \frac{512 \,{\left (99 \, \cos \left (d x + c\right )^{4} - 154 \, \cos \left (d x + c\right )^{2} + 63\right )} b^{5}}{\cos \left (d x + c\right )^{11}}}{354816 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.668564, size = 720, normalized size = 1.53 \begin{align*} \frac{3465 \,{\left (16 \, a^{5} - 20 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{11} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3465 \,{\left (16 \, a^{5} - 20 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{11} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 32256 \, b^{5} + 50688 \,{\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + 78848 \,{\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} + 462 \,{\left (15 \,{\left (16 \, a^{5} - 20 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{9} + 10 \,{\left (16 \, a^{5} - 20 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{7} + 384 \, a b^{4} \cos \left (d x + c\right ) + 8 \,{\left (16 \, a^{5} - 20 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 48 \,{\left (20 \, a^{3} b^{2} - 11 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{354816 \, d \cos \left (d x + c\right )^{11}} \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 [B] time = 1.32433, size = 1480, normalized size = 3.14 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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