Optimal. Leaf size=204 \[ -\frac{10 a^3 b^2 \sin (c+d x)}{d}+\frac{10 a^2 b^3 \cos (c+d x)}{d}+\frac{10 a^2 b^3 \sec (c+d x)}{d}+\frac{10 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{5 a^4 b \cos (c+d x)}{d}+\frac{a^5 \sin (c+d x)}{d}+\frac{15 a b^4 \sin (c+d x)}{2 d}+\frac{5 a b^4 \sin (c+d x) \tan ^2(c+d x)}{2 d}-\frac{15 a b^4 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{b^5 \cos (c+d x)}{d}+\frac{b^5 \sec ^3(c+d x)}{3 d}-\frac{2 b^5 \sec (c+d x)}{d} \]
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Rubi [A] time = 0.208001, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {3090, 2637, 2638, 2592, 321, 206, 2590, 14, 288, 270} \[ -\frac{10 a^3 b^2 \sin (c+d x)}{d}+\frac{10 a^2 b^3 \cos (c+d x)}{d}+\frac{10 a^2 b^3 \sec (c+d x)}{d}+\frac{10 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{5 a^4 b \cos (c+d x)}{d}+\frac{a^5 \sin (c+d x)}{d}+\frac{15 a b^4 \sin (c+d x)}{2 d}+\frac{5 a b^4 \sin (c+d x) \tan ^2(c+d x)}{2 d}-\frac{15 a b^4 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{b^5 \cos (c+d x)}{d}+\frac{b^5 \sec ^3(c+d x)}{3 d}-\frac{2 b^5 \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3090
Rule 2637
Rule 2638
Rule 2592
Rule 321
Rule 206
Rule 2590
Rule 14
Rule 288
Rule 270
Rubi steps
\begin{align*} \int \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx &=\int \left (a^5 \cos (c+d x)+5 a^4 b \sin (c+d x)+10 a^3 b^2 \sin (c+d x) \tan (c+d x)+10 a^2 b^3 \sin (c+d x) \tan ^2(c+d x)+5 a b^4 \sin (c+d x) \tan ^3(c+d x)+b^5 \sin (c+d x) \tan ^4(c+d x)\right ) \, dx\\ &=a^5 \int \cos (c+d x) \, dx+\left (5 a^4 b\right ) \int \sin (c+d x) \, dx+\left (10 a^3 b^2\right ) \int \sin (c+d x) \tan (c+d x) \, dx+\left (10 a^2 b^3\right ) \int \sin (c+d x) \tan ^2(c+d x) \, dx+\left (5 a b^4\right ) \int \sin (c+d x) \tan ^3(c+d x) \, dx+b^5 \int \sin (c+d x) \tan ^4(c+d x) \, dx\\ &=-\frac{5 a^4 b \cos (c+d x)}{d}+\frac{a^5 \sin (c+d x)}{d}+\frac{\left (10 a^3 b^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}-\frac{\left (10 a^2 b^3\right ) \operatorname{Subst}\left (\int \frac{1-x^2}{x^2} \, dx,x,\cos (c+d x)\right )}{d}+\frac{\left (5 a b^4\right ) \operatorname{Subst}\left (\int \frac{x^4}{\left (1-x^2\right )^2} \, dx,x,\sin (c+d x)\right )}{d}-\frac{b^5 \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{x^4} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{5 a^4 b \cos (c+d x)}{d}+\frac{a^5 \sin (c+d x)}{d}-\frac{10 a^3 b^2 \sin (c+d x)}{d}+\frac{5 a b^4 \sin (c+d x) \tan ^2(c+d x)}{2 d}+\frac{\left (10 a^3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}-\frac{\left (10 a^2 b^3\right ) \operatorname{Subst}\left (\int \left (-1+\frac{1}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac{\left (15 a b^4\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-x^2} \, dx,x,\sin (c+d x)\right )}{2 d}-\frac{b^5 \operatorname{Subst}\left (\int \left (1+\frac{1}{x^4}-\frac{2}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{10 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{5 a^4 b \cos (c+d x)}{d}+\frac{10 a^2 b^3 \cos (c+d x)}{d}-\frac{b^5 \cos (c+d x)}{d}+\frac{10 a^2 b^3 \sec (c+d x)}{d}-\frac{2 b^5 \sec (c+d x)}{d}+\frac{b^5 \sec ^3(c+d x)}{3 d}+\frac{a^5 \sin (c+d x)}{d}-\frac{10 a^3 b^2 \sin (c+d x)}{d}+\frac{15 a b^4 \sin (c+d x)}{2 d}+\frac{5 a b^4 \sin (c+d x) \tan ^2(c+d x)}{2 d}-\frac{\left (15 a b^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{2 d}\\ &=\frac{10 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{15 a b^4 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{5 a^4 b \cos (c+d x)}{d}+\frac{10 a^2 b^3 \cos (c+d x)}{d}-\frac{b^5 \cos (c+d x)}{d}+\frac{10 a^2 b^3 \sec (c+d x)}{d}-\frac{2 b^5 \sec (c+d x)}{d}+\frac{b^5 \sec ^3(c+d x)}{3 d}+\frac{a^5 \sin (c+d x)}{d}-\frac{10 a^3 b^2 \sin (c+d x)}{d}+\frac{15 a b^4 \sin (c+d x)}{2 d}+\frac{5 a b^4 \sin (c+d x) \tan ^2(c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 5.95975, size = 397, normalized size = 1.95 \[ \frac{12 a \left (-10 a^2 b^2+a^4+5 b^4\right ) \sin (c+d x)-12 b \left (-10 a^2 b^2+5 a^4+b^4\right ) \cos (c+d x)+\frac{2 b^3 \left (60 a^2-11 b^2\right ) \sin \left (\frac{1}{2} (c+d x)\right )}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}+\frac{2 b^3 \left (11 b^2-60 a^2\right ) \sin \left (\frac{1}{2} (c+d x)\right )}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}-30 a b^2 \left (4 a^2-3 b^2\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+30 a b^2 \left (4 a^2-3 b^2\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+120 a^2 b^3+\frac{b^4 (15 a+b)}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{b^4 (b-15 a)}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{2 b^5 \sin \left (\frac{1}{2} (c+d x)\right )}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3}-\frac{2 b^5 \sin \left (\frac{1}{2} (c+d x)\right )}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3}-22 b^5}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.259, size = 327, normalized size = 1.6 \begin{align*}{\frac{{a}^{5}\sin \left ( dx+c \right ) }{d}}-5\,{\frac{{a}^{4}b\cos \left ( dx+c \right ) }{d}}-10\,{\frac{{a}^{3}{b}^{2}\sin \left ( dx+c \right ) }{d}}+10\,{\frac{{a}^{3}{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+10\,{\frac{{a}^{2}{b}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d\cos \left ( dx+c \right ) }}+10\,{\frac{\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}{a}^{2}{b}^{3}}{d}}+20\,{\frac{{a}^{2}{b}^{3}\cos \left ( dx+c \right ) }{d}}+{\frac{5\,a{b}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{5\,a{b}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{2\,d}}+{\frac{15\,a{b}^{4}\sin \left ( dx+c \right ) }{2\,d}}-{\frac{15\,a{b}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{{b}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{3\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{\frac{{b}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{d\cos \left ( dx+c \right ) }}-{\frac{8\,{b}^{5}\cos \left ( dx+c \right ) }{3\,d}}-{\frac{{b}^{5}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d}}-{\frac{4\,\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}{b}^{5}}{3\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08271, size = 244, normalized size = 1.2 \begin{align*} -\frac{15 \, a b^{4}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) - 4 \, \sin \left (d x + c\right )\right )} - 120 \, a^{2} b^{3}{\left (\frac{1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} + 4 \, b^{5}{\left (\frac{6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )} - 60 \, a^{3} b^{2}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right ) - 2 \, \sin \left (d x + c\right )\right )} + 60 \, a^{4} b \cos \left (d x + c\right ) - 12 \, a^{5} \sin \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.5368, size = 455, normalized size = 2.23 \begin{align*} \frac{4 \, b^{5} - 12 \,{\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + 15 \,{\left (4 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left (4 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 24 \,{\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} + 6 \,{\left (5 \, a b^{4} \cos \left (d x + c\right ) + 2 \,{\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \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 [A] time = 1.29411, size = 379, normalized size = 1.86 \begin{align*} \frac{15 \,{\left (4 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 15 \,{\left (4 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{12 \,{\left (a^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 10 \, a^{3} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 5 \, a b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 5 \, a^{4} b + 10 \, a^{2} b^{3} - b^{5}\right )}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1} + \frac{2 \,{\left (15 \, a b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 60 \, a^{2} b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 6 \, b^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 120 \, a^{2} b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 24 \, b^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, a b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 60 \, a^{2} b^{3} + 10 \, b^{5}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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