Optimal. Leaf size=222 \[ \frac{4 a b \left (4 a^2+5 b^2\right ) \tan ^3(c+d x)}{15 d}+\frac{4 a b \left (4 a^2+5 b^2\right ) \tan (c+d x)}{5 d}+\frac{\left (36 a^2 b^2+5 a^4+8 b^4\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^2 \left (5 a^2+32 b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{\left (36 a^2 b^2+5 a^4+8 b^4\right ) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{7 a^3 b \tan (c+d x) \sec ^4(c+d x)}{15 d}+\frac{a^2 \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^2}{6 d} \]
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Rubi [A] time = 0.379992, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2792, 3031, 3021, 2748, 3767, 3768, 3770} \[ \frac{4 a b \left (4 a^2+5 b^2\right ) \tan ^3(c+d x)}{15 d}+\frac{4 a b \left (4 a^2+5 b^2\right ) \tan (c+d x)}{5 d}+\frac{\left (36 a^2 b^2+5 a^4+8 b^4\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^2 \left (5 a^2+32 b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{\left (36 a^2 b^2+5 a^4+8 b^4\right ) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{7 a^3 b \tan (c+d x) \sec ^4(c+d x)}{15 d}+\frac{a^2 \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^2}{6 d} \]
Antiderivative was successfully verified.
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Rule 2792
Rule 3031
Rule 3021
Rule 2748
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^4 \sec ^7(c+d x) \, dx &=\frac{a^2 (a+b \cos (c+d x))^2 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{1}{6} \int (a+b \cos (c+d x)) \left (14 a^2 b+a \left (5 a^2+18 b^2\right ) \cos (c+d x)+3 b \left (a^2+2 b^2\right ) \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx\\ &=\frac{7 a^3 b \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac{a^2 (a+b \cos (c+d x))^2 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac{1}{30} \int \left (-5 a^2 \left (5 a^2+32 b^2\right )-24 a b \left (4 a^2+5 b^2\right ) \cos (c+d x)-15 b^2 \left (a^2+2 b^2\right ) \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx\\ &=\frac{a^2 \left (5 a^2+32 b^2\right ) \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac{7 a^3 b \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac{a^2 (a+b \cos (c+d x))^2 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac{1}{120} \int \left (-96 a b \left (4 a^2+5 b^2\right )-15 \left (5 a^4+36 a^2 b^2+8 b^4\right ) \cos (c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac{a^2 \left (5 a^2+32 b^2\right ) \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac{7 a^3 b \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac{a^2 (a+b \cos (c+d x))^2 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{1}{5} \left (4 a b \left (4 a^2+5 b^2\right )\right ) \int \sec ^4(c+d x) \, dx-\frac{1}{8} \left (-5 a^4-36 a^2 b^2-8 b^4\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac{\left (5 a^4+36 a^2 b^2+8 b^4\right ) \sec (c+d x) \tan (c+d x)}{16 d}+\frac{a^2 \left (5 a^2+32 b^2\right ) \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac{7 a^3 b \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac{a^2 (a+b \cos (c+d x))^2 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac{1}{16} \left (-5 a^4-36 a^2 b^2-8 b^4\right ) \int \sec (c+d x) \, dx-\frac{\left (4 a b \left (4 a^2+5 b^2\right )\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 d}\\ &=\frac{\left (5 a^4+36 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{4 a b \left (4 a^2+5 b^2\right ) \tan (c+d x)}{5 d}+\frac{\left (5 a^4+36 a^2 b^2+8 b^4\right ) \sec (c+d x) \tan (c+d x)}{16 d}+\frac{a^2 \left (5 a^2+32 b^2\right ) \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac{7 a^3 b \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac{a^2 (a+b \cos (c+d x))^2 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{4 a b \left (4 a^2+5 b^2\right ) \tan ^3(c+d x)}{15 d}\\ \end{align*}
Mathematica [A] time = 0.982429, size = 154, normalized size = 0.69 \[ \frac{15 \left (36 a^2 b^2+5 a^4+8 b^4\right ) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (64 a b \left (5 \left (2 a^2+b^2\right ) \tan ^2(c+d x)+15 \left (a^2+b^2\right )+3 a^2 \tan ^4(c+d x)\right )+10 a^2 \left (5 a^2+36 b^2\right ) \sec ^3(c+d x)+15 \left (36 a^2 b^2+5 a^4+8 b^4\right ) \sec (c+d x)+40 a^4 \sec ^5(c+d x)\right )}{240 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.072, size = 302, normalized size = 1.4 \begin{align*}{\frac{{a}^{4} \left ( \sec \left ( dx+c \right ) \right ) ^{5}\tan \left ( dx+c \right ) }{6\,d}}+{\frac{5\,{a}^{4} \left ( \sec \left ( dx+c \right ) \right ) ^{3}\tan \left ( dx+c \right ) }{24\,d}}+{\frac{5\,{a}^{4}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{16\,d}}+{\frac{5\,{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{16\,d}}+{\frac{32\,{a}^{3}b\tan \left ( dx+c \right ) }{15\,d}}+{\frac{4\,{a}^{3}b \left ( \sec \left ( dx+c \right ) \right ) ^{4}\tan \left ( dx+c \right ) }{5\,d}}+{\frac{16\,{a}^{3}b \left ( \sec \left ( dx+c \right ) \right ) ^{2}\tan \left ( dx+c \right ) }{15\,d}}+{\frac{3\,{a}^{2}{b}^{2}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{2\,d}}+{\frac{9\,{a}^{2}{b}^{2}\tan \left ( dx+c \right ) \sec \left ( dx+c \right ) }{4\,d}}+{\frac{9\,{a}^{2}{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{4\,d}}+{\frac{8\,a{b}^{3}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{4\,a{b}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{{b}^{4}\tan \left ( dx+c \right ) \sec \left ( dx+c \right ) }{2\,d}}+{\frac{{b}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.992319, size = 371, normalized size = 1.67 \begin{align*} \frac{128 \,{\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} a^{3} b + 640 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a b^{3} - 5 \, a^{4}{\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 )} - 180 \, a^{2} b^{2}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \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 )\right )} - 120 \, b^{4}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.11043, size = 528, normalized size = 2.38 \begin{align*} \frac{15 \,{\left (5 \, a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left (5 \, a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (128 \,{\left (4 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (d x + c\right )^{5} + 192 \, a^{3} b \cos \left (d x + c\right ) + 15 \,{\left (5 \, a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 40 \, a^{4} + 64 \,{\left (4 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} + 10 \,{\left (5 \, a^{4} + 36 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \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 = 1.49498, size = 799, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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