Optimal. Leaf size=162 \[ -\frac{c^5 \tan (e+f x)}{a^3 f}+\frac{128 c^5 \cot ^5(e+f x)}{5 a^3 f}+\frac{128 c^5 \cot ^3(e+f x)}{3 a^3 f}+\frac{32 c^5 \cot (e+f x)}{a^3 f}-\frac{128 c^5 \csc ^5(e+f x)}{5 a^3 f}+\frac{64 c^5 \csc ^3(e+f x)}{3 a^3 f}-\frac{16 c^5 \csc (e+f x)}{a^3 f}+\frac{8 c^5 \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac{c^5 x}{a^3} \]
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Rubi [A] time = 0.443483, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 29, number of rules used = 15, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.577, Rules used = {3904, 3886, 3473, 8, 2606, 194, 2607, 30, 14, 3767, 2621, 302, 207, 2620, 270} \[ -\frac{c^5 \tan (e+f x)}{a^3 f}+\frac{128 c^5 \cot ^5(e+f x)}{5 a^3 f}+\frac{128 c^5 \cot ^3(e+f x)}{3 a^3 f}+\frac{32 c^5 \cot (e+f x)}{a^3 f}-\frac{128 c^5 \csc ^5(e+f x)}{5 a^3 f}+\frac{64 c^5 \csc ^3(e+f x)}{3 a^3 f}-\frac{16 c^5 \csc (e+f x)}{a^3 f}+\frac{8 c^5 \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac{c^5 x}{a^3} \]
Antiderivative was successfully verified.
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Rule 3904
Rule 3886
Rule 3473
Rule 8
Rule 2606
Rule 194
Rule 2607
Rule 30
Rule 14
Rule 3767
Rule 2621
Rule 302
Rule 207
Rule 2620
Rule 270
Rubi steps
\begin{align*} \int \frac{(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx &=-\frac{\int \cot ^6(e+f x) (c-c \sec (e+f x))^8 \, dx}{a^3 c^3}\\ &=-\frac{\int \left (c^8 \cot ^6(e+f x)-8 c^8 \cot ^5(e+f x) \csc (e+f x)+28 c^8 \cot ^4(e+f x) \csc ^2(e+f x)-56 c^8 \cot ^3(e+f x) \csc ^3(e+f x)+70 c^8 \cot ^2(e+f x) \csc ^4(e+f x)-56 c^8 \cot (e+f x) \csc ^5(e+f x)+28 c^8 \csc ^6(e+f x)-8 c^8 \csc ^6(e+f x) \sec (e+f x)+c^8 \csc ^6(e+f x) \sec ^2(e+f x)\right ) \, dx}{a^3 c^3}\\ &=-\frac{c^5 \int \cot ^6(e+f x) \, dx}{a^3}-\frac{c^5 \int \csc ^6(e+f x) \sec ^2(e+f x) \, dx}{a^3}+\frac{\left (8 c^5\right ) \int \cot ^5(e+f x) \csc (e+f x) \, dx}{a^3}+\frac{\left (8 c^5\right ) \int \csc ^6(e+f x) \sec (e+f x) \, dx}{a^3}-\frac{\left (28 c^5\right ) \int \cot ^4(e+f x) \csc ^2(e+f x) \, dx}{a^3}-\frac{\left (28 c^5\right ) \int \csc ^6(e+f x) \, dx}{a^3}+\frac{\left (56 c^5\right ) \int \cot ^3(e+f x) \csc ^3(e+f x) \, dx}{a^3}+\frac{\left (56 c^5\right ) \int \cot (e+f x) \csc ^5(e+f x) \, dx}{a^3}-\frac{\left (70 c^5\right ) \int \cot ^2(e+f x) \csc ^4(e+f x) \, dx}{a^3}\\ &=\frac{c^5 \cot ^5(e+f x)}{5 a^3 f}+\frac{c^5 \int \cot ^4(e+f x) \, dx}{a^3}-\frac{c^5 \operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^3}{x^6} \, dx,x,\tan (e+f x)\right )}{a^3 f}-\frac{\left (8 c^5\right ) \operatorname{Subst}\left (\int \frac{x^6}{-1+x^2} \, dx,x,\csc (e+f x)\right )}{a^3 f}-\frac{\left (8 c^5\right ) \operatorname{Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\csc (e+f x)\right )}{a^3 f}-\frac{\left (28 c^5\right ) \operatorname{Subst}\left (\int x^4 \, dx,x,-\cot (e+f x)\right )}{a^3 f}+\frac{\left (28 c^5\right ) \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (e+f x)\right )}{a^3 f}-\frac{\left (56 c^5\right ) \operatorname{Subst}\left (\int x^4 \, dx,x,\csc (e+f x)\right )}{a^3 f}-\frac{\left (56 c^5\right ) \operatorname{Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\csc (e+f x)\right )}{a^3 f}-\frac{\left (70 c^5\right ) \operatorname{Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,-\cot (e+f x)\right )}{a^3 f}\\ &=\frac{28 c^5 \cot (e+f x)}{a^3 f}+\frac{55 c^5 \cot ^3(e+f x)}{3 a^3 f}+\frac{57 c^5 \cot ^5(e+f x)}{5 a^3 f}-\frac{56 c^5 \csc ^5(e+f x)}{5 a^3 f}-\frac{c^5 \int \cot ^2(e+f x) \, dx}{a^3}-\frac{c^5 \operatorname{Subst}\left (\int \left (1+\frac{1}{x^6}+\frac{3}{x^4}+\frac{3}{x^2}\right ) \, dx,x,\tan (e+f x)\right )}{a^3 f}-\frac{\left (8 c^5\right ) \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\csc (e+f x)\right )}{a^3 f}-\frac{\left (8 c^5\right ) \operatorname{Subst}\left (\int \left (1+x^2+x^4+\frac{1}{-1+x^2}\right ) \, dx,x,\csc (e+f x)\right )}{a^3 f}-\frac{\left (56 c^5\right ) \operatorname{Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,\csc (e+f x)\right )}{a^3 f}-\frac{\left (70 c^5\right ) \operatorname{Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,-\cot (e+f x)\right )}{a^3 f}\\ &=\frac{32 c^5 \cot (e+f x)}{a^3 f}+\frac{128 c^5 \cot ^3(e+f x)}{3 a^3 f}+\frac{128 c^5 \cot ^5(e+f x)}{5 a^3 f}-\frac{16 c^5 \csc (e+f x)}{a^3 f}+\frac{64 c^5 \csc ^3(e+f x)}{3 a^3 f}-\frac{128 c^5 \csc ^5(e+f x)}{5 a^3 f}-\frac{c^5 \tan (e+f x)}{a^3 f}+\frac{c^5 \int 1 \, dx}{a^3}-\frac{\left (8 c^5\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\csc (e+f x)\right )}{a^3 f}\\ &=\frac{c^5 x}{a^3}+\frac{8 c^5 \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac{32 c^5 \cot (e+f x)}{a^3 f}+\frac{128 c^5 \cot ^3(e+f x)}{3 a^3 f}+\frac{128 c^5 \cot ^5(e+f x)}{5 a^3 f}-\frac{16 c^5 \csc (e+f x)}{a^3 f}+\frac{64 c^5 \csc ^3(e+f x)}{3 a^3 f}-\frac{128 c^5 \csc ^5(e+f x)}{5 a^3 f}-\frac{c^5 \tan (e+f x)}{a^3 f}\\ \end{align*}
Mathematica [B] time = 5.6226, size = 557, normalized size = 3.44 \[ -\frac{c^5 \sec \left (\frac{e}{2}\right ) (\cos (e+f x)-1)^5 \cot \left (\frac{1}{2} (e+f x)\right ) \csc ^4\left (\frac{1}{2} (e+f x)\right ) \left (1016 \sin \left (\frac{f x}{2}\right ) \cot ^6\left (\frac{1}{2} (e+f x)\right ) \csc \left (\frac{1}{2} (e+f x)\right )+\sec ^2\left (\frac{e}{2}\right ) \sin \left (\frac{f x}{2}\right ) (-210 \cos (e+f x)-84 \cos (2 (e+f x))-14 \cos (3 (e+f x))+131 \cos (2 e+f x)+66 \cos (e+2 f x)+66 \cos (3 e+2 f x)+21 \cos (2 e+3 f x)+21 \cos (4 e+3 f x)+76 \cos (e)+131 \cos (f x)-140) \csc ^7\left (\frac{1}{2} (e+f x)\right )+48 \left (\sin \left (\frac{e}{2}\right )-\sin \left (\frac{3 e}{2}\right )\right ) \sec ^2\left (\frac{e}{2}\right ) \cot \left (\frac{1}{2} (e+f x)\right ) \csc ^4\left (\frac{1}{2} (e+f x)\right )-60 \cos (e) \sec \left (\frac{e}{2}\right ) \cot ^7\left (\frac{1}{2} (e+f x)\right ) \left (-8 \log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )+8 \log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )+f x\right )+8 \left (\sin \left (\frac{e}{2}\right )-\sin \left (\frac{3 e}{2}\right )\right ) \sec ^2\left (\frac{e}{2}\right ) (\cos (e+f x)-7) \cot ^3\left (\frac{1}{2} (e+f x)\right ) \csc ^4\left (\frac{1}{2} (e+f x)\right )+2 \sec \left (\frac{e}{2}\right ) \cot ^5\left (\frac{1}{2} (e+f x)\right ) \left (30 \cos (e) \left (-8 \log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )+8 \log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )+f x\right )-\tan \left (\frac{e}{2}\right ) (15 (\cos (e+f x)+\cos (f x)-1)+\cos (e)) \csc ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )}{240 a^3 f \left (\tan \left (\frac{e}{2}\right )-1\right ) \left (\tan \left (\frac{e}{2}\right )+1\right ) (\cos (e+f x)+1)^3 \left (\cot \left (\frac{1}{2} (e+f x)\right )-1\right ) \left (\cot \left (\frac{1}{2} (e+f x)\right )+1\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.099, size = 179, normalized size = 1.1 \begin{align*} -{\frac{8\,{c}^{5}}{5\,f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{5}}-{\frac{8\,{c}^{5}}{3\,f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}-16\,{\frac{{c}^{5}\tan \left ( 1/2\,fx+e/2 \right ) }{f{a}^{3}}}+2\,{\frac{{c}^{5}\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) }{f{a}^{3}}}+8\,{\frac{{c}^{5}\ln \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) }{f{a}^{3}}}+{\frac{{c}^{5}}{f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) +1 \right ) ^{-1}}+{\frac{{c}^{5}}{f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) -1 \right ) ^{-1}}-8\,{\frac{{c}^{5}\ln \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) }{f{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.58554, size = 759, normalized size = 4.69 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.18968, size = 721, normalized size = 4.45 \begin{align*} \frac{15 \, c^{5} f x \cos \left (f x + e\right )^{4} + 45 \, c^{5} f x \cos \left (f x + e\right )^{3} + 45 \, c^{5} f x \cos \left (f x + e\right )^{2} + 15 \, c^{5} f x \cos \left (f x + e\right ) + 60 \,{\left (c^{5} \cos \left (f x + e\right )^{4} + 3 \, c^{5} \cos \left (f x + e\right )^{3} + 3 \, c^{5} \cos \left (f x + e\right )^{2} + c^{5} \cos \left (f x + e\right )\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 60 \,{\left (c^{5} \cos \left (f x + e\right )^{4} + 3 \, c^{5} \cos \left (f x + e\right )^{3} + 3 \, c^{5} \cos \left (f x + e\right )^{2} + c^{5} \cos \left (f x + e\right )\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) -{\left (239 \, c^{5} \cos \left (f x + e\right )^{3} + 477 \, c^{5} \cos \left (f x + e\right )^{2} + 349 \, c^{5} \cos \left (f x + e\right ) + 15 \, c^{5}\right )} \sin \left (f x + e\right )}{15 \,{\left (a^{3} f \cos \left (f x + e\right )^{4} + 3 \, a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + a^{3} f \cos \left (f x + e\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{c^{5} \left (\int \frac{5 \sec{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec{\left (e + f x \right )} + 1}\, dx + \int - \frac{10 \sec ^{2}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec{\left (e + f x \right )} + 1}\, dx + \int \frac{10 \sec ^{3}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec{\left (e + f x \right )} + 1}\, dx + \int - \frac{5 \sec ^{4}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec{\left (e + f x \right )} + 1}\, dx + \int \frac{\sec ^{5}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec{\left (e + f x \right )} + 1}\, dx + \int - \frac{1}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec{\left (e + f x \right )} + 1}\, dx\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.54874, size = 219, normalized size = 1.35 \begin{align*} \frac{\frac{15 \,{\left (f x + e\right )} c^{5}}{a^{3}} + \frac{120 \, c^{5} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1 \right |}\right )}{a^{3}} - \frac{120 \, c^{5} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1 \right |}\right )}{a^{3}} + \frac{30 \, c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right )} a^{3}} - \frac{8 \,{\left (3 \, a^{12} c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 5 \, a^{12} c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 30 \, a^{12} c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{a^{15}}}{15 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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