Optimal. Leaf size=205 \[ \frac{\tan (e+f x) \left (-d^2 \left (2 c^2+10 c d-27 d^2\right ) \sec (e+f x)+21 c^2 d^2+8 c^3 d+2 c^4-88 c d^3+72 d^4\right )}{15 f \left (a^3 \sec (e+f x)+a^3\right )}+\frac{d^3 (4 c-3 d) \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac{(c-d) \tan (e+f x) (c+d \sec (e+f x))^3}{5 f (a \sec (e+f x)+a)^3}+\frac{(c-d) (2 c+9 d) \tan (e+f x) (c+d \sec (e+f x))^2}{15 a f (a \sec (e+f x)+a)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.280187, antiderivative size = 265, normalized size of antiderivative = 1.29, number of steps used = 7, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {3987, 98, 150, 143, 63, 217, 203} \[ \frac{\tan (e+f x) \left (-d^2 \left (2 c^2+10 c d-27 d^2\right ) \sec (e+f x)+21 c^2 d^2+8 c^3 d+2 c^4-88 c d^3+72 d^4\right )}{15 f \left (a^3 \sec (e+f x)+a^3\right )}+\frac{2 d^3 (4 c-3 d) \tan (e+f x) \tan ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a (\sec (e+f x)+1)}}\right )}{a^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{(c-d) \tan (e+f x) (c+d \sec (e+f x))^3}{5 f (a \sec (e+f x)+a)^3}+\frac{(c-d) (2 c+9 d) \tan (e+f x) (c+d \sec (e+f x))^2}{15 a f (a \sec (e+f x)+a)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3987
Rule 98
Rule 150
Rule 143
Rule 63
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{\sec (e+f x) (c+d \sec (e+f x))^4}{(a+a \sec (e+f x))^3} \, dx &=-\frac{\left (a^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{(c+d x)^4}{\sqrt{a-a x} (a+a x)^{7/2}} \, dx,x,\sec (e+f x)\right )}{f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{(c-d) (c+d \sec (e+f x))^3 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac{\tan (e+f x) \operatorname{Subst}\left (\int \frac{(c+d x)^2 \left (-a^2 \left (2 c^2+6 c d-3 d^2\right )+a^2 (c-6 d) d x\right )}{\sqrt{a-a x} (a+a x)^{5/2}} \, dx,x,\sec (e+f x)\right )}{5 a f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{(c-d) (2 c+9 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac{(c-d) (c+d \sec (e+f x))^3 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac{\tan (e+f x) \operatorname{Subst}\left (\int \frac{(c+d x) \left (-a^4 \left (2 c^3+8 c^2 d+23 c d^2-18 d^3\right )+a^4 d \left (2 c^2+10 c d-27 d^2\right ) x\right )}{\sqrt{a-a x} (a+a x)^{3/2}} \, dx,x,\sec (e+f x)\right )}{15 a^4 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{(c-d) (2 c+9 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac{(c-d) (c+d \sec (e+f x))^3 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac{\left (2 c^4+8 c^3 d+21 c^2 d^2-88 c d^3+72 d^4-d^2 \left (2 c^2+10 c d-27 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{15 f \left (a^3+a^3 \sec (e+f x)\right )}-\frac{\left ((4 c-3 d) d^3 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-a x} \sqrt{a+a x}} \, dx,x,\sec (e+f x)\right )}{a f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{(c-d) (2 c+9 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac{(c-d) (c+d \sec (e+f x))^3 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac{\left (2 c^4+8 c^3 d+21 c^2 d^2-88 c d^3+72 d^4-d^2 \left (2 c^2+10 c d-27 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{15 f \left (a^3+a^3 \sec (e+f x)\right )}+\frac{\left (2 (4 c-3 d) d^3 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2 a-x^2}} \, dx,x,\sqrt{a-a \sec (e+f x)}\right )}{a^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{(c-d) (2 c+9 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac{(c-d) (c+d \sec (e+f x))^3 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac{\left (2 c^4+8 c^3 d+21 c^2 d^2-88 c d^3+72 d^4-d^2 \left (2 c^2+10 c d-27 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{15 f \left (a^3+a^3 \sec (e+f x)\right )}+\frac{\left (2 (4 c-3 d) d^3 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a+a \sec (e+f x)}}\right )}{a^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{2 (4 c-3 d) d^3 \tan ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a+a \sec (e+f x)}}\right ) \tan (e+f x)}{a^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{(c-d) (2 c+9 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac{(c-d) (c+d \sec (e+f x))^3 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac{\left (2 c^4+8 c^3 d+21 c^2 d^2-88 c d^3+72 d^4-d^2 \left (2 c^2+10 c d-27 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{15 f \left (a^3+a^3 \sec (e+f x)\right )}\\ \end{align*}
Mathematica [A] time = 2.37366, size = 292, normalized size = 1.42 \[ \frac{2 \cos \left (\frac{1}{2} (e+f x)\right ) \left (4 (c-d)^2 \left (7 c^2+26 c d+57 d^2\right ) \sec \left (\frac{e}{2}\right ) \sin \left (\frac{f x}{2}\right ) \cos ^4\left (\frac{1}{2} (e+f x)\right )-60 d^3 \cos ^5\left (\frac{1}{2} (e+f x)\right ) \left ((4 c-3 d) \left (\log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )\right )-d \sec (e) \sin (f x) \sec (e+f x)\right )-8 (c-d)^3 (2 c+3 d) \tan \left (\frac{e}{2}\right ) \cos ^3\left (\frac{1}{2} (e+f x)\right )+3 (c-d)^4 \tan \left (\frac{e}{2}\right ) \cos \left (\frac{1}{2} (e+f x)\right )+3 (c-d)^4 \sec \left (\frac{e}{2}\right ) \sin \left (\frac{f x}{2}\right )-8 (c-d)^3 (2 c+3 d) \sec \left (\frac{e}{2}\right ) \sin \left (\frac{f x}{2}\right ) \cos ^2\left (\frac{1}{2} (e+f x)\right )\right )}{15 a^3 f (\cos (e+f x)+1)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.079, size = 454, normalized size = 2.2 \begin{align*}{\frac{{c}^{4}}{20\,f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{5}}-{\frac{{c}^{3}d}{5\,f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{5}}+{\frac{3\,{c}^{2}{d}^{2}}{10\,f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{5}}-{\frac{c{d}^{3}}{5\,f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{5}}+{\frac{{d}^{4}}{20\,f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{5}}-{\frac{{c}^{4}}{6\,f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}+{\frac{{c}^{2}{d}^{2}}{f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}-{\frac{4\,c{d}^{3}}{3\,f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}+{\frac{{d}^{4}}{2\,f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}+{\frac{{c}^{4}}{4\,f{a}^{3}}\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) }+{\frac{{c}^{3}d}{f{a}^{3}}\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) }+{\frac{3\,{c}^{2}{d}^{2}}{2\,f{a}^{3}}\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) }-7\,{\frac{c\tan \left ( 1/2\,fx+e/2 \right ){d}^{3}}{f{a}^{3}}}+{\frac{17\,{d}^{4}}{4\,f{a}^{3}}\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) }-{\frac{{d}^{4}}{f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) +1 \right ) ^{-1}}+4\,{\frac{{d}^{3}\ln \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) c}{f{a}^{3}}}-3\,{\frac{{d}^{4}\ln \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) }{f{a}^{3}}}-{\frac{{d}^{4}}{f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) -1 \right ) ^{-1}}-4\,{\frac{{d}^{3}\ln \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) c}{f{a}^{3}}}+3\,{\frac{{d}^{4}\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.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.05262, size = 641, normalized size = 3.13 \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 = 0.533384, size = 910, normalized size = 4.44 \begin{align*} \frac{15 \,{\left ({\left (4 \, c d^{3} - 3 \, d^{4}\right )} \cos \left (f x + e\right )^{4} + 3 \,{\left (4 \, c d^{3} - 3 \, d^{4}\right )} \cos \left (f x + e\right )^{3} + 3 \,{\left (4 \, c d^{3} - 3 \, d^{4}\right )} \cos \left (f x + e\right )^{2} +{\left (4 \, c d^{3} - 3 \, d^{4}\right )} \cos \left (f x + e\right )\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 15 \,{\left ({\left (4 \, c d^{3} - 3 \, d^{4}\right )} \cos \left (f x + e\right )^{4} + 3 \,{\left (4 \, c d^{3} - 3 \, d^{4}\right )} \cos \left (f x + e\right )^{3} + 3 \,{\left (4 \, c d^{3} - 3 \, d^{4}\right )} \cos \left (f x + e\right )^{2} +{\left (4 \, c d^{3} - 3 \, d^{4}\right )} \cos \left (f x + e\right )\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \,{\left (15 \, d^{4} +{\left (7 \, c^{4} + 12 \, c^{3} d + 12 \, c^{2} d^{2} - 88 \, c d^{3} + 72 \, d^{4}\right )} \cos \left (f x + e\right )^{3} + 3 \,{\left (2 \, c^{4} + 12 \, c^{3} d + 12 \, c^{2} d^{2} - 68 \, c d^{3} + 57 \, d^{4}\right )} \cos \left (f x + e\right )^{2} +{\left (2 \, c^{4} + 12 \, c^{3} d + 42 \, c^{2} d^{2} - 128 \, c d^{3} + 117 \, d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{30 \,{\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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{c^{4} \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{d^{4} \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{4 c d^{3} \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{6 c^{2} d^{2} \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{4 c^{3} d \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}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.22684, size = 529, normalized size = 2.58 \begin{align*} -\frac{\frac{120 \, d^{4} \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{60 \,{\left (4 \, c d^{3} - 3 \, d^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1 \right |}\right )}{a^{3}} + \frac{60 \,{\left (4 \, c d^{3} - 3 \, d^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1 \right |}\right )}{a^{3}} - \frac{3 \, a^{12} c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 12 \, a^{12} c^{3} d \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 18 \, a^{12} c^{2} d^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 12 \, a^{12} c d^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 3 \, a^{12} d^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 10 \, a^{12} c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 60 \, a^{12} c^{2} d^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 80 \, a^{12} c d^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 30 \, a^{12} d^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 15 \, a^{12} c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 60 \, a^{12} c^{3} d \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 90 \, a^{12} c^{2} d^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 420 \, a^{12} c d^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 255 \, a^{12} d^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{a^{15}}}{60 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]