Optimal. Leaf size=229 \[ \frac{2 c^4 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{a^{5/2} f}-\frac{11 c^4 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right )}{\sqrt{2} a^{5/2} f}+\frac{7 c^4 \tan (e+f x)}{2 a^2 f \sqrt{a \sec (e+f x)+a}}-\frac{c^4 \sin ^2(e+f x) \tan ^3(e+f x) \sec ^4\left (\frac{1}{2} (e+f x)\right )}{4 f (a \sec (e+f x)+a)^{5/2}}-\frac{c^4 \sin (e+f x) \tan ^2(e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right )}{4 a f (a \sec (e+f x)+a)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.295997, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3904, 3887, 470, 578, 582, 522, 203} \[ \frac{2 c^4 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{a^{5/2} f}-\frac{11 c^4 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right )}{\sqrt{2} a^{5/2} f}+\frac{7 c^4 \tan (e+f x)}{2 a^2 f \sqrt{a \sec (e+f x)+a}}-\frac{c^4 \sin ^2(e+f x) \tan ^3(e+f x) \sec ^4\left (\frac{1}{2} (e+f x)\right )}{4 f (a \sec (e+f x)+a)^{5/2}}-\frac{c^4 \sin (e+f x) \tan ^2(e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right )}{4 a f (a \sec (e+f x)+a)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3904
Rule 3887
Rule 470
Rule 578
Rule 582
Rule 522
Rule 203
Rubi steps
\begin{align*} \int \frac{(c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^{5/2}} \, dx &=\left (a^4 c^4\right ) \int \frac{\tan ^8(e+f x)}{(a+a \sec (e+f x))^{13/2}} \, dx\\ &=-\frac{\left (2 a^2 c^4\right ) \operatorname{Subst}\left (\int \frac{x^8}{\left (1+a x^2\right ) \left (2+a x^2\right )^3} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{f}\\ &=-\frac{c^4 \sec ^4\left (\frac{1}{2} (e+f x)\right ) \sin ^2(e+f x) \tan ^3(e+f x)}{4 f (a+a \sec (e+f x))^{5/2}}-\frac{c^4 \operatorname{Subst}\left (\int \frac{x^4 \left (10+6 a x^2\right )}{\left (1+a x^2\right ) \left (2+a x^2\right )^2} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{2 f}\\ &=-\frac{c^4 \sec ^2\left (\frac{1}{2} (e+f x)\right ) \sin (e+f x) \tan ^2(e+f x)}{4 a f (a+a \sec (e+f x))^{3/2}}-\frac{c^4 \sec ^4\left (\frac{1}{2} (e+f x)\right ) \sin ^2(e+f x) \tan ^3(e+f x)}{4 f (a+a \sec (e+f x))^{5/2}}+\frac{c^4 \operatorname{Subst}\left (\int \frac{x^2 \left (-6 a-14 a^2 x^2\right )}{\left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{4 a^2 f}\\ &=\frac{7 c^4 \tan (e+f x)}{2 a^2 f \sqrt{a+a \sec (e+f x)}}-\frac{c^4 \sec ^2\left (\frac{1}{2} (e+f x)\right ) \sin (e+f x) \tan ^2(e+f x)}{4 a f (a+a \sec (e+f x))^{3/2}}-\frac{c^4 \sec ^4\left (\frac{1}{2} (e+f x)\right ) \sin ^2(e+f x) \tan ^3(e+f x)}{4 f (a+a \sec (e+f x))^{5/2}}-\frac{c^4 \operatorname{Subst}\left (\int \frac{-28 a^2-36 a^3 x^2}{\left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{4 a^4 f}\\ &=\frac{7 c^4 \tan (e+f x)}{2 a^2 f \sqrt{a+a \sec (e+f x)}}-\frac{c^4 \sec ^2\left (\frac{1}{2} (e+f x)\right ) \sin (e+f x) \tan ^2(e+f x)}{4 a f (a+a \sec (e+f x))^{3/2}}-\frac{c^4 \sec ^4\left (\frac{1}{2} (e+f x)\right ) \sin ^2(e+f x) \tan ^3(e+f x)}{4 f (a+a \sec (e+f x))^{5/2}}-\frac{\left (2 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{1+a x^2} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{a^2 f}+\frac{\left (11 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{2+a x^2} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{a^2 f}\\ &=\frac{2 c^4 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{a^{5/2} f}-\frac{11 c^4 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{2} \sqrt{a+a \sec (e+f x)}}\right )}{\sqrt{2} a^{5/2} f}+\frac{7 c^4 \tan (e+f x)}{2 a^2 f \sqrt{a+a \sec (e+f x)}}-\frac{c^4 \sec ^2\left (\frac{1}{2} (e+f x)\right ) \sin (e+f x) \tan ^2(e+f x)}{4 a f (a+a \sec (e+f x))^{3/2}}-\frac{c^4 \sec ^4\left (\frac{1}{2} (e+f x)\right ) \sin ^2(e+f x) \tan ^3(e+f x)}{4 f (a+a \sec (e+f x))^{5/2}}\\ \end{align*}
Mathematica [A] time = 2.59924, size = 164, normalized size = 0.72 \[ \frac{c^4 \cot \left (\frac{1}{2} (e+f x)\right ) \sec (e+f x) \left ((19 \cos (e+f x)-12 \cos (2 (e+f x))-3 \cos (3 (e+f x))-4) \sec ^4\left (\frac{1}{2} (e+f x)\right )+32 \cos (e+f x) \sqrt{\sec (e+f x)-1} \tan ^{-1}\left (\sqrt{\sec (e+f x)-1}\right )-88 \sqrt{2} \cos (e+f x) \sqrt{\sec (e+f x)-1} \tan ^{-1}\left (\frac{\sqrt{\sec (e+f x)-1}}{\sqrt{2}}\right )\right )}{16 a^2 f \sqrt{a (\sec (e+f x)+1)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.272, size = 550, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 11.8767, size = 1673, normalized size = 7.31 \begin{align*} \text{result too large to display} \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*} c^{4} \left (\int - \frac{4 \sec{\left (e + f x \right )}}{a^{2} \sqrt{a \sec{\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )} + 2 a^{2} \sqrt{a \sec{\left (e + f x \right )} + a} \sec{\left (e + f x \right )} + a^{2} \sqrt{a \sec{\left (e + f x \right )} + a}}\, dx + \int \frac{6 \sec ^{2}{\left (e + f x \right )}}{a^{2} \sqrt{a \sec{\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )} + 2 a^{2} \sqrt{a \sec{\left (e + f x \right )} + a} \sec{\left (e + f x \right )} + a^{2} \sqrt{a \sec{\left (e + f x \right )} + a}}\, dx + \int - \frac{4 \sec ^{3}{\left (e + f x \right )}}{a^{2} \sqrt{a \sec{\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )} + 2 a^{2} \sqrt{a \sec{\left (e + f x \right )} + a} \sec{\left (e + f x \right )} + a^{2} \sqrt{a \sec{\left (e + f x \right )} + a}}\, dx + \int \frac{\sec ^{4}{\left (e + f x \right )}}{a^{2} \sqrt{a \sec{\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )} + 2 a^{2} \sqrt{a \sec{\left (e + f x \right )} + a} \sec{\left (e + f x \right )} + a^{2} \sqrt{a \sec{\left (e + f x \right )} + a}}\, dx + \int \frac{1}{a^{2} \sqrt{a \sec{\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )} + 2 a^{2} \sqrt{a \sec{\left (e + f x \right )} + a} \sec{\left (e + f x \right )} + a^{2} \sqrt{a \sec{\left (e + f x \right )} + a}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [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.
[In]
[Out]