Optimal. Leaf size=109 \[ \frac {2 \sec ^3(e+f x) \sqrt {d \tan (e+f x)}}{7 d f}+\frac {4 \sec (e+f x) \sqrt {d \tan (e+f x)}}{7 d f}+\frac {4 \sqrt {\sin (2 e+2 f x)} \sec (e+f x) F\left (\left .e+f x-\frac {\pi }{4}\right |2\right )}{7 f \sqrt {d \tan (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2613, 2614, 2573, 2641} \[ \frac {2 \sec ^3(e+f x) \sqrt {d \tan (e+f x)}}{7 d f}+\frac {4 \sec (e+f x) \sqrt {d \tan (e+f x)}}{7 d f}+\frac {4 \sqrt {\sin (2 e+2 f x)} \sec (e+f x) F\left (\left .e+f x-\frac {\pi }{4}\right |2\right )}{7 f \sqrt {d \tan (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2573
Rule 2613
Rule 2614
Rule 2641
Rubi steps
\begin {align*} \int \frac {\sec ^5(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx &=\frac {2 \sec ^3(e+f x) \sqrt {d \tan (e+f x)}}{7 d f}+\frac {6}{7} \int \frac {\sec ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx\\ &=\frac {4 \sec (e+f x) \sqrt {d \tan (e+f x)}}{7 d f}+\frac {2 \sec ^3(e+f x) \sqrt {d \tan (e+f x)}}{7 d f}+\frac {4}{7} \int \frac {\sec (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx\\ &=\frac {4 \sec (e+f x) \sqrt {d \tan (e+f x)}}{7 d f}+\frac {2 \sec ^3(e+f x) \sqrt {d \tan (e+f x)}}{7 d f}+\frac {\left (4 \sqrt {\sin (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)} \sqrt {\sin (e+f x)}} \, dx}{7 \sqrt {\cos (e+f x)} \sqrt {d \tan (e+f x)}}\\ &=\frac {4 \sec (e+f x) \sqrt {d \tan (e+f x)}}{7 d f}+\frac {2 \sec ^3(e+f x) \sqrt {d \tan (e+f x)}}{7 d f}+\frac {\left (4 \sec (e+f x) \sqrt {\sin (2 e+2 f x)}\right ) \int \frac {1}{\sqrt {\sin (2 e+2 f x)}} \, dx}{7 \sqrt {d \tan (e+f x)}}\\ &=\frac {4 F\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sec (e+f x) \sqrt {\sin (2 e+2 f x)}}{7 f \sqrt {d \tan (e+f x)}}+\frac {4 \sec (e+f x) \sqrt {d \tan (e+f x)}}{7 d f}+\frac {2 \sec ^3(e+f x) \sqrt {d \tan (e+f x)}}{7 d f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.54, size = 79, normalized size = 0.72 \[ \frac {2 \sin (e+f x) \left (4 \sqrt {\sec ^2(e+f x)} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\tan ^2(e+f x)\right )+(\cos (2 (e+f x))+2) \sec ^4(e+f x)\right )}{7 f \sqrt {d \tan (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \tan \left (f x + e\right )} \sec \left (f x + e\right )^{5}}{d \tan \left (f x + e\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (f x + e\right )^{5}}{\sqrt {d \tan \left (f x + e\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.71, size = 224, normalized size = 2.06 \[ -\frac {\left (-1+\cos \left (f x +e \right )\right ) \left (4 \sin \left (f x +e \right ) \EllipticF \left (\sqrt {-\frac {-\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {-\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \left (\cos ^{3}\left (f x +e \right )\right )-2 \left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {2}+2 \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {2}-\cos \left (f x +e \right ) \sqrt {2}+\sqrt {2}\right ) \left (1+\cos \left (f x +e \right )\right )^{2} \sqrt {2}}{7 f \sin \left (f x +e \right )^{3} \cos \left (f x +e \right )^{4} \sqrt {\frac {d \sin \left (f x +e \right )}{\cos \left (f x +e \right )}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (f x + e\right )^{5}}{\sqrt {d \tan \left (f x + e\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\cos \left (e+f\,x\right )}^5\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{5}{\left (e + f x \right )}}{\sqrt {d \tan {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________