Optimal. Leaf size=72 \[ \frac {(a+b) \tan ^7(e+f x)}{7 f}+\frac {(3 a+2 b) \tan ^5(e+f x)}{5 f}+\frac {(3 a+b) \tan ^3(e+f x)}{3 f}+\frac {a \tan (e+f x)}{f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3191, 373} \[ \frac {(a+b) \tan ^7(e+f x)}{7 f}+\frac {(3 a+2 b) \tan ^5(e+f x)}{5 f}+\frac {(3 a+b) \tan ^3(e+f x)}{3 f}+\frac {a \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 373
Rule 3191
Rubi steps
\begin {align*} \int \sec ^8(e+f x) \left (a+b \sin ^2(e+f x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \left (1+x^2\right )^2 \left (a+(a+b) x^2\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \left (a+(3 a+b) x^2+(3 a+2 b) x^4+(a+b) x^6\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {a \tan (e+f x)}{f}+\frac {(3 a+b) \tan ^3(e+f x)}{3 f}+\frac {(3 a+2 b) \tan ^5(e+f x)}{5 f}+\frac {(a+b) \tan ^7(e+f x)}{7 f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.31, size = 86, normalized size = 1.19 \[ \frac {\tan (e+f x) \left (15 a \tan ^6(e+f x)+63 a \tan ^4(e+f x)+105 a \tan ^2(e+f x)+105 a+15 b \sec ^6(e+f x)-3 b \sec ^4(e+f x)-4 b \sec ^2(e+f x)-8 b\right )}{105 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 77, normalized size = 1.07 \[ \frac {{\left (8 \, {\left (6 \, a - b\right )} \cos \left (f x + e\right )^{6} + 4 \, {\left (6 \, a - b\right )} \cos \left (f x + e\right )^{4} + 3 \, {\left (6 \, a - b\right )} \cos \left (f x + e\right )^{2} + 15 \, a + 15 \, b\right )} \sin \left (f x + e\right )}{105 \, f \cos \left (f x + e\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.22, size = 88, normalized size = 1.22 \[ \frac {15 \, a \tan \left (f x + e\right )^{7} + 15 \, b \tan \left (f x + e\right )^{7} + 63 \, a \tan \left (f x + e\right )^{5} + 42 \, b \tan \left (f x + e\right )^{5} + 105 \, a \tan \left (f x + e\right )^{3} + 35 \, b \tan \left (f x + e\right )^{3} + 105 \, a \tan \left (f x + e\right )}{105 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.50, size = 104, normalized size = 1.44 \[ \frac {-a \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (f x +e \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (f x +e \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (f x +e \right )\right )}{35}\right ) \tan \left (f x +e \right )+b \left (\frac {\sin ^{3}\left (f x +e \right )}{7 \cos \left (f x +e \right )^{7}}+\frac {4 \left (\sin ^{3}\left (f x +e \right )\right )}{35 \cos \left (f x +e \right )^{5}}+\frac {8 \left (\sin ^{3}\left (f x +e \right )\right )}{105 \cos \left (f x +e \right )^{3}}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.38, size = 60, normalized size = 0.83 \[ \frac {15 \, {\left (a + b\right )} \tan \left (f x + e\right )^{7} + 21 \, {\left (3 \, a + 2 \, b\right )} \tan \left (f x + e\right )^{5} + 35 \, {\left (3 \, a + b\right )} \tan \left (f x + e\right )^{3} + 105 \, a \tan \left (f x + e\right )}{105 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 13.84, size = 59, normalized size = 0.82 \[ \frac {\left (\frac {a}{7}+\frac {b}{7}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^7+\left (\frac {3\,a}{5}+\frac {2\,b}{5}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^5+\left (a+\frac {b}{3}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^3+a\,\mathrm {tan}\left (e+f\,x\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________