Optimal. Leaf size=73 \[ a^3 x+\frac {b \left (3 a^2+3 a b+b^2\right ) \tan (c+d x)}{d}+\frac {b^2 (3 a+2 b) \tan ^3(c+d x)}{3 d}+\frac {b^3 \tan ^5(c+d x)}{5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4128, 390, 203} \[ \frac {b \left (3 a^2+3 a b+b^2\right ) \tan (c+d x)}{d}+a^3 x+\frac {b^2 (3 a+2 b) \tan ^3(c+d x)}{3 d}+\frac {b^3 \tan ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 390
Rule 4128
Rubi steps
\begin {align*} \int \left (a+b \sec ^2(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b+b x^2\right )^3}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (b \left (3 a^2+3 a b+b^2\right )+b^2 (3 a+2 b) x^2+b^3 x^4+\frac {a^3}{1+x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {b \left (3 a^2+3 a b+b^2\right ) \tan (c+d x)}{d}+\frac {b^2 (3 a+2 b) \tan ^3(c+d x)}{3 d}+\frac {b^3 \tan ^5(c+d x)}{5 d}+\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=a^3 x+\frac {b \left (3 a^2+3 a b+b^2\right ) \tan (c+d x)}{d}+\frac {b^2 (3 a+2 b) \tan ^3(c+d x)}{3 d}+\frac {b^3 \tan ^5(c+d x)}{5 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 1.02, size = 268, normalized size = 3.67 \[ \frac {\sec (c) \sec ^5(c+d x) \left (150 a^3 d x \cos (2 c+d x)+75 a^3 d x \cos (2 c+3 d x)+75 a^3 d x \cos (4 c+3 d x)+15 a^3 d x \cos (4 c+5 d x)+15 a^3 d x \cos (6 c+5 d x)+150 a^3 d x \cos (d x)-360 a^2 b \sin (2 c+d x)+360 a^2 b \sin (2 c+3 d x)-90 a^2 b \sin (4 c+3 d x)+90 a^2 b \sin (4 c+5 d x)+540 a^2 b \sin (d x)-180 a b^2 \sin (2 c+d x)+300 a b^2 \sin (2 c+3 d x)+60 a b^2 \sin (4 c+5 d x)+420 a b^2 \sin (d x)+80 b^3 \sin (2 c+3 d x)+16 b^3 \sin (4 c+5 d x)+160 b^3 \sin (d x)\right )}{480 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.75, size = 90, normalized size = 1.23 \[ \frac {15 \, a^{3} d x \cos \left (d x + c\right )^{5} + {\left ({\left (45 \, a^{2} b + 30 \, a b^{2} + 8 \, b^{3}\right )} \cos \left (d x + c\right )^{4} + 3 \, b^{3} + {\left (15 \, a b^{2} + 4 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{15 \, d \cos \left (d x + c\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.33, size = 91, normalized size = 1.25 \[ \frac {3 \, b^{3} \tan \left (d x + c\right )^{5} + 15 \, a b^{2} \tan \left (d x + c\right )^{3} + 10 \, b^{3} \tan \left (d x + c\right )^{3} + 15 \, {\left (d x + c\right )} a^{3} + 45 \, a^{2} b \tan \left (d x + c\right ) + 45 \, a b^{2} \tan \left (d x + c\right ) + 15 \, b^{3} \tan \left (d x + c\right )}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.25, size = 84, normalized size = 1.15 \[ \frac {a^{3} \left (d x +c \right )+3 a^{2} b \tan \left (d x +c \right )-3 b^{2} a \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )-b^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.35, size = 83, normalized size = 1.14 \[ a^{3} x + \frac {{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a b^{2}}{d} + \frac {{\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} b^{3}}{15 \, d} + \frac {3 \, a^{2} b \tan \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.51, size = 73, normalized size = 1.00 \[ \frac {\mathrm {tan}\left (c+d\,x\right )\,\left (3\,b\,{\left (a+b\right )}^2-3\,b^2\,\left (a+b\right )+b^3\right )+\frac {b^3\,{\mathrm {tan}\left (c+d\,x\right )}^5}{5}+{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (b^2\,\left (a+b\right )-\frac {b^3}{3}\right )+a^3\,d\,x}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sec ^{2}{\left (c + d x \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________