Optimal. Leaf size=38 \[ \frac {a \tan (c+d x)}{d}-a x+\frac {b \cos (c+d x)}{d}+\frac {b \sec (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2722, 3473, 8, 2590, 14} \[ \frac {a \tan (c+d x)}{d}-a x+\frac {b \cos (c+d x)}{d}+\frac {b \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 14
Rule 2590
Rule 2722
Rule 3473
Rubi steps
\begin {align*} \int (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx &=\int \left (a \tan ^2(c+d x)+b \sin (c+d x) \tan ^2(c+d x)\right ) \, dx\\ &=a \int \tan ^2(c+d x) \, dx+b \int \sin (c+d x) \tan ^2(c+d x) \, dx\\ &=\frac {a \tan (c+d x)}{d}-a \int 1 \, dx-\frac {b \operatorname {Subst}\left (\int \frac {1-x^2}{x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-a x+\frac {a \tan (c+d x)}{d}-\frac {b \operatorname {Subst}\left (\int \left (-1+\frac {1}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-a x+\frac {b \cos (c+d x)}{d}+\frac {b \sec (c+d x)}{d}+\frac {a \tan (c+d x)}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 47, normalized size = 1.24 \[ -\frac {a \tan ^{-1}(\tan (c+d x))}{d}+\frac {a \tan (c+d x)}{d}+\frac {b \cos (c+d x)}{d}+\frac {b \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.53, size = 47, normalized size = 1.24 \[ -\frac {a d x \cos \left (d x + c\right ) - b \cos \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - b}{d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 58, normalized size = 1.53 \[ -\frac {{\left (d x + c\right )} a + \frac {2 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, b\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.31, size = 59, normalized size = 1.55 \[ \frac {a \left (\tan \left (d x +c \right )-d x -c \right )+b \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.43, size = 39, normalized size = 1.03 \[ -\frac {{\left (d x + c - \tan \left (d x + c\right )\right )} a - b {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 12.48, size = 55, normalized size = 1.45 \[ -a\,x-\frac {2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+4\,b}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sin {\left (c + d x \right )}\right ) \sin ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________