Optimal. Leaf size=61 \[ -\frac {a^2+b^2}{b^3 d (a+b \tan (c+d x))}-\frac {2 a \log (a+b \tan (c+d x))}{b^3 d}+\frac {\tan (c+d x)}{b^2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 61, 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 = {3506, 697} \[ -\frac {a^2+b^2}{b^3 d (a+b \tan (c+d x))}-\frac {2 a \log (a+b \tan (c+d x))}{b^3 d}+\frac {\tan (c+d x)}{b^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 697
Rule 3506
Rubi steps
\begin {align*} \int \frac {\sec ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1+\frac {x^2}{b^2}}{(a+x)^2} \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{b^2}+\frac {a^2+b^2}{b^2 (a+x)^2}-\frac {2 a}{b^2 (a+x)}\right ) \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=-\frac {2 a \log (a+b \tan (c+d x))}{b^3 d}+\frac {\tan (c+d x)}{b^2 d}-\frac {a^2+b^2}{b^3 d (a+b \tan (c+d x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 51, normalized size = 0.84 \[ \frac {-\frac {a^2+b^2}{a+b \tan (c+d x)}-2 a \log (a+b \tan (c+d x))+b \tan (c+d x)}{b^3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.59, size = 178, normalized size = 2.92 \[ -\frac {2 \, b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) - b^{2} + {\left (a^{2} \cos \left (d x + c\right )^{2} + a b \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - {\left (a^{2} \cos \left (d x + c\right )^{2} + a b \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \log \left (\cos \left (d x + c\right )^{2}\right )}{a b^{3} d \cos \left (d x + c\right )^{2} + b^{4} d \cos \left (d x + c\right ) \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 5.25, size = 71, normalized size = 1.16 \[ -\frac {\frac {2 \, a \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{b^{3}} - \frac {\tan \left (d x + c\right )}{b^{2}} - \frac {2 \, a b \tan \left (d x + c\right ) + a^{2} - b^{2}}{{\left (b \tan \left (d x + c\right ) + a\right )} b^{3}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.48, size = 78, normalized size = 1.28 \[ \frac {\tan \left (d x +c \right )}{b^{2} d}-\frac {2 a \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{3} d}-\frac {a^{2}}{d \,b^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {1}{b d \left (a +b \tan \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 60, normalized size = 0.98 \[ -\frac {\frac {a^{2} + b^{2}}{b^{4} \tan \left (d x + c\right ) + a b^{3}} + \frac {2 \, a \log \left (b \tan \left (d x + c\right ) + a\right )}{b^{3}} - \frac {\tan \left (d x + c\right )}{b^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.72, size = 67, normalized size = 1.10 \[ \frac {\mathrm {tan}\left (c+d\,x\right )}{b^2\,d}-\frac {a^2+b^2}{b\,d\,\left (\mathrm {tan}\left (c+d\,x\right )\,b^3+a\,b^2\right )}-\frac {2\,a\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}{b^3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{4}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________