Optimal. Leaf size=38 \[ \frac {\sec ^3(a+b x)}{3 b}+\frac {\sec (a+b x)}{b}-\frac {\tanh ^{-1}(\cos (a+b x))}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2622, 302, 207} \[ \frac {\sec ^3(a+b x)}{3 b}+\frac {\sec (a+b x)}{b}-\frac {\tanh ^{-1}(\cos (a+b x))}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 207
Rule 302
Rule 2622
Rubi steps
\begin {align*} \int \csc (a+b x) \sec ^4(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\sec (a+b x)\right )}{b}\\ &=\frac {\sec (a+b x)}{b}+\frac {\sec ^3(a+b x)}{3 b}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{b}\\ &=-\frac {\tanh ^{-1}(\cos (a+b x))}{b}+\frac {\sec (a+b x)}{b}+\frac {\sec ^3(a+b x)}{3 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 57, normalized size = 1.50 \[ \frac {\sec ^3(a+b x)}{3 b}+\frac {\sec (a+b x)}{b}+\frac {\log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{b}-\frac {\log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 67, normalized size = 1.76 \[ -\frac {3 \, \cos \left (b x + a\right )^{3} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 3 \, \cos \left (b x + a\right )^{3} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 6 \, \cos \left (b x + a\right )^{2} - 2}{6 \, b \cos \left (b x + a\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.21, size = 101, normalized size = 2.66 \[ \frac {\frac {8 \, {\left (\frac {3 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {3 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 2\right )}}{{\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}^{3}} + 3 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{6 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 47, normalized size = 1.24 \[ \frac {1}{3 b \cos \left (b x +a \right )^{3}}+\frac {1}{b \cos \left (b x +a \right )}+\frac {\ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 50, normalized size = 1.32 \[ \frac {\frac {2 \, {\left (3 \, \cos \left (b x + a\right )^{2} + 1\right )}}{\cos \left (b x + a\right )^{3}} - 3 \, \log \left (\cos \left (b x + a\right ) + 1\right ) + 3 \, \log \left (\cos \left (b x + a\right ) - 1\right )}{6 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.39, size = 33, normalized size = 0.87 \[ -\frac {\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )-\frac {{\cos \left (a+b\,x\right )}^2+\frac {1}{3}}{{\cos \left (a+b\,x\right )}^3}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{4}{\left (a + b x \right )}}{\sin {\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________