Optimal. Leaf size=27 \[ \frac {b \sec (e+f x)}{f}-\frac {(a+b) \tanh ^{-1}(\cos (e+f x))}{f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4133, 453, 206} \[ \frac {b \sec (e+f x)}{f}-\frac {(a+b) \tanh ^{-1}(\cos (e+f x))}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 453
Rule 4133
Rubi steps
\begin {align*} \int \csc (e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {b+a x^2}{x^2 \left (1-x^2\right )} \, dx,x,\cos (e+f x)\right )}{f}\\ &=\frac {b \sec (e+f x)}{f}-\frac {(a+b) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac {(a+b) \tanh ^{-1}(\cos (e+f x))}{f}+\frac {b \sec (e+f x)}{f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.05, size = 84, normalized size = 3.11 \[ \frac {a \log \left (\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f}-\frac {a \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f}+\frac {b \sec (e+f x)}{f}+\frac {b \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{f}-\frac {b \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.74, size = 60, normalized size = 2.22 \[ -\frac {{\left (a + b\right )} \cos \left (f x + e\right ) \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) - {\left (a + b\right )} \cos \left (f x + e\right ) \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) - 2 \, b}{2 \, f \cos \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.49, size = 57, normalized size = 2.11 \[ \frac {a \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{f}+\frac {b}{f \cos \left (f x +e \right )}+\frac {b \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 44, normalized size = 1.63 \[ -\frac {{\left (a + b\right )} \log \left (\cos \left (f x + e\right ) + 1\right ) - {\left (a + b\right )} \log \left (\cos \left (f x + e\right ) - 1\right ) - \frac {2 \, b}{\cos \left (f x + e\right )}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.09, size = 29, normalized size = 1.07 \[ \frac {b}{f\,\cos \left (e+f\,x\right )}-\frac {\mathrm {atanh}\left (\cos \left (e+f\,x\right )\right )\,\left (a+b\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sec ^{2}{\left (e + f x \right )}\right ) \csc {\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________