Optimal. Leaf size=40 \[ -\frac {(a+2 b) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3012, 3770} \[ -\frac {(a+2 b) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3012
Rule 3770
Rubi steps
\begin {align*} \int \csc ^3(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx &=-\frac {a \cot (c+d x) \csc (c+d x)}{2 d}+\frac {1}{2} (a+2 b) \int \csc (c+d x) \, dx\\ &=-\frac {(a+2 b) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.04, size = 118, normalized size = 2.95 \[ -\frac {a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {b \log \left (\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}-\frac {b \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.46, size = 95, normalized size = 2.38 \[ \frac {2 \, a \cos \left (d x + c\right ) - {\left ({\left (a + 2 \, b\right )} \cos \left (d x + c\right )^{2} - a - 2 \, b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left ({\left (a + 2 \, b\right )} \cos \left (d x + c\right )^{2} - a - 2 \, b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{4 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.16, size = 121, normalized size = 3.02 \[ \frac {2 \, {\left (a + 2 \, b\right )} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) + \frac {{\left (a - \frac {2 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {4 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}{\cos \left (d x + c\right ) - 1} - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.52, size = 63, normalized size = 1.58 \[ -\frac {a \cot \left (d x +c \right ) \csc \left (d x +c \right )}{2 d}+\frac {a \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2 d}+\frac {b \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 58, normalized size = 1.45 \[ -\frac {{\left (a + 2 \, b\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) - {\left (a + 2 \, b\right )} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac {2 \, a \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 13.39, size = 42, normalized size = 1.05 \[ \frac {a\,\cos \left (c+d\,x\right )}{2\,d\,\left ({\cos \left (c+d\,x\right )}^2-1\right )}-\frac {\mathrm {atanh}\left (\cos \left (c+d\,x\right )\right )\,\left (\frac {a}{2}+b\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sin ^{2}{\left (c + d x \right )}\right ) \csc ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________