Optimal. Leaf size=43 \[ -\frac {a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b^2 \sec (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3598, 3855,
2686, 8} \begin {gather*} -\frac {a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b^2 \sec (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2686
Rule 3598
Rule 3855
Rubi steps
\begin {align*} \int \csc (c+d x) (a+b \tan (c+d x))^2 \, dx &=\int \left (a^2 \csc (c+d x)+2 a b \sec (c+d x)+b^2 \sec (c+d x) \tan (c+d x)\right ) \, dx\\ &=a^2 \int \csc (c+d x) \, dx+(2 a b) \int \sec (c+d x) \, dx+b^2 \int \sec (c+d x) \tan (c+d x) \, dx\\ &=-\frac {a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b^2 \text {Subst}(\int 1 \, dx,x,\sec (c+d x))}{d}\\ &=-\frac {a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b^2 \sec (c+d x)}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(97\) vs. \(2(43)=86\).
time = 0.27, size = 97, normalized size = 2.26 \begin {gather*} \frac {a \left (-a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+b^2 \sec (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.23, size = 56, normalized size = 1.30
method | result | size |
derivativedivides | \(\frac {a^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )+2 a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+\frac {b^{2}}{\cos \left (d x +c \right )}}{d}\) | \(56\) |
default | \(\frac {a^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )+2 a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+\frac {b^{2}}{\cos \left (d x +c \right )}}{d}\) | \(56\) |
risch | \(\frac {2 b^{2} {\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {2 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {2 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.33, size = 60, normalized size = 1.40 \begin {gather*} \frac {a b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - a^{2} \log \left (\cot \left (d x + c\right ) + \csc \left (d x + c\right )\right ) + \frac {b^{2}}{\cos \left (d x + c\right )}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 102 vs.
\(2 (43) = 86\).
time = 0.35, size = 102, normalized size = 2.37 \begin {gather*} -\frac {a^{2} \cos \left (d x + c\right ) \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - a^{2} \cos \left (d x + c\right ) \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, a b \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) + 2 \, a b \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, b^{2}}{2 \, d \cos \left (d x + c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \tan {\left (c + d x \right )}\right )^{2} \csc {\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.57, size = 74, normalized size = 1.72 \begin {gather*} \frac {2 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 2 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2 \, b^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.72, size = 125, normalized size = 2.91 \begin {gather*} \frac {a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {2\,b^2}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}-\frac {4\,a\,b\,\mathrm {atanh}\left (\frac {16\,a^2\,b^2}{8\,a^3\,b-16\,a^2\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {8\,a^3\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a^3\,b-16\,a^2\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________