Optimal. Leaf size=62 \[ \frac {2 \, _2F_1\left (1,\frac {2-p}{4};\frac {6-p}{4};-\tan ^2(c+d x)\right ) \tan (c+d x)}{d (2-p) \sqrt {b \tan ^p(c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3740, 3557,
371} \begin {gather*} \frac {2 \tan (c+d x) \, _2F_1\left (1,\frac {2-p}{4};\frac {6-p}{4};-\tan ^2(c+d x)\right )}{d (2-p) \sqrt {b \tan ^p(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 371
Rule 3557
Rule 3740
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {b \tan ^p(c+d x)}} \, dx &=\frac {\tan ^{\frac {p}{2}}(c+d x) \int \tan ^{-\frac {p}{2}}(c+d x) \, dx}{\sqrt {b \tan ^p(c+d x)}}\\ &=\frac {\tan ^{\frac {p}{2}}(c+d x) \text {Subst}\left (\int \frac {x^{-p/2}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d \sqrt {b \tan ^p(c+d x)}}\\ &=\frac {2 \, _2F_1\left (1,\frac {2-p}{4};\frac {6-p}{4};-\tan ^2(c+d x)\right ) \tan (c+d x)}{d (2-p) \sqrt {b \tan ^p(c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 60, normalized size = 0.97 \begin {gather*} -\frac {2 \, _2F_1\left (1,\frac {2-p}{4};\frac {6-p}{4};-\tan ^2(c+d x)\right ) \tan (c+d x)}{d (-2+p) \sqrt {b \tan ^p(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.24, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {b \left (\tan ^{p}\left (d x +c \right )\right )}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 \frac {1}{\sqrt {b \tan ^{p}{\left (c + d x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {b\,{\mathrm {tan}\left (c+d\,x\right )}^p}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________