Optimal. Leaf size=110 \[ -\frac {b \tan (c+d x) \sqrt {b \tan ^4(c+d x)}}{3 d}+\frac {b \tan ^3(c+d x) \sqrt {b \tan ^4(c+d x)}}{5 d}-b x \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}+\frac {b \cot (c+d x) \sqrt {b \tan ^4(c+d x)}}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3658, 3473, 8} \[ \frac {b \tan ^3(c+d x) \sqrt {b \tan ^4(c+d x)}}{5 d}-\frac {b \tan (c+d x) \sqrt {b \tan ^4(c+d x)}}{3 d}-b x \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}+\frac {b \cot (c+d x) \sqrt {b \tan ^4(c+d x)}}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3473
Rule 3658
Rubi steps
\begin {align*} \int \left (b \tan ^4(c+d x)\right )^{3/2} \, dx &=\left (b \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int \tan ^6(c+d x) \, dx\\ &=\frac {b \tan ^3(c+d x) \sqrt {b \tan ^4(c+d x)}}{5 d}-\left (b \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int \tan ^4(c+d x) \, dx\\ &=-\frac {b \tan (c+d x) \sqrt {b \tan ^4(c+d x)}}{3 d}+\frac {b \tan ^3(c+d x) \sqrt {b \tan ^4(c+d x)}}{5 d}+\left (b \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int \tan ^2(c+d x) \, dx\\ &=\frac {b \cot (c+d x) \sqrt {b \tan ^4(c+d x)}}{d}-\frac {b \tan (c+d x) \sqrt {b \tan ^4(c+d x)}}{3 d}+\frac {b \tan ^3(c+d x) \sqrt {b \tan ^4(c+d x)}}{5 d}-\left (b \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int 1 \, dx\\ &=\frac {b \cot (c+d x) \sqrt {b \tan ^4(c+d x)}}{d}-b x \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}-\frac {b \tan (c+d x) \sqrt {b \tan ^4(c+d x)}}{3 d}+\frac {b \tan ^3(c+d x) \sqrt {b \tan ^4(c+d x)}}{5 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.77, size = 66, normalized size = 0.60 \[ \frac {\cot (c+d x) \left (b \tan ^4(c+d x)\right )^{3/2} \left (15 \cot ^4(c+d x)-5 \cot ^2(c+d x)-15 \tan ^{-1}(\tan (c+d x)) \cot ^5(c+d x)+3\right )}{15 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.14, size = 62, normalized size = 0.56 \[ \frac {{\left (3 \, b \tan \left (d x + c\right )^{5} - 5 \, b \tan \left (d x + c\right )^{3} - 15 \, b d x + 15 \, b \tan \left (d x + c\right )\right )} \sqrt {b \tan \left (d x + c\right )^{4}}}{15 \, d \tan \left (d x + c\right )^{2}} \]
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 [A] time = 0.07, size = 64, normalized size = 0.58 \[ -\frac {\left (b \left (\tan ^{4}\left (d x +c \right )\right )\right )^{\frac {3}{2}} \left (-3 \left (\tan ^{5}\left (d x +c \right )\right )+5 \left (\tan ^{3}\left (d x +c \right )\right )+15 \arctan \left (\tan \left (d x +c \right )\right )-15 \tan \left (d x +c \right )\right )}{15 d \tan \left (d x +c \right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.50, size = 53, normalized size = 0.48 \[ \frac {3 \, b^{\frac {3}{2}} \tan \left (d x + c\right )^{5} - 5 \, b^{\frac {3}{2}} \tan \left (d x + c\right )^{3} - 15 \, {\left (d x + c\right )} b^{\frac {3}{2}} + 15 \, b^{\frac {3}{2}} \tan \left (d x + c\right )}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^4\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \tan ^{4}{\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________