Optimal. Leaf size=83 \[ -\frac {2 \left (a^2-b^2\right ) (a+b \sin (c+d x))^{5/2}}{5 b^3 d}+\frac {4 a (a+b \sin (c+d x))^{7/2}}{7 b^3 d}-\frac {2 (a+b \sin (c+d x))^{9/2}}{9 b^3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2747, 711}
\begin {gather*} -\frac {2 \left (a^2-b^2\right ) (a+b \sin (c+d x))^{5/2}}{5 b^3 d}-\frac {2 (a+b \sin (c+d x))^{9/2}}{9 b^3 d}+\frac {4 a (a+b \sin (c+d x))^{7/2}}{7 b^3 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 711
Rule 2747
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2} \, dx &=\frac {\text {Subst}\left (\int (a+x)^{3/2} \left (b^2-x^2\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac {\text {Subst}\left (\int \left (\left (-a^2+b^2\right ) (a+x)^{3/2}+2 a (a+x)^{5/2}-(a+x)^{7/2}\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=-\frac {2 \left (a^2-b^2\right ) (a+b \sin (c+d x))^{5/2}}{5 b^3 d}+\frac {4 a (a+b \sin (c+d x))^{7/2}}{7 b^3 d}-\frac {2 (a+b \sin (c+d x))^{9/2}}{9 b^3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.18, size = 58, normalized size = 0.70 \begin {gather*} \frac {(a+b \sin (c+d x))^{5/2} \left (-16 a^2+91 b^2+35 b^2 \cos (2 (c+d x))+40 a b \sin (c+d x)\right )}{315 b^3 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.97, size = 55, normalized size = 0.66
method | result | size |
default | \(-\frac {2 \left (a +b \sin \left (d x +c \right )\right )^{\frac {5}{2}} \left (-35 b^{2} \left (\cos ^{2}\left (d x +c \right )\right )-20 a b \sin \left (d x +c \right )+8 a^{2}-28 b^{2}\right )}{315 b^{3} d}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 61, normalized size = 0.73 \begin {gather*} -\frac {2 \, {\left (35 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} - 90 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a + 63 \, {\left (a^{2} - b^{2}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}\right )}}{315 \, b^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.34, size = 111, normalized size = 1.34 \begin {gather*} -\frac {2 \, {\left (35 \, b^{4} \cos \left (d x + c\right )^{4} + 8 \, a^{4} - 60 \, a^{2} b^{2} - 28 \, b^{4} - {\left (3 \, a^{2} b^{2} + 7 \, b^{4}\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (25 \, a b^{3} \cos \left (d x + c\right )^{2} + 2 \, a^{3} b + 38 \, a b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}}{315 \, b^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 314 vs.
\(2 (76) = 152\).
time = 5.62, size = 314, normalized size = 3.78 \begin {gather*} \begin {cases} a^{\frac {3}{2}} x \cos ^{3}{\left (c \right )} & \text {for}\: b = 0 \wedge d = 0 \\a^{\frac {3}{2}} \cdot \left (\frac {2 \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {\sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d}\right ) & \text {for}\: b = 0 \\x \left (a + b \sin {\left (c \right )}\right )^{\frac {3}{2}} \cos ^{3}{\left (c \right )} & \text {for}\: d = 0 \\- \frac {16 a^{4} \sqrt {a + b \sin {\left (c + d x \right )}}}{315 b^{3} d} + \frac {8 a^{3} \sqrt {a + b \sin {\left (c + d x \right )}} \sin {\left (c + d x \right )}}{315 b^{2} d} + \frac {8 a^{2} \sqrt {a + b \sin {\left (c + d x \right )}} \sin ^{2}{\left (c + d x \right )}}{21 b d} + \frac {2 a^{2} \sqrt {a + b \sin {\left (c + d x \right )}} \cos ^{2}{\left (c + d x \right )}}{5 b d} + \frac {152 a \sqrt {a + b \sin {\left (c + d x \right )}} \sin ^{3}{\left (c + d x \right )}}{315 d} + \frac {4 a \sqrt {a + b \sin {\left (c + d x \right )}} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {8 b \sqrt {a + b \sin {\left (c + d x \right )}} \sin ^{4}{\left (c + d x \right )}}{45 d} + \frac {2 b \sqrt {a + b \sin {\left (c + d x \right )}} \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} & \text {otherwise} \end {cases} \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 = -0.01 \begin {gather*} \int {\cos \left (c+d\,x\right )}^3\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________