Optimal. Leaf size=292 \[ \frac {\sqrt {6 \pi } b^{3/2} \sin \left (3 a-\frac {3 b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}-\frac {\sqrt {2 \pi } b^{3/2} \sin \left (a-\frac {b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}-\frac {\sqrt {2 \pi } b^{3/2} \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}+\frac {\sqrt {6 \pi } b^{3/2} \cos \left (3 a-\frac {3 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}-\frac {4 b \sin ^2(a+b x) \cos (a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \sin ^3(a+b x)}{3 d (c+d x)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.71, antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3314, 3306, 3305, 3351, 3304, 3352, 3312} \[ \frac {\sqrt {6 \pi } b^{3/2} \sin \left (3 a-\frac {3 b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}-\frac {\sqrt {2 \pi } b^{3/2} \sin \left (a-\frac {b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}-\frac {\sqrt {2 \pi } b^{3/2} \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}+\frac {\sqrt {6 \pi } b^{3/2} \cos \left (3 a-\frac {3 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}-\frac {4 b \sin ^2(a+b x) \cos (a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \sin ^3(a+b x)}{3 d (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3304
Rule 3305
Rule 3306
Rule 3312
Rule 3314
Rule 3351
Rule 3352
Rubi steps
\begin {align*} \int \frac {\sin ^3(a+b x)}{(c+d x)^{5/2}} \, dx &=-\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \sin ^3(a+b x)}{3 d (c+d x)^{3/2}}+\frac {\left (8 b^2\right ) \int \frac {\sin (a+b x)}{\sqrt {c+d x}} \, dx}{d^2}-\frac {\left (12 b^2\right ) \int \frac {\sin ^3(a+b x)}{\sqrt {c+d x}} \, dx}{d^2}\\ &=-\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \sin ^3(a+b x)}{3 d (c+d x)^{3/2}}-\frac {\left (12 b^2\right ) \int \left (\frac {3 \sin (a+b x)}{4 \sqrt {c+d x}}-\frac {\sin (3 a+3 b x)}{4 \sqrt {c+d x}}\right ) \, dx}{d^2}+\frac {\left (8 b^2 \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{d^2}+\frac {\left (8 b^2 \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{d^2}\\ &=-\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \sin ^3(a+b x)}{3 d (c+d x)^{3/2}}+\frac {\left (3 b^2\right ) \int \frac {\sin (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{d^2}-\frac {\left (9 b^2\right ) \int \frac {\sin (a+b x)}{\sqrt {c+d x}} \, dx}{d^2}+\frac {\left (16 b^2 \cos \left (a-\frac {b c}{d}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^3}+\frac {\left (16 b^2 \sin \left (a-\frac {b c}{d}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^3}\\ &=\frac {8 b^{3/2} \sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}+\frac {8 b^{3/2} \sqrt {2 \pi } C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{d^{5/2}}-\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \sin ^3(a+b x)}{3 d (c+d x)^{3/2}}+\frac {\left (3 b^2 \cos \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {3 b c}{d}+3 b x\right )}{\sqrt {c+d x}} \, dx}{d^2}-\frac {\left (9 b^2 \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{d^2}+\frac {\left (3 b^2 \sin \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {3 b c}{d}+3 b x\right )}{\sqrt {c+d x}} \, dx}{d^2}-\frac {\left (9 b^2 \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{d^2}\\ &=\frac {8 b^{3/2} \sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}+\frac {8 b^{3/2} \sqrt {2 \pi } C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{d^{5/2}}-\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \sin ^3(a+b x)}{3 d (c+d x)^{3/2}}+\frac {\left (6 b^2 \cos \left (3 a-\frac {3 b c}{d}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {3 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^3}-\frac {\left (18 b^2 \cos \left (a-\frac {b c}{d}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^3}+\frac {\left (6 b^2 \sin \left (3 a-\frac {3 b c}{d}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {3 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^3}-\frac {\left (18 b^2 \sin \left (a-\frac {b c}{d}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^3}\\ &=-\frac {b^{3/2} \sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}+\frac {b^{3/2} \sqrt {6 \pi } \cos \left (3 a-\frac {3 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}+\frac {b^{3/2} \sqrt {6 \pi } C\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{d^{5/2}}-\frac {b^{3/2} \sqrt {2 \pi } C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{d^{5/2}}-\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \sin ^3(a+b x)}{3 d (c+d x)^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.42, size = 496, normalized size = 1.70 \[ \frac {6 \sqrt {6 \pi } b d x \sqrt {\frac {b}{d}} \sqrt {c+d x} \sin \left (3 a-\frac {3 b c}{d}\right ) C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right )+6 \sqrt {6 \pi } b c \sqrt {\frac {b}{d}} \sqrt {c+d x} \sin \left (3 a-\frac {3 b c}{d}\right ) C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right )-6 \sqrt {2 \pi } b d x \sqrt {\frac {b}{d}} \sqrt {c+d x} \sin \left (a-\frac {b c}{d}\right ) C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}\right )-6 \sqrt {2 \pi } b c \sqrt {\frac {b}{d}} \sqrt {c+d x} \sin \left (a-\frac {b c}{d}\right ) C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}\right )-6 \sqrt {2 \pi } b \sqrt {\frac {b}{d}} (c+d x)^{3/2} \cos \left (a-\frac {b c}{d}\right ) S\left (\sqrt {\frac {b}{d}} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}\right )+6 \sqrt {6 \pi } b \sqrt {\frac {b}{d}} (c+d x)^{3/2} \cos \left (3 a-\frac {3 b c}{d}\right ) S\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right )-6 b c \cos (a+b x)+6 b c \cos (3 (a+b x))-3 d \sin (a+b x)+d \sin (3 (a+b x))-6 b d x \cos (a+b x)+6 b d x \cos (3 (a+b x))}{6 d^2 (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.93, size = 388, normalized size = 1.33 \[ \frac {3 \, \sqrt {6} {\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 3 \, \sqrt {2} {\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 3 \, \sqrt {2} {\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + 3 \, \sqrt {6} {\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 2 \, {\left (6 \, {\left (b d x + b c\right )} \cos \left (b x + a\right )^{3} - 6 \, {\left (b d x + b c\right )} \cos \left (b x + a\right ) + {\left (d \cos \left (b x + a\right )^{2} - d\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{3 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 368, normalized size = 1.26 \[ \frac {-\frac {\sin \left (\frac {\left (d x +c \right ) b}{d}+\frac {d a -c b}{d}\right )}{2 \left (d x +c \right )^{\frac {3}{2}}}+\frac {b \left (-\frac {\cos \left (\frac {\left (d x +c \right ) b}{d}+\frac {d a -c b}{d}\right )}{\sqrt {d x +c}}-\frac {b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {d a -c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{d}+\frac {\sin \left (\frac {3 \left (d x +c \right ) b}{d}+\frac {3 d a -3 c b}{d}\right )}{6 \left (d x +c \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {\cos \left (\frac {3 \left (d x +c \right ) b}{d}+\frac {3 d a -3 c b}{d}\right )}{\sqrt {d x +c}}-\frac {b \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 d a -3 c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {3 d a -3 c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{d}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [C] time = 1.02, size = 253, normalized size = 0.87 \[ \frac {3 \, \sqrt {3} {\left ({\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, \frac {3 i \, {\left (d x + c\right )} b}{d}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -\frac {3 i \, {\left (d x + c\right )} b}{d}\right )\right )} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + {\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, \frac {3 i \, {\left (d x + c\right )} b}{d}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -\frac {3 i \, {\left (d x + c\right )} b}{d}\right )\right )} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {3}{2}} + {\left ({\left (\left (3 i + 3\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) - \left (3 i - 3\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) + {\left (-\left (3 i - 3\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) + \left (3 i + 3\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )\right )} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {3}{2}}}{16 \, {\left (d x + c\right )}^{\frac {3}{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\sin \left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________